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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf,len=726
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='13792v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='FA]  31 Jan 2023  Linear extension operators for Sobolev spaces on  uniform trees  Charles Fefferman1 and Bo’az Klartag2  Dedicated in friendship to David Jerison  Abstract  Let 1 < p < ∞ and suppose that we are given a function f defined on the leaves of a  weighted tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We would like to extend f to a function F defined on the entire tree, so as  to minimize the weighted W 1,p-Sobolev norm of the extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' An easy situation is when  p = 2, where the harmonic extension operator provides such a function F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In this note  we record our analysis of the particular case of a uniform binary tree, which is a complete,  finite, binary tree with weights that depend only on the distance from the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Neither the  averaging operator nor the harmonic extension operator work here in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Nevertheless,  we prove the existence of a linear extension operator whose norm is bounded by a constant  depending solely on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This operator is a variant of the standard harmonic extension operator,  and in fact it is harmonic extension with respect to a certain Markov kernel determined by p  and by the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  1  Introduction  Consider a full binary tree of height N, whose set of vertices is denoted by  V =  N  �  k=0  {0, 1}k,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', the vertices are strings of zeroes and ones of length at most N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For x ∈ {0, 1}k and ℓ ≤ k  we write πℓ(x) ∈ {0, 1}ℓ for the prefix of x of length ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Thus for k ≥ 1, the parent of a vertex  x ∈ {0, 1}k ⊆ V is the vertex πk−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The set {0, 1}0 is a singleton whose unique element  is denoted by ∅, the empty string, which is the root of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The set of leaves of the tree is  1Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544,  USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Email: cf@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Supported by the Air Force Office of Scientific Research, grant number  FA9950-18-1-0069 and the National Science Foundation (NSF), grant number DMS-1700180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  2Department of Mathematics,  Weizmann Institute of Science,  Rehovot 7610001,  Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Email:  boaz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='klartag@weizmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='il.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Supported by a grant from the Israel Science Foundation (ISF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  1  {0, 1}N, and all other vertices in V are internal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' A vertex x ∈ {0, 1}k is said to have  depth  d(x) = k,  thus leaves have depth N and the root has depth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The collections of bijections from V to  V that preserve depth and parenthood relations form a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This group is referred to as the  symmetry group of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' It has 22N−1 elements, and it is a 2-Sylow subgroup of the group of  all permutations of the leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The set  E =  N  �  k=1  {0, 1}k = V \\ {∅}  is referred to as the set of edges of the tree, and the depth of an edge e ∈ {0, 1}k is d(e) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  That is, we think of e ∈ E as an undirected edge connecting the vertex whose string corresponds  to e to its unique parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Each internal vertex other than the root is connected to three vertices,  which are its parent and its two children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Assume that we are given edge weights  W1, W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , WN > 0,  where we view Wk as the weight of all edges of depth k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For 1 < p < ∞, the associated  ˙W 1,p-seminorm is defined, for F : V → R, via  ∥F∥ ˙W 1,p(V ) =  \uf8eb  \uf8ed  N  �  k=1  Wk  �  x∈{0,1}k  |F(x) − F(πk−1(x))|p  \uf8f6  \uf8f8  1/p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (1)  We write ∂V = {0, 1}N ⊆ V for the set of leaves of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The trace of the ∥·∥ ˙W 1,p-seminorm  is defined, for f : ∂V → R, via  ∥f∥ ˙W 1,p(∂V ) = inf  �  ∥F∥ ˙W 1,p(V ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' F|∂V = f  �  ,  (2)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', the infimum of the ˙W 1,p-seminorm over all extensions of f from the leaves to the entire tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  We write RV for the collection of all functions f : V → R, and similarly R∂V is the collection  of all functions f : ∂V → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Our main result is the following:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let 1 < p < ∞ and let W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , WN > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Then there exists a linear operator  H : R∂V → RV with the following properties:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' It is a linear extension operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', (Hf)(x) = f(x) for any x ∈ ∂V and any function  f : ∂V → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Its norm is bounded by a constant ¯Cp depending only on p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', for any f : ∂V → R,  ∥Hf∥ ˙W 1,p(V ) ≤ ¯Cp∥f∥ ˙W 1,p(∂V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  2  In fact, we have the bound  ¯Cp ≤ 4p1/pq1/q ·  �  1 + max{(p − 1)−1/p, (q − 1)−1/q}  �  ≤ C · max  �  1  p − 1,  1  q − 1  �  ,  (3)  where q = p/(p − 1) and where C > 0 is a universal constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2 is constructive, and the extension operator H that we construct is  in fact a harmonic extension operator with respect to a certain random walk defined on the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  At each step the random walk jumps from a vertex to one of its neighbors, where of course the  neighbors of a vertex are its parent and its children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The Markov kernel corresponding to the  random walk is invariant under the symmetries of the tree, thus the probability to move from a  vertex to its neighbor depends only on the weights of the vertex and of its neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The Markov  kernel of our random walk is determined by the following requirement: For any s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  the probability that a random walk starting at some vertex of depth s will reach a leaf before  reaching a vertex of depth s − 1 equals  qs =  (2sWs)−1/(p−1)  �N  k=s(2kWk)−1/(p−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (4)  Thus the weights of our random walk typically depend on p ∈ (1, ∞), except for the case where  Ws is proportional to 2−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This seems inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Indeed, in