-9E3T4oBgHgl3EQfSwmi/content/tmp_files/2301.04436v1.pdf.txt

arXiv:2301.04436v1  [math.CA]  11 Jan 2023
OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS
WITH TWO VARIABLES
ISROIL A. IKROMOV, MICHAEL RUZHANSKY, AKBAR R. SAFAROV∗
Abstract. In this paper we consider the problem of estimation of oscillatory in-
tegrals with Mittag-Leffler functions in two variables. The generalisation is that
we replace the exponential function with the Mittag-Leffler-type function, to study
oscillatory type integrals.
Contents
1.
Introduction
1
2.
Preliminaries
2
3.
Auxiliary statements
4
4.
Proof of the main result
7
Acknowledgements
9
Data availability
9
References
9
1. Introduction
The function Eα(z) is named after the Swedish mathematican G¨osta Magnus
Mittag-Leffler (1846-1927) who defined it by a power series
Eα(z) =
∞
�
k=0
zk
Γ(αk + 1),
α ∈ C, Re(α) > 0,
(1.1)
and studied its properties in 1902-1905 in several subsequent notes [18, 19, 20, 21] in
connection with his summation method for divergent series.
A classical generalization of the Mittag-Leffler function, namely the two-parametric
Mittag-Leffler function is
Eα,β(z) =
∞
�
k=0
zk
Γ(αk + β),
α, β ∈ C, Re(α) > 0,
(1.2)
which was deeply investigated independently by Humbert and Agarval in 1953 ([1,
10, 11]) and by Dzherbashyan in 1954 ([4, 5, 6]) as well as in [9].
∗Corresponding author
2010 Mathematics Subject Classification. 35D10, 42B20, 26D10.
Key words and phrases. Mittag-Leffler functions, phase function, amplitude.
All authors contributed equally to the writing of this paper. All authors read and approved the
final manuscript.
1

2
I.A.IKROMOV, M.RUZHANSKY, A.R.SAFAROV
It has the property that
E1,1(x) = ex, and we can refer to [23] for other properties.
(1.3)
In harmonic analysis one of the most important estimates for oscillatory integral is
van der Corput lemma [24, 25, 26, 34]. Estimates for oscillatory integrals with poly-
nomial phases can be found, for instance, in papers [2, 15, 29, 30, 31]. In the current
paper we replace the exponential function with the Mittag-Leffler-type function and
study oscillatory type integrals (2.3). In the papers [26] and [27] analogues of the van
der Corput lemmas involving Mittag-Leffler functions for one dimensional integrals
have been considered. We extend results of [26] and [27] for two-dimensional inte-
grals with phase having some simple singularities. Analogous problem on estimates
for Mittag-Leffler functions with the smooth phase functions of two variables having
simple singularities was considered in [28] and [32].
2. Preliminaries
Definition 2.1. An oscillatory integral with phase f and amplitude a is an integral
of the form
J(λ, f, a) =
�
Rn a(x)eiλf(x)dx,
(2.1)
where a ∈ C∞
0 (Rn) and λ ∈ R.
If the support of a lies in a sufficiently small neighborhood of the origin and f is
an analytic function at x = 0, then for λ → ∞ the following asymptotic expansion
holds ([17]):
J(λ, f, a) ≈ eiλf(0) �
s
n−1
�
k=0
bs,k(a)λs(ln λ)k,
(2.2)
where s belongs to a finite number of arithmetic progressions, independent of a,
composed of negative rational numbers, bs,k is a distribution with support in the
critical set {x : ∇f(x) = 0}.
Inspired by the terminology from [3], we refer to the maximal value of s, denoting
it by α in this case, as the growth index of f, or the oscillation index at the origin,
and the corresponding value of k is referred to as the multiplicity.
More precisely, the multiplicity of the oscillation index of an analytic phase at a
critical point is the maximal number k possessing the property: for any neighbour-
hood of the critical point there is an amplitude with support in this neighbourhood
for which in the asymptotic series (2.2) the coefficient bs,k(a) is not equal to zero.
The multiplicity of the oscillation index will be denoted by m (see [3]).
Let f be a smooth real-valued function defined on a neighborhood of the origin in
R2 with f(0, 0) = 0, ∇f(0, 0) = 0, and consider the associated Taylor series
f(x1, x2) ∼
∞
�
j,k=0
cjkxj
1xk
2
of f centered at the origin. The set
ℑ(f) := {(j, k) ∈ N2 : cjk =
1
j!k!∂j
x1∂k
x2f(0, 0) ̸= 0}

OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS
3
is called the Taylor support of f at (0, 0). We shall always assume that
ℑ(f) ̸= ∅,
i.e., that the function f is of finite type at the origin. If f is real analytic, so that the
Taylor series converges to f near the origin, this just means that f ̸= 0. The Newton
polyhedron ℵ(f) of f at the origin is defined to be the convex hull of the union of
all the quadrants (j, k) + R2
+, with (j, k) ∈ ℑ(f). The associated Newton diagram
ℵd(f) in the sense of Varchenko [33] is the union of all compact faces of the Newton
polyhedron; here, by a face, we mean an edge or a vertex.
We shall use coordinates (t1, t2) for points in the plane containing the Newton
polyhedron, in order to distinguish this plane from the (x1, x2) - plane.
The distance d = d(f) between the Newton polyhedron and the origin in the sense
of Varchenko is given by the coordinate d of the point (d, d) at which the bisectrix
t1 = t2 intersects the boundary of the Newton polyhedron.
The principal face π(f) of the Newton polyhedron of f is the face of minimal
dimension containing the point (d, d). Deviating from the notation in [33], we shall
call the series
fp(x1, x2) :=
�
j,k∈π(f)
cjkxj
1xk
2
the principal part of f. In the case that π(f) is compact, fπ is a mixed homogeneous
polynomial; otherwise, we shall consider fπ as a formal power series.
Note that the distance between the Newton polyhedron and the origin depends
on the chosen local coordinate system in which f is expressed. By a local analytic
(respectively smooth) coordinate system at the origin we shall mean an analytic (re-
spectively smooth) coordinate system defined near the origin which preserves 0. If
we work in the category of smooth functions f, we shall always consider smooth co-
ordinate systems, and if f is analytic, then one usually restricts oneself to analytic
coordinate systems (even though this will not really be necessary for the questions we
are going to study, as we will see). The height of the analytic (respectively smooth)
function f is defined by
h := h(f) := sup{dx},
where the supremum is taken over all local analytic (respectively smooth) coordinate
systems x at the origin, and where dx is the distance between the Newton polyhedron
and the origin in the coordinates x.
A given coordinate system x is said to be adapted to f if h(f) = dx.
Let π be the principal face. We assume that π is a point or a compact edge, then
fπ is a weighted homogeneous polynomial. Denote by ν the maximal order of roots
of fπ on the unit circle at the origin, so
ν := max
S1 ord(fπ).
If there exists a coordinate system x such that ν = dx then we set m = 1. It can
be shown that in this case x is adapted to f (see [12]). Otherwise we take m = 0.
Following A. N. Varchenko we call m the multiplicity of the Newton polyhedron.
In the classical paper by A. N. Varchenko [33], he obtained the sharp estimates
for oscillatory integrals in terms of the height. Also in the paper [13] the height was
used to get the sharp bound for maximal operators associated to smooth surfaces in

4
I.A.IKROMOV, M.RUZHANSKY, A.R.SAFAROV
R3. It turns out that analogous quantities can be used for oscillatory integrals with
the Mittag-Leffler function.
We consider the following integral with phase f and amplitude ψ, of the form
Iα,β =
�
U
Eα,β(iλf(x))ψ(x)dx,
(2.3)
where 0 < α < 1, β > 0, U is a sufficiently small neighborhood of the origin. We
are interested in particular in the behavior of Iα,β when λ is large, as for small λ the
integral is just bounded. In particular if α = 1 and β = 1 we have oscillatory integral
(2.1).
The main result of the work is the following.
Theorem 2.2. Let f be a smooth finite type function of two variables defined in a
sufficiently small neighborhood of the origin and let ψ ∈ C∞
0 (U).
Let h be the height of the function f, and let m = 0, 1 be the multiplicity of its
Newton polyhedron. If 0 < α < 1, β > 0, h > 1, and λ ≫ 1 then we have the
estimate
����
�
U
Eα,β(iλf(x1, x2))ψ(x)dx
���� ≤
C| ln λ|m∥ψ∥L∞(U)
λ
1
h
.
(2.4)
If 0 < α < 1, β > 0, h = 1 and λ ≫ 1, then we have following estimate
����
�
U
Eα,β(iλf(x1, x2))ψ(x)dx
���� ≤
C| ln λ|2∥ψ∥L∞(U)
λ
,
(2.5)
where the constants C are independent of the phase, amplitude and λ.
3. Auxiliary statements
We first recall some useful properties.
Proposition 3.1. If 0 < α < 2, β is an arbitrary real number, µ is such that πα/2 <
µ < min{π, πα}, then there is C > 0, such that we have
|Eα,β(z)| ≤
C
1 + |z|, z ∈ C, µ ≤ | arg(z)| ≤ π.
(3.1)
See [4], [9], [23].
Proposition 3.2. Let Ω be an open, bounded subset of
R2, and let f : Ω → R be a
measurable function such that for all λ ≫ 1 and for some positive δ ̸= 1, we have
����
�
Ω
eiλf(x)dx
���� ≤ C|λ|−δ| ln λ|m,
(3.2)
with m ≥ 0. Then, we have
��x ∈ Ω : |f(x)| ≤ ε| ≤ Cδεδ| ln ε|m, for δ < 1,
for 0 < ε ≪ 1, and for δ > 1, |x ∈ Ω : |f(x)| ≤ ε| ≤ Cδε ,
for δ = 1,
��x ∈ Ω : |f(x)| ≤ ε| ≤ Cδε| ln ε|m+1,
where Cδ depends only on δ, |A| means the Lebesgue measure of a set A. See [7].

OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS
5
Proof. For the convenience of the reader we give an independent proof of Proposition
3.2. We consider an even non-negative smooth function
ω(x) =
�
1,
when |x| ≤ 1,
0,
when |x| ≥ 2.
For the characteristic function of Ω with Ω ⊂ U, the following inequality holds true
|x ∈ Ω : |f(x)| ≤ ε| =
�
Ω
χ[0,1]
�|f(x)|
ε
�
dx ≤
�
Ω
ω
�f(x)
ε
�
dx.
Now we will use the Fourier inversion formula, and rewrite the last integral as
�
Ω
ω
�f(x)
ε
�
dx = 1
2π
�
Ω
� ∞
−∞
ˇω(ξ)eiξ f(x)
ε dξdx.
As ˇω(ξ) is a Schwartz function, we can use Fubini theorem and change the order of
integration. So we have
�
Ω
� ∞
−∞
ˇω(ξ)eiξ f(x)
ε dξdx =
� ∞
−∞
ˇω(ξ)
�
Ω
eiξ f(x)
ε dxdξ.
We use inequality (3.2) for the inner integral and get
����
�
Ω
eiξ f(x)
ε dx
���� ≤ C| ln(2 + ξ
ε)|m
(1 + | ξ
ε|)δ
.
As ˇω(ξ) is a Schwartz function, we also have
|ˇω(ξ)| ≤
C
1 + |ξ|.
So
�����
� ∞
−∞
C ˇω(ξ)| ln(2 + ξ
ε)|m
(2 + | ξ
ε|)δ
dξ
����� ≲
� ∞
0
2C| ln( ξ
ε)|m
(1 + |ξ|)(2 + | ξ
ε|)δ dξ.
Now we change the variable as ξ = ηε, and we get
� ∞
0
| ln( ξ
ε)|m
(1 + |ξ|)(2 + | ξ
ε|)δ dξ =
� ∞
0
ε| ln η|m
(1 + |εη|)(2 + |η|)δ dη.
Now we estimate the last integral for different values of δ.
If δ < 1 then we have
� ∞
0
ε| ln η|m
(1 + |εη|)(2 + |η|)δ dη ≤ Cε
�
1
ε
0
| ln η|mdη
(2 + η)δ + Cε
� ∞
1
ε
| ln η|mdη
εηδ+1
.
We represent
1
(2+η)δ =
1
ηδ(1+ 2
η )δ =
1
ηδ + O(
1
ηδ+1). So
Cε
�
1
ε
0
| ln η|mdη
(2 + η)δ = ε
� 2
0
| ln η|mdη
(2 + η)δ + ε
�
1
ε
2
| ln η|mdη
(2 + η)δ .
Integrating by parts we obtain
ε
�
1
ε
2
| ln η|mdη
(2 + η)δ ≤ ε
�
1
ε
2
| ln η|mdη
ηδ
≤ Cεδ| ln ε|m.