some examples such as the 3rd  example in Section 2 below, the linear extension operator H : R∂V → RV that corresponds to  the parameter value p = p0, is not uniformly bounded for any p ∈ (1, ∞) \\ {p0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' When p = 2,  our random walk coincides with the usual random walk corresponding to the given weights on  the edges of the binary tree, and thus in this case H is the standard harmonic extension operator  (hence ¯Cp = 1 for p = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  There is a certain range of weights where the averaging operator yields a uniformly bounded  linear extension operator, as proven by Bj¨orn ×2, Gill and Shanmugalingam [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The averaging  operator is the extension operator that assigns to each internal vertex v the average of the function  values on the leaves of the subtree whose root is v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This averaging operator seems natural also  from the point of view of Whitney’s extension theory, see the work by Shvartsman [5] on Sobolev  extension in W 1,p(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' However, there are examples of uniform binary trees where the averaging  operator does not provide a uniformly bounded operator, such as the case where Wk = 2−k for  all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  What about non-uniform trees?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Suppose that the edge weights are arbitrary positive numbers  (We)e∈E that are not necessarily determined by the depth of the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' When p = 2, there is still  a harmonic extension operator of norm one from ˙W 1,p(∂V ) to ˙W 1,p(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' However, for p ̸= 2 the  situation seems subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We conjecture that in the general case of non-uniform tree weights there  is no linear extension operator whose norm is bounded by a function of p alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This conjecture  is closely related to questions about well-complemented subspaces of ℓn  p that are beyond the  scope of this note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  3  In order to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1 we reformulate the problem in a way that brings us closer to  analysis in ℓp-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For 1 < p < ∞, the associated Lp(E)-norm is defined, for f : E → R,  via  ∥f∥p = ∥f∥Lp(E) =  \uf8eb  \uf8ed  N  �  k=1  Wk  �  x∈{0,1}k  |f(x)|p  \uf8f6  \uf8f8  1/p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (5)  We think of a function f : E → R as the gradient of a function ˜f : V → R, uniquely determined  up to an additive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Given f : E → R we thus define a function ˜f : V → R as follows:  For any x ∈ V other then the root,  ˜f(x) =  d(x)  �  i=1  f(πix),  (6)  while ˜f(x) = 0 if x = ∅ is the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The only property of ˜f that matters is that for all x ∈ E,  f(x) = ˜f(x) − ˜f(πd(x)−1(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  We are interested in finding a linear operator T : Lp(E) → Lp(E), with a uniform bound on its  operator norm, that has the following properties:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The operator T takes the form  Tf(x) = ˜T ˜f(x) − ˜T ˜f(πd(x)−1x)  (7)  for some linear operator ˜T : RV → RV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' That is, ˜T takes functions on V to functions on  V , and T is induces from ˜T via formula (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The operator ˜T is equivariant with respect to the tree symmetries and it satisfies ˜T(1) ≡ 1,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', it maps the constant function 1 to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The function ˜Tg coincides with the function g on the leaves of the tree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' ( ˜Tg)|∂V = g|∂V  for any g : V → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The function ˜Tg is determined by the values of the function g on the leaves of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  The operator norm of ˜T with respect to the ˙W 1,p-seminorm equals to the operator norm  of T with respect to the Lp(E)-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Defining H(f|∂V ) = ˜Tf, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1 may thus be  reformulated as follows:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let 1 < p < ∞ and let W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , WN > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Then there exists a linear operator  T : Lp(E) → Lp(E) with the above properties, whose operator norm is at most a certain  constant ¯Cp depending only on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In fact, we have the bound (3) for the constant ¯Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  4  The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2 occupies the next three sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In Section 2 we discuss invariant  random walks on a full binary tree and describe the corresponding harmonic extension operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  In Section 3 we deal with the problem of bounding the norm of this operator, and use the sym-  metries of the problem in order to reduce it to a one-dimensional question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This one-dimensional  question is then answered in Section 4 using the Muckenhoupt criterion [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  When analyzing the binary tree we use the following notation: We write dlca(x, y) for the  length of the maximal prefix shared by the strings x ∈ {0, 1}k and y ∈ {0, 1}ℓ, while lca(x, y)  is the maximal prefix itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Thus for two vertices x, y ∈ V , their least common ancestor is  lca(x, y) ∈ V and its depth is dlca(x, y) ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Note that for any x ∈ E and ω ∈ ∂V ,  dlca(πd(x)−1x, ω) = min{d(x) − 1, dlca(x, ω)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (8)  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We would like to thank Jacob Carruth, Arie Israel, Anna Skorobogatova  and Ignacio Uriarte-Tuero for helpful conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This research was conducted while BK  was visiting Princeton University’s Department of Mathematics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' he is grateful for their gracious  hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  2  Invariant random walks  A Markov chain on V is a sequence of random variables R1, R2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' ∈ V such that the distribution  of Ri+1 conditioned on R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , Ri is the same as its distribution conditioned on Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' A Markov  chain is time-homogeneous if for any x, y ∈ V , the probability that Ri+1 = x conditioned on the  event Ri = y does not depend on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' A random walk on V is a time-homogeneous Markov chain  R1, R2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' ∈ V such that Ri+1 is a neighbor of Ri with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  We say that the random walk is invariant if the probability to jump from a vertex x to a vertex  y depends only on the depths d(x) and d(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Our random walk will be invariant, and it will stop  when it reaches a leaf, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', we have the stopping time  τ = min{i ≥ 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Ri is a leaf}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  For s ≥ 1 we define qs to be the probability of the following event: Assuming that R1 is a vertex  of depth s, the event is that Ri will remain at the subtree whose root is R1 for all 1 ≤ i ≤ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Equivalently, define  Xi = d(Ri) ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Then X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N} is a random walk, since |Xi+1 − Xi| = 1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Furthermore,  qs = P(∀1 ≤ i ≤ τ, Xi ≥ s | X1 = s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (9)  Clearly  q0 = qN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (10)  For r, s ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N} with r ≤ s we set  ps,r = P(min{Xi ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' i ≤ τ} = r | X1 = s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  5  That is, the number ps,r is the probability that r is the minimal node that the walker visits when  starting from node s, before reaching the terminal node N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Clearly �s  r=0 ps,r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For 0 ≤ r ≤ s ≤ N − 1,  ps,r = qr ·  s  �  k=r+1  (1 − qk)  (11)  where an empty product equals one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Moreover, pN,r = δrN, where δrN is the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The expression on the right-hand side of (11) is the probability to ever reach s − 1 when  starting from X1 = s, and from s − 1 to ever reach s − 2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' until we finally reach r, yet from  r we require to never reach r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Alternatively, when 0 ≤ r ≤ s, s ≥ 1 we have the recurrence  relation  ps,r = qsδs,r + (1 − qs) · ps−1,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (12)  This recurrence relation leads to another proof of (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Suppose that our random walk R1, R2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' ∈ V begins at a vertex R1 = x with d(x) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Consider a leaf y ∈ ∂V with dlca(x, y) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' What is the probability that our random walk will  reach the leaf y?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We claim that this probability is  bs,r := P(Rτ = y) =  r  �  k=0  2k−Nps,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (13)  Indeed, conditioning on the value of k = mini≤τ d(Ri), by symmetry we know that Rτ is dis-  tributed uniformly among the 2N−k leaf-descendants of the vertex πk(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' When k ≤ r, exactly  one of these leaf-descendants is the leaf y, since the vertex of minimal depth that (Ri) visits must  be the vertex πk(x), which is a prefix of y as k ≤ r = dlca(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Hence the probability that  Rτ = y, conditioning on the value of k, equals to 1/2N−k when k ≤ r and it vanishes other-  wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By using the definition of ps,k and the complete probability formula, we obtain (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The  harmonic extension operator associated with our invariant random walk is given by  ˜Tg(x) =  �  ω∈{0,1}N  bd(x),dlca(x,ω) · g(ω)  (x ∈ V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (14)  The operator T is induced from ˜T via formula (7) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Requirements 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=',4 from Section 1 are  clearly satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  We stipulate that the collection of descendants of a vertex x ∈ V , denoted by D(x) ⊆ V ,  includes the vertex x itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Abbreviate a ∧ b = min{a, b} and a ∨ b = max{a, b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The operator  T takes the form  Tf(x) =  �  y∈E  K(x, y)f(y),  (15)  where the kernel K is described next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  6  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let x, y ∈ E and denote s = d(x), t = d(y), r = dlca(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Then the following  hold: If x ̸∈ D(y) and y ̸∈ D(x), then r ≤ s ∧ t − 1 and  K(x, y) = −qs · 2−t ·  r  �  k=0  2kps−1,k ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (16)  Otherwise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', if y ∈ D(x) or if x ∈ D(y) then r = s ∧ t and  K(x, y) = qs · 2−t ·  r−1  �  k=0  (2r − 2k)ps−1,k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (17)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By (7) and (14) we have, for any x ∈ E,  Tf(x) = ˜T ˜f(x) − ˜T ˜f(πd(x)−1x)  =  �  ω∈{0,1}N  bd(x),dlca(x,ω) ˜f(ω) −  �  ω∈{0,1}N  bd(x)−1,dlca(πd(x)−1x,ω) ˜f(ω)  =  �  ω∈{0,1}N  ad(x),dlca(x,w) ˜f(ω)  (18)  where for any 0 ≤ r ≤ s, by (8) and (13),  as,r = bs,r − bs−1,min{s−1,r} =  r  �  k=0  2k−Nps,k −  min{s−1,r}  �  k=0  2k−Nps−1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (19)  Hence, by (6) and (18),  Tf(x) =  �  ω∈{0,1}N  ad(x),dlca(x,ω)  N  �  i=1  f(πiω)  =  �  y∈E  \uf8ee  \uf8f0  �  ω∈{0,1}N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='πd(y)ω=y  ad(x),dlca(x,ω)  \uf8f9  \uf8fb f(y) =  �  y∈E  K(x, y)f(y),  where the kernel K of the operator T satisfies, for any x, y ∈ E,  K(x, y) =  �  ω∈{0,1}N ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='πd(y)ω=y  ad(x),dlca(x,ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (20)  Fix x, y ∈ E with s = d(x), t = d(y) and r = dlca(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let us consider first the case where x  is not a descendant of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This means that the prefix of y that is shared by x, is not the entire string  y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Hence for any ω ∈ {0, 1}N with πd(y)ω = y we have dlca(x, ω) = dlca(x, y) ≤ d(y) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Therefore, from (20) and (19),  K(x, y) = 2N−d(y)ad(x),dlca(x,y)  = 2N−t �  bs,r − bs−1,(s−1)∧r  �  =  r  �  k=0  2k−tps,k −  (s−1)∧r  �  k=0  2k−tps−1,k,  7  as bs,r = �r  k=0 2k−Nps,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Since ps,s = qs, we have  K(x, y) = δrs2s−tqs +  (s−1)∧r  �  k=0  2k−t [ps,k − ps−1,k] = qs ·  \uf8ee  \uf8f0δrs2s−t −  (s−1)∧r  �  k=0  2k−tps−1,k  \uf8f9  \uf8fb ,  (21)  where we used the relation (12), which implies that when k ≤ s − 1,  ps,k − ps−1,k = −qs · ps−1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (22)  We may now prove the conclusion of the proposition in the case where x ̸∈ D(y) and y ̸∈ D(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Indeed, in this case r ≤ s ∧ t − 1 and formula (21) applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Since δrs = 0 in this case, we deduce  formula (16) from (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  The next case we consider is the case where r = s ≤ t − 1, or equivalently, where y ∈  D(x) \\ {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Thus x ̸∈ D(y) and formula (21) applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Recalling that �s−1  k=0 ps−1,k = 1 we obtain  from (21) that  K(x, y) = qs ·  s−1  �  k=0  (2s−t − 2k−t)ps−1,k = qs · 2−t ·  s−1  �  k=0  (2r − 2k)ps−1,k,  proving formula (17) in the case y ∈ D(x) \\ {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  We move on to the case where x ∈ D(y) \\ {y}, thus dlca(x, y) = t ≤ s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In this case, by  applying (20), (19), (13) and then (22),  K(x, y) =  �  ω∈{0,1}N ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='πd(y)ω=y  ad(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='dlca(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='ω) = 2N−d(x)ad(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='d(x) +  d(x)−1  �  k=d(y)  2N−k−1ad(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k  = 2N−sas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='s +  s−1  �  k=t  2N−k−1as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k = 2N−s(bs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='s − bs−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='s−1) +  s−1  �  k=t  2N−k−1(bs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k − bs−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k)  = 2N−s  �  s  �  k=0  2k−Nps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k −  s−1  �  k=0  2k−Nps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k  �  +  s−1  �  k=t  2N−k−1  k  �  ℓ=0  2ℓ−N(ps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='ℓ − ps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='ℓ)  = qs − qs  s−1  �  k=0  2k−sps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k − qs  s−1  �  ℓ=0  s−1  �  k=ℓ∨t  2ℓ−k−1ps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='ℓ  = qs  �  1 −  s−1  �  k=0  2k−sps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k −  s−1  �  ℓ=0  [2ℓ−ℓ∨t − 2ℓ−s]ps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='ℓ  �  = qs  �  1 −  s−1  �  k=0  2k−k∨tps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k  �  = qs ·  �  1 −  t−1  �  k=0  2k−tps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k −  s−1  �  k=t  ps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k  �  = qs ·  t−1  �  k=0  (1 − 2k−t)ps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k = qs · 2−t ·  t−1  �  k=0  (2t − 2k)ps−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  8  Since r = t in this case, we have proved formula (17) in the case where y ∈ D(x) \\ {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Finally,  the last case that remains is when x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In this case r = s = t and  K(x, y) =  �  ω∈{0,1}N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='πd(y)ω=y  ad(x),dlca(x,ω) = 2N−d(x)ad(x),d(x) = 2N−sas,s = 2N−s(bs,s − bs−1,s−1)  = 2N−s  �  s  �  k=0  2k−Nps,k −  s−1  �  k=0  2k−Nps−1,k  �  = qs − qs  s−1  �  k=0  2k−sps−1,k  = qs ·  s−1  �  k=0  (1 − 2k−s)ps−1,k,  completing the proof of formula (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The simplest example is when qs = 1 for all s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In this case the operator ˜T is the  familiar averaging operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' That is, the extension operator ˜T is the operator that assigns  to each vertex the average of the values at the leaves of its subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In this case  ps,r = δs,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Consider the case where the invariant random walk is such that Xi := d(Ri) is a symmetric  random walk on {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', the probability to jump from i to i + 1 is exactly 1/2 for  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Recall that qs is the probability to never leave the subtree when starting  at a vertex of depth s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We claim that in this example, for s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  qs =  1  N − s + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (23)  Indeed, the function f(i) = i is harmonic on {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N} and hence f(Xi) is a mar-  tingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Thus for any stopping time ˜τ we have f(X1) = Ef(X˜τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We pick the stopping  time  ˜τ = min{ i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Xi ∈ {s − 1, N} }  and obtain (23) since  N · qs + (s − 1) · (1 − qs) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Next we use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1 and find a formula for ps,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Since formula (23) is valid for any  s ≥ 1, we conclude that for any r ≥ 0 and s ≥ r + 1,  s�  k=r+1  (1 − qk) =  s�  k=r+1  N − k  N − k + 1 = N − s  N − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (24)  9  Formula (24) is actually valid for any 0 ≤ r ≤ s, since an empty product equals one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Recall that q0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We thus conclude from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1 that for s = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N − 1,  ps,r = N − s  N − r ·  �  1  r = 0  1  N−r+1  1 ≤ r ≤ s  while pN,r = δN,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let 0 < δ < 1, and consider the case where (Xi) is a random walk on {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N} such  that the probability to jump from k to k + 1 equals 1/2 if k < N − 1, and it equals δ if  k = N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' A harmonic function here is  f(k) =  �  k  k ≤ N − 1  N − 2 + 1/δ  k = N  Therefore, for s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N − 1,  (N − 2 + 1/δ) · qs + (s − 1)(1 − qs) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Thus q0 = qN = 1 while for s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N − 1,  qs =  1  N − s + 1/δ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Hence for any s ≤ N − 1 and r ≤ s − 1,  s  �  k=r+1  (1 − qk) =  s  �  k=r+1  N − k + δ−1 − 2  N − k + δ−1 − 1 = N − s + δ−1 − 2  N − r + δ−1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  We conclude from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1 that for s = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N − 1 and 0 ≤ r ≤ s,  ps,r = qr ·  s�  k=r+1  (1 − qk) = N − s + δ−1 − 2  N − r + δ−1 − 2 ·  �  1  r = 0  1  N−r+1/δ−1  1 ≤ r ≤ s  while pN,r = δN,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  We conclude this section with the following:  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For any numbers q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , qN−1 ∈ (0, 1) there exists a random walk  X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N}  satisfying (9) with τ = min{i ≥ 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Xi = N} for s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Write xk for the probability that the random walk jumps from k to k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Then for  s = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N − 1,  qs = xs(qs+1 + (1 − qs+1)xs),  (25)  where we set q0 = qN = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We claim that xs ∈ [0, 1] is determined by equation (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Indeed, the  right-hand side of (25) is a continuous, increasing function of xs in the interval [0, 1], that maps  this interval to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  10  3  The ancestral and non-ancestral parts of the kernel  We need to bound the operator norm in Lp(E) of the operator T whose kernel is described in  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let us consider first the non-ancestral part of the operator, given by the kernel  K0(x, y) = K(x, y) · 1{x̸∈D(y),y̸∈D(x)} = −1{r≤s∧t−1} · qs · 2−t ·  r  �  k=0  2kps−1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (26)  Here as usual s = d(x), t = d(y) and r = dlca(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Write