6
I.A.IKROMOV, M.RUZHANSKY, A.R.SAFAROV
As δ < 1, the integrals
� 2
0
| ln η|mdη
(2+η)δ
and
� ∞
1
ε
| ln η|mdη
εηδ+1
convergence.
If δ > 1 then we trivially obtain
����
� ∞
0
Cε| ln η|m
(1 + |εη|)(2 + |η|)δ dη
���� ≤ Cε.
If δ = 1 then assuming 0 < ε < 1
2 we get |εη| < 1 (for |η| < 2), then write the integral
as the sum of three integrals and obtain
����
� ∞
0
Cε| ln η|m
(1 + |εη|)(1 + |η|)dη
���� ≤
����
� 2
0
Cε| ln η|mdη
���� +
�����
�
1
ε
2
Cε| ln η|m
η
dη
����� +
�����
� ∞
1
ε
Cε| lnη|m
η
dη
����� .
Then we have
����
� 2
0
Cε| ln η|mdη
���� ≤ Cε,
and we get with simple calculating that
�����
�
1
ε
2
Cε| lnη|m
η
dη
����� ≤ Cε| ln ε|m+1.
We use the formula of integrating by parts several times, to get
�����
� ∞
1
ε
Cε| ln η|m
η
dη
����� ≤ Cε| ln ε|m,
completing the proof.
□
From Proposition 3.2 we get the following corollaries.
Corollary 3.3. Let f(x1, x2) be a smooth function with f(0, 0) = 0, ∇f(0, 0) = 0,
and h be the height of the function f(x1, x2), and let m = 0, 1 be the multiplicity of
its Newton polyhedron. Let also a(x) =
�
1,
when |x| ≤ σ,
0,
when |x| ≥ 2σ,
σ > 0, and a(x) ≥ 0
with a ∈ C∞
0 (R2). If for all real λ ≫ 1 and for any positive δ ̸= 1, the following
inequality holds
����
�
R2 eiλf(x)a(x)dx
���� ≤ C|λ|−δ| ln λ|m,
(3.3)
then we have
||x| ≤ σ : |f(x)| ≤ ε| ≤ Cεδ| ln ε|m,
where m ≥ 0. See [8, 12, 14, 22].
Corollary 3.4. Let f(x1, x2) be a smooth function with f(0, 0) = 0, ∇f(0, 0) = 0,
and let Ω be a sufficiently small compact set around the origin. Let also h be the
height of the function f(x1, x2), and let m = 0, 1 be the multiplicity of its Newton
polyhedron. Then for all 0 < ε ≪ 1 we have
|x ∈ Ω : |f(x)| ≤ ε| ≤ Cε
1
h| ln ε|m,
where h is the height of f and m is its multiplicity [8].

OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS
7
4. Proof of the main result
Proof of Theorem 2.2. As for λ < 2 the integral (2.3) is just bounded, we
consider the case λ ≥ 2. Without loss of generality, we can consider the integral over
U. Using inequality (3.1), we have
|Eα,β(iλf(x))| ≤
C
1 + λ|f(x)|.
(4.1)
We then use (4.1) for the integral (2.3), and get that
|Iα,β| ≤
����
�
U
Eα,β(iλf(x))ψ(x)dx
���� ≤ C
�
U
|ψ(x)|dx
1 + λ|f(x)|.
(4.2)
Now we represent the integral Iα,β over the union of sets Ω1 := Ω ∩ {λ|f(x1, x2)| <
M} and Ω2 := Ω ∩ {λ|f(x1, x2)| ≥ M} respectively, where M is a positive real
number.
We estimate the integral Iα,β over the sets Ω1 and Ω2, respectively,
|Iα,β| ≤ C
�
U
|ψ(x)|dx
1 + λ|f(x)| = J1 + J2 := C
�
Ω1
|ψ(x)|dx
1 + λ|f(x)| + C
�
Ω2
|ψ(x)|dx
1 + λ|f(x)|.
First we estimate the integral over the set Ω1. Using the results of the paper ([17]
page 31) (see also Corollary 3.4) we obtain
|J1| = C
�
Ω1
|ψ(x)|dx
1 + λ|f(x)| ≤
C| ln λ|m∥ψ∥L∞(Ω1)
λ
1
h
.
Lemma 4.1. Let f ∈ C∞ and h be the height of the function f, and let m = 0, 1 be
the multiplicity of its Newton polyhedron. For any smooth function a = a(x, y) with
sufficiently small support and for h > 1 the following inequality holds
I :=
�
{|f(x,y)|≥ M
λ }
a(x, y)
1 + λ|f(x, y)|dxdy ≤
C| ln λ|m∥a∥L∞(U)
λ
1
h
,
(4.3)
where supp{a(x, y)} = U.
Proof. Let h > 1. Consider the sets
Ak =
�
x ∈ U : 2k
λ ≤ |f(x)| ≤ 2k+1
λ
�
.
For the measure of a set of smaller values we use Lemma 1
′ in the paper [16] (see also
Corollary 3.4), and we have
µ
�
|f(x)| ≤ 2k+1
λ , x ∈ U
�
≤ C
�2k+1
λ
� 1
h �
ln
����
λ
2k+1
����
�m
.
Let
Ik :=
�
Ak
a(x, y)
1 + λ|f(x, y)|dxdy.
For the integral
�
2k≤λ|f(x)|≤2k+1
Ik =
�
Ω2
a(x, y)
1 + λ|f(x, y)|dxdy,