T0 : Lp(E) → Lp(E) for the  operator whose kernel is K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' A function f : E → R is invariant under the symmetries of the  tree, or invariant in short, if it takes the form  f(x) = F(d(x))  for some function F : {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N} → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The operator T0 is equivariant under the symmetries  of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Therefore, if f(x) = F(d(x)) is an invariant function, then so is T0f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In fact, in the  case where f(x) = F(d(x)) we can write  T0f(x) =  N  �  t=1  L0(d(x), t)F(t)  (27)  for a certain kernel L0(s, t) defined for s, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For s, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  L0(s, t) = −qs ·  m−1  �  k=0  (1 − 2k−m)ps−1,k ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let r ≤ min{t, s} − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' A moment of reflection reveals that for x ∈ E with d(x) = s,  n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' s, r) := #{y ∈ E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' d(y) = t, dlca(x, y) = r} = 2t−r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  By (26), (27) and the definition of T0, for any x ∈ E with d(x) = s,  L0(s, t) =  �  y∈E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='d(y)=t  K0(x, y) = −  t∧s−1  �  r=0  n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' s, r) · qs · 2−t ·  r  �  k=0  2kps−1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Denote m = s ∧ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Then for s, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  L0(s, t) = −qs ·  t∧s−1  �  r=0  r  �  k=0  2k−r−1ps−1,k = −qs ·  m−1  �  k=0  m−1  �  r=k  2k−r−1ps−1,k  = −qs ·  m−1  �  k=0  (1 − 2k−m)ps−1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  11  Write ΩN = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N} and for F : ΩN → R define  ∥F∥p = ∥F∥Lp(ΩN) =  � N  �  k=1  2k · Wk · |F(k)|p  �1/p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (28)  Observe that ∥f∥Lp(E) = ∥F∥p if f(x) = F(d(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let  S0F(s) =  N  �  t=1  L0(s, t)F(t)  so that by (27),  T0(F ◦ d) = (S0F) ◦ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (29)  For f, g : E → R we consider the scalar product  ⟨f, g⟩ =  �  x∈E  Wd(x)f(x)g(x)  while for F, G : ΩN → R we set  ⟨F, G⟩ := ⟨F ◦ d, G ◦ d⟩ =  N  �  k=1  2kWkF(k)G(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  The adjoint operators T ∗  0 and S∗  0 are defined with respect to these scalar products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The follow-  ing lemma is probably well-known to experts (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Howard and Schep [2] for a related  argument), and its proof is provided for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Then the norm of the operator T0 : Lp(E) → Lp(E) is attained at  an invariant, non-negative function f, and it equals to the norm of the operator S0 : Lp(Ωn) →  Lp(Ωn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Denote momentarily T = −T0 and S = −S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1 the kernel −L0 of the  operator S is non-negative, and by (26) the kernel of the operator T is non-negative as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  By approximation, we may assume that these two kernels are strictly positive, while keeping  condition (29), thus  T (F ◦ d) = (SF) ◦ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (30)  We deduce that T ∗(F ◦ d) = (S∗F) ◦ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By compactness,  sup  0̸≡F ∈Lp(ΩN )  ∥SF∥p  ∥F∥p  is attained at some function F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Since the kernel of S is non-negative, we may assume that  the extremal function F is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By the Lagrange multipliers theorem, the function F  satisfies a certain eigenvalue equation, and in fact there exists λ ∈ R such that  S∗(SF)p−1 = λF p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (31)  12  Since the kernel of S is positive and F is non-negative and not identically zero, it follows from  (31) that F is actually positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The norm of the operator S : Lp(Ωn) → Lp(Ωn) equals λ1/p > 0,  since  ∥SF∥p  p = ⟨(SF)p−1, SF⟩ = ⟨S∗(SF)p−1, F⟩ = λ⟨F p−1, F⟩ = λ∥F∥p  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Denoting f = F ◦ d, we find from (30) that f is a positive invariant function satisfying  T ∗(T f)p−1 = λf p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Since the kernel of T is non-negative, we have the pointwise H¨older inequality  T (uv) ≤ T (up)1/p · T (vq)1/q,  valid for any non-negative functions u, v ∈ Lp(E), where q = p/(p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The operator T has  a non-negative kernel, and hence its norm is attained at a non-negative function u ∈ Lp(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By  the pointwise H¨older inequality,  (T u)p ≤ T (upf −p/q) · (T f)p/q = T (upf 1−p) · (T f)p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Therefore,  ∥T u∥p  Lp(E) ≤ ⟨T (upf 1−p), (T f)p−1⟩ = ⟨upf 1−p, T ∗(T f)p−1⟩ = λ⟨upf 1−p, f p−1⟩ = λ∥u∥p  Lp(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Thus the norm of T : Lp(E) → Lp(E) is at most λ1/p, which is the norm of S : Lp(ΩN) →  Lp(ΩN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The two norms must therefore be equal, since the operator S is equivalent to the  restriction of T to the space of invariant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  We move on to the ancestral part of the operator, which according to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2 is given  by  K1(x, y) = K(x, y) − K0(x, y) = 1{r=s∧t} · qs · 2−t ·  r−1  �  k=0  (2r − 2k)ps−1,k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (32)  Write T1 for the operator whose kernel is K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' As before, for an invariant function f(x) =  F(d(x)) we may write  T1f(x) =  N  �  t=1  L1(d(x), t)F(t)  (33)  for a certain kernel L1(s, t) defined for s, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We also write  S1F(s) =  N  �  t=1  L1(s, t)F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  It is possible to use formula (7) for the operator T and the definition (14) of the harmonic exten-  sion operator ˜T and deduce that  S0 + S1 ≡ 0,  (34)  essentially because the only invariant, harmonic function on the vertices of the tree is the constant  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' An alternative, more direct proof of (34) is provided in the following:  13  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Then the norm of the operator T1 : Lp(E) → Lp(E) is equal to  the norm of S1 : Lp(Ωn) → Lp(Ωn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Additionally, for s, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N with m = s ∧ t we have  L1(s, t) = qs ·  m−1  �  k=0  (1 − 2k−m)ps−1,k = −L0(s, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The first assertion of the lemma follows from fact that the kernel of T1 is non-negative  and invariant under the symmetries of the tree, as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For the second part, let s, t =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N and denote m = s ∧ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We claim that for x ∈ E with d(x) = s,  n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' s, m) := #{y ∈ E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' d(y) = t, dlca(x, y) = m} = max{1, 2t−s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (35)  Indeed, assume first that t ≥ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' How many y’s are there with d(y) = t and dlca(x, y) = m?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Since m = s, the answer is 2t−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Next, if s ≥ t, then the number of such y’s is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This proves  (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Therefore,  L1(s, t) =  �  y∈E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='d(y)=t  K1(x, y) = n(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' s, m) · qs · 2−t ·  m−1  �  k=0  (2m − 2k)ps−1,k  = max{2−t, 2−s} · qs ·  m−1  �  k=0  (2m − 2k)ps−1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  = 2−m · qs ·  m−1  �  k=0  (2m − 2k)ps−1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We have  ∥T∥Lp(E)→Lp(E) ≤ 2∥S0∥Lp(ΩN)→Lp(ΩN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This follows from the fact that T = T0 + T1 together with the facts that ∥T0∥ = ∥S0∥ and  ∥T1∥ = ∥S1∥ while S1 = −S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  In view of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='4, we are interested in bounds for the norm of the operator S = −S0 =  S1 : Lp(ΩN) → Lp(ΩN) whose non-negative kernel is  L(s, t) = qs ·  s∧t−1  �  k=0  (1 − 2k−s∧t)ps−1,k ≤ qs ·  s∧t−1  �  k=0  ps−1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (36)  14  From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1 we know that ps,r = qr · �s  k=r+1(1 − qk) for s ≤ N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' A little exercise in  probability shows that for any 1 ≤ m ≤ s ≤ N,  m−1  �  k=0  ps−1,k =  s−1  �  k=m  (1 − qk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (37)  Alternatively, (37) holds true for m = 0 as q0 = 1, and it may be proven by induction on m since  ps−1,m +  s−1  �  k=m  (1 − qk) = qm ·  s−1  �  k=m+1  (1 − qk) +  s−1  �  k=m  (1 − qk) =  s�  k=m+1  (1 − qk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  From (36) and (37) we thus obtain  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For s, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N, with m = min{s, t},  0 ≤ L(s, t) ≤ qs  s−1  �  k=m  (1 − qk),  where an empty product equals one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Some examples (parallel to the ones discussed in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For the averaging operator, where qs = 1 and ps,r = δs,r, we have  L(s, t) = −1  2 · 1{s≤t},  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', this is the matrix whose entries equal 0 below the diagonal and −1/2 on and above  the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' This is a rather simple matrix, and it is bounded with respect to the weighted  Lp-norm for quite a few sequences of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For the symmetric random walk matrix, we have q0 = 1 while for 1 ≤ s ≤ N,  qs =  1  N − s + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Hence in view of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='5, with m = min{s, t},  0 ≤ L(s, t) ≤  1  N − s + 1  s−1  �  k=m  N − k  N − k + 1 =  1  N − m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In the case where  qs =  1  N − s + 1/δ − 1  for some 0 < δ < 1, we have  0 ≤ L(s, t) ≤  1  N − s + 1/δ − 1  s−1  �  k=m  N − k + 1/δ − 2  N − k + 1/δ − 1 =  1  N − m + 1/δ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  15  All that remains is to bound the Lp(ΩN)-norm of the operator whose kernel is discussed  in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Recall from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='3 that we have the freedom to choose the parameters  q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , qN−1 ∈ (0, 1) as we please.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  How should we choose these parameters?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Since L(N, t) ≤ �N−1  k=t (1−qk) and we are looking  for upper bounds for the norm, the qs should not be too tiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' On the other hand, L(s, t) ≤ qs for  s ≤ N − 1 and hence it is beneficial to choose qs rather small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We would therefore need some  balance for the qs, which is the subject of the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  4  One-dimensional analysis  Let 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' It will be slightly more convenient to denote  K(s, t) = L(N + 1 − s, N + 1 − t)  and  Qs = qN+1−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Recalling from (10) that qN = 1, we see that  Q1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  From Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='5 we know that for s, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  K(s, t) = L(N + 1 − s, N + 1 − t) ≤ qN+1−s  N−s  �  k=N+1−max{s,t}  (1 − qk) = Qs  max{s,t}  �  k=s+1  (1 − Qk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  From Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='5 we know that L ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Consequently, for s, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  0 ≤ K(s, t) ≤  �  Qs  t ≤ s  Qs · �t  k=s+1(1 − Qk)  t ≥ s + 1  (38)  Recall that we are given edge weights W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , WN > 0, and that the associated Lp(E)-norm is  given by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Denote  ws := 2N+1−s · WN+1−s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Consider the weighted ℓp-norm  ∥f∥p,w =  � N  �  k=1  wk|f(k)|p  �1/p  (39)  and the operator T whose kernel is K(s, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We are allowed to choose the weights q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , qN−1 ∈  (0, 1) as we please, or equivalently, we have the freedom to determine Q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , QN ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We  must keep Q1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Based on considerations related to the Muckenhoupt criterion discussed  below, we set  Qs =  w−1/(p−1)  s  �s  k=1 w−1/(p−1)  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (40)  It is clear that Q1 = 1 and that Qs ∈ (0, 1) for all s ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Recall that q = p/(p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  16  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In order to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2, it suffices to show that the operator norm of T with  respect to the ∥ · ∥p,w-norm is bounded by a constant ˆCp > 0 depending only on p, where in fact  ˆCp ≤ 2p1/pq1/q ·  �  1 + max{(p − 1)−1/p, (q − 1)−1/q}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (41)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In view of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='4, it suffices to bound the operator norm of −S0, whose kernel is  L, with respect to the Lp(ΩN)-norm defined in (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Under the transformation  s �→ N + 1 − s  the operator −S0 whose kernel is L transforms to the operator T whose kernel is K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The Lp(ΩN)-  norm from (28) transforms to the ∥·∥p,w-norm defined in (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Hence Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2 would follow  once we obtain the bound (41), where ¯Cp ≤ 2 ˆCp by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  The remainder of this section is devoted to the proof of the following:  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The operator norm of T with respect to the norm (39) is bounded by a number  ˆCp depending only on p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In fact, we have the bound (41) for the constant ˆCp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Our main tool in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2 is the Muckenhoupt criterion [4], which is an  indispensable tool for proving one-dimensional inequalities of Poincar´e-Sobolev type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For the  reader’s convenience, we include here a statement and a proof of a straightforward modification  of the Muckenhoupt criterion, with sums in place of integrals:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' (Muckenhoupt) Let 1 < p < ∞ and write ΩN = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let U, V : ΩN →  (0, ∞) and let A > 0 be such that for all r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  � N  �  k=r  |U(k)|p  �1/p  ≤ A  �  r  �  k=1  |V (k)|−q  �−1/q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (42)  Then for any function f : ΩN → R,  � N  �  k=1  �����U(k)  k  �  ℓ=1  f(ℓ)  �����  p�1/p  ≤ CpA  � N  �  k=1  |V (k)f(k)|p  �1/p  ,  (43)  with Cp = p1/pq1/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  By continuity, the analog of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='3 for p = 1, ∞ holds true with C1 = C∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We  remark that as in [4], this criterion is tight, in the sense that the infimum over all A > 0 satisfying  (42) is equivalent to the best constant in inequality (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='3 we require  the following:  17  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , αN > 0 and r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  r  �  k=1  αk  �  k  �  ℓ=1  αℓ  �−1/p  ≤ q ·  �  r  �  k=1  αk  �1/q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (44)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We will use the simple inequality  (a + b)1/q − a1/q ≥ b ·  min  ξ∈(a,a+b)  ξ1/q−1  q  = 1  q(a + b)1/q−1 · b,  (45)  valid for any a, b ≥ 0 with a + b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' From (45), for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  � k  �  ℓ=1  αℓ  �1/q  −  �k−1  �  ℓ=1  αℓ  �1/q  ≥ 1  q · αk ·  �  k  �  ℓ=1  αℓ  �1/q−1  ,  where an empty sum equals zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By summing this for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , r we obtain (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='3 (Muckenhoupt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By the H¨older inequality, for any function f : ΩN → R  and weights h : ΩN → (0, ∞),  N  �  k=1  �����U(k)  k  �  ℓ=1  f(ℓ)  �����  p  ≤  N  �  k=1  Up(k) ·  k  �  ℓ=1  |f(ℓ)V (ℓ)h(ℓ)|p ·  �  k  �  j=1  |V (j)h(j)|−q  �p/q  =  N  �  ℓ=1  |f(ℓ)V (ℓ)h(ℓ)|p  N  �  k=ℓ  Up(k)  � k  �  j=1  |V (j)h(j)|−q  �p/q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (46)  Set h(k) =  ��k  ℓ=1 V (ℓ)−q�1/(pq)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='4 with αk = V (k)−q we obtain  r  �  k=1  |V (k)h(k)|−q =  r  �  k=1  αk  � k  �  ℓ=1  αℓ  �−1/p  ≤ q ·  �  r  �  k=1  αk  �1/q  = q ·  �  r  �  k=1  V (k)−q  �1/q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Hence for any f : ΩN → R, the expression in (46) is at most  qp/q ·  N  �  ℓ=1  |f(ℓ)V (ℓ)h(ℓ)|p  N  �  k=ℓ  Up(k)  �  k  �  j=1  |V (j)|−q  �p/q2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (47)  By applying (42) and then Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='4 with αk = Up(N + 1 − k) and with p ∈ (1, ∞) playing  the rˆole of q ∈ (1, ∞), we see that  N  �  k=ℓ  |U(k)|p  �  k  �  j=1  |V (j)|−q  �p/q2  ≤ Ap/q  N  �  k=ℓ  |U(k)|p  � N  �  j=k  |U(j)|p  �−1/q  = Ap/q  N+1−ℓ  �  k=1  αk  � k  �  j=1  αj  �−1/q  ≤ p · Ap/q  �N+1−ℓ  �  k=1  αk  �1/p  = p · Ap/q  � N  �  k=ℓ  Up(k)  �1/p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  18  Hence the expression in (47) is at most  p · (qA)p/q ·  N  �  ℓ=1  |f(ℓ)V (ℓ)h(ℓ)|p ·  � N  �  k=ℓ  Up(k)  �1/p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Applying (42) again we bound the last expression from above by  p · (qA)p/q · A ·  N  �  ℓ=1  |f(ℓ)V (ℓ)h(ℓ)|p ·  �  ℓ  �  k=1  V −q(k)  �−1/q  = pqp/qAp  N  �  ℓ=1  |f(ℓ)V (ℓ)|p,  completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let 1 < p < ∞ and ΩN = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Let U, V : ΩN → (0, ∞) and let A > 0  be such that for all r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  �  r  �  k=1  |U(k)|p  �1/p  ≤ A  � N  �  k=r  |V (k)|−q  �−1/q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (48)  Then for any f : ΩN → R,  � N  �  k=1  �����U(k)  N  �  ℓ=k  f(ℓ)  �����  p�1/p  ≤ CpA  � N  �  k=1  |V (k)f(k)|p  �1/p  ,  (49)  with Cp = p1/pq1/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Denote ˜U(k) = U(N + 1 − k) and ˜V (k) = V (N + 1 − k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Then from (48), for all  r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  � N  �  k=r  | ˜U(k)|p  �1/p  ≤ A  �  r  �  k=1  | ˜V (k)|−q  �−1/q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='3, this implies that for any g : ΩN → R, denoting f(r) = g(N + 1 − r),  � N  �  k=1  �����  ˜U(k)  k  �  ℓ=1  f(N + 1 − ℓ)  �����  p�1/p  ≤ CpA  � N  �  k=1  | ˜V (k)f(N + 1 − k)|p  �1/p  ,  or equivalently,  � N  �  k=1  �����  ˜U(N + 1 − k)  N  �  ℓ=k  f(ℓ)  �����  p�1/p  ≤ CpA  � N  �  k=1  | ˜V (N + 1 − k)f(k)|p  �1/p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  This implies (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  19  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For f : Ωn → R and s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N denote  T0f(s) = Qs  s  �  t=1  f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Then the operator norm of T0 with respect to the norm ∥ · ∥p,w defined in (39) is bounded by a  number ˜Cp depending only on p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In fact, ˜Cp ≤ 21/pp1/pq1/q · max{1, (p − 1)−1/p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  The proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='6 requires the following:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For any α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , αN > 0 and r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  N  �  k=r  αk  � k  �  ℓ=1  αℓ  �−p  ≤  2  min{1, p − 1} ·  �  r  �  k=1  αk  �−p+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (50)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We use the inequality  (p − 1) · b(a + b)−p ≤ a−p+1 − (a + b)−p+1,  which is valid for any a, b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Then for k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  (p − 1) · αk  � k  �  ℓ=1  αℓ  �−p  ≤  �k−1  �  ℓ=1  αℓ  �−p+1  −  �  k  �  ℓ=1  αℓ  �−p+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  By summing this for k = r + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N we obtain  (p − 1) ·  N  �  k=r+1  αk  � k  �  ℓ=1  αℓ  �−p  ≤  �  r  �  k=1  αk  �−p+1  −  � N  �  ℓ=1  αℓ  �−p+1  ≤  �  r  �  k=1  αk  �−p+1  ,  where an empty sum equals zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We conclude (50) by summing this with the trivial inequality  min{1, p − 1} · αr  �  r  �  ℓ=1  αℓ  �−p  ≤  �  r  �  k=1  αk  �−p+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Define  U(k) = w1/p  k  Qk  and  V (k) = w1/p  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Let us verify the condition of the Muckenhoupt criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We need to find A > 0 such that for  all r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N inequality (42) holds true, that is,  N  �  k=r  wkQp  k ≤ Ap  �  r  �  k=1  w−q/p  k  �−p/q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (51)  20  Recall that p/q = p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' From the definition (40) of Qk, we need  N  �  k=r  w−1/(p−1)  k  �  k  �  ℓ=1  w−1/(p−1)  ℓ  �−p  ≤ Ap  �  r  �  k=1  w−1/(p−1)  k  �−(p−1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Setting αk = w−1/(p−1)  k  and using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='7, we see that (51) holds true with  A = 21/p · max{1, (p − 1)−1/p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='3 we thus conclude that for any f : ΩN → R,  � N  �  k=1  wk  �����Qk  k  �  ℓ=1  f(ℓ)  �����  p�1/p  ≤ p1/pq1/qA ·  � N  �  k=1  wk|f(k)|p  �1/p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  This implies the required bound for the operator norm of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For f : Ωn → R and s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N denote  T1f(s) =  N  �  t=s+1  �  Qs  t�  k=s+1  (1 − Qk)  �  f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Then the operator norm of T1 with respect to the norm ∥ · ∥p,w defined in (39) is bounded by  21/qp1/pq1/q · max{1, (q − 1)−1/q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Denote αk = w−1/(p−1)  k  and recall from (40) that Qk = αk/ �k  ℓ=1 αℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' We claim that for  any t ≥ s + 1,  Qs  t�  k=s+1  (1 − Qk) =  αs  �t  k=1 αk  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (52)  Indeed, (52) holds true for t = s, since an empty product equals one, and for t ≥ s + 1 it is  proven by an easy induction on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Consequently,  T1f(s) = αs  N  �  t=s+1  1  �t  j=1 αj  f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Since p, q ∈ (1, ∞), the elementary inequality of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='7 is valid also when p is replaced by  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' It implies that for r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  �  r  �  k=1  αk  �−q+1  ≥ min{1, q − 1}  2 N  �  k=r  αk  �  k  �  j=1  αj  �−q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (53)  21  Set A = 21/q min{1, q − 1}−1/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Since αk = w−1/(p−1)  k  and q/p = q − 1 = 1/(p − 1), it follows  from (53) that for r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' , N,  �  r  �  k=1  wkαp  k  �1/p  ≤ A  \uf8eb  \uf8ed  N  �  k=r  w−q/p  k  � k  �  j=1  αj  �−q\uf8f6  \uf8f8  −1/q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  This is precisely the Muckenhoupt criterion from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='5, with  U(k) = w1/p  k αk  and  V (k) = w1/p  k  k  �  j=1  αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Thus, by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='5, for any g : ΩN → R,  N  �  k=1  wk  �����αk  N  �  ℓ=k  g(ℓ)  �����  p  ≤ (CpA)p  N  �  k=1  wk  �����  �  k  �  j=1  αj  �  g(k)  �����  p  ,  (54)  with Cp = p1/pq1/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' By restricting attention to non-negative functions g in (54), we may alter  (54) and replace �N  ℓ=k by the shorter sum �N  ℓ=k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Inequality (54) remains correct, for non-  negative g, also after this modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Denoting g(k) = f(k)/ �k  j=1 αj, we conclude that for  any non-negative function f : ΩN → R,  N  �  k=1  wk  �����αk  N  �  ℓ=k+1  1  �ℓ  j=1 αj  f(ℓ)  �����  p  ≤ (CpA)p  N  �  k=1  wk |f(k)|p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  (55)  Since the kernel of T1 is non-negative, its operator norm is attained at a non-negative function  f : ΩN → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Therefore (55) implies the required bound for the operator norm of T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The kernel of the operator T is given in (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' It is a non-negative  kernel, and therefore the operator norm of T is at most the operator norm of the operator whose  kernel is the expression on the right-hand side of (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The latter operator equals  T0 + T1  with T0 from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='6 and T1 from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' From these two propositions it follows  that the operator norm of T is at most  21/pp1/pq1/q· max{1, (p − 1)−1/p} + 21/qp1/pq1/q · max{1, (q − 1)−1/q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2 follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  22  Remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' In this paper we have left open several natural questions, including the existence of linear  extension operators for ˙W 1,p(T) for weighted trees T in the extreme cases p = 1, p = ∞,  as well as the analog of our result for the inhomogeneous Sobolev space W 1,p(T) in place  of ˙W 1,p(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' The problem of existence of linear Sobolev extension operators for weighted trees arose in  connection with an extension problem for W 1,p(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' More precisely, given E ⊆ R2, let  ˙W 1,p(E) denote the space of restrictions to E of functions in ˙W 1,p(R2), endowed with the  natural seminorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Does there exist a linear extension operator from ˙W 1,p(E) to ˙W 1,p(R2)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  The answer is affirmative for p ≥ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' see A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Israel [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For 1 < p < 2, the answer  is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' For a particular class of examples E, the problem reduces to the question  answered by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  References  [1] Bj¨orn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Bj¨orn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Gill, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Shanmugalingam, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Geometric analysis on Cantor sets  and trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' 725, (2017), 63–114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  [2] Howard, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Schep, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Norms of positive operators on Lp-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' 109, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' 1, (1990), 135–146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  [3] Israel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', A bounded linear extension operator for L2,p(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' (2), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' 178,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' 1, (2013), 183–230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  [4] Muckenhoupt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Hardy’s inequality with weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Studia Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' 44, (1972), 31–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  [5] Shvartsman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Sobolev W 1  p -spaces on closed subsets of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' 220, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content=' 6,  (2009), 1842–1922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}
+page_content='  23' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFST4oBgHgl3EQfZjjS/content/2301.13792v1.pdf'}