8
I.A.IKROMOV, M.RUZHANSKY, A.R.SAFAROV
we find the following estimate:
|Ik| =
����
�
Ak
a(x, y)
1 + λ|f(x, y)|dxdy
���� ≤ C∥a∥L∞(U)
�2k+1
λ
� 1
h ����ln 2k+1
λ
����
m
2−k.
From here we find the sum of Ik and, by estimating the integral I, we get
I ≤ ∥a∥L∞(U)
∞
�
k=1
Ik ≤ ∥a∥L∞(U)
∞
�
k=1
�2k+1
λ
� 1
h ����ln 2k+1
λ
����
m
2−k
≤ ∥a∥L∞(U)
| ln λ|m
λ
1
h
∞
�
k=1
2
k+1
h −kkm.
As h > 1, the last series is convergent, proving the lemma.
□
Remark 4.2. Consider the case h = 1.
The smooth function has non-degenerate
critical point at the origin if and only if h = 1. As f(x, y) is a smooth function with
∇f(0, 0) = 0, using Morse lemma we have f ∼ x2 ± y2. So in this case we estimate
two sets ∆ = ∆1 ∪ ∆2, where ∆1 := {(x, y) : λ|x2 ± y2| ≤ M, |x| ≤ 1, |y| ≤ 1} and
∆2 := {(x, y) : λ|x2 ± y2| > M, |x| ≤ 1, |y| ≤ 1}. First we consider the integral over
the set ∆1. Then we have
����
�
∆1
a(x, y)
1 + λ|x2 ± y2|dxdy
���� ≤ C∥a∥L∞(∆1)
����
�
∆1
dxdy
����.
Now we estimate the last integral as
����
�
λ|x2+y2|≤M
dxdy
���� ≤ C
λ .
Then we estimate the measure of the set {|x2 − y2| ≤ εM}, where ε = 1
λ. We have,
for simplicity putting M = 1,
����
�
|x2−y2|≤εM
dxdy
���� ≤ C
�����
� √1−ε
√ε
dy
� √
y2+ε
√
y2−ε
dx
����� =
�����
� √1−ε
√ε
��
y2 + ε −
�
y2 − ε
�
dy
����� =
=
�y
2
�
y2 + ε + ε
2 ln |y +
�
y2 + ε|
� ���
√1−ε
√ε
−
�y
2
�
y2 − ε − ε
2 ln |y +
�
y2 − ε|
� ���
√1−ε
√ε
=
=
�����
√1 − ε
2
+ ε
2 ln
√1 − ε + 1
√ε
−
√
2
2 ε − ε
2 ln |√ε(1 +
√
2)|−
−
��
(1 − ε)(1 − 2ε)
2
− ε
2 ln |
√
1 − ε +
√
1 − 2ε| + ε
2 ln √ε|
������ ≤ Cε ln ε.
Now we consider the integral over the set ∆2. In this case we change the variables
to polar coordinate system and with easy calculating we get
����
�
{λ|x2+y2|≥M}
a(x, y)
1 + λ|x2 + y2|dxdy
���� ≤ C| ln λ|∥a∥L∞(∆2)
λ
(4.4)
and
����
�
{λ|x2−y2|≥M}
a(x, y)
1 + λ|x2 − y2|dxdy
���� ≤ C| ln λ|2∥a∥L∞(∆2)
λ
.
(4.5)

OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS
9
Now we continue the proof of Theorem 2.2. Let h > 1. We use Proposition 3.2 for
the integral J1, to get
|J1| ≤
C| ln λ|m∥a∥L∞(U)
λ
1
h
.
Let consider the integral J2. If h > 1, then using Lemma 4.1 we get
|J2| ≤
C| ln λ|∥a∥L∞(U)
λ
1
h
.
If h = 1, using the Remark 4.2 we get the inequality (2.5). The proof is complete.
The proof of Theorem 2.2 shows that if h = 1, we can get a more precise result.
Proposition 4.3. If h = 1 and f has an extremal point at the point (0,0) (then f is
diffeomorhic equivalent to x2
1 + x2
2 or −x2
1 − x2
2), then we have
|Iα,β| ≤
C| ln λ|∥ψ∥L∞(U)
λ
,
for all λ ≥ 2.
Declaration of competing interest
This work does not have any conflicts of interest.
Acknowledgements
The second author was supported in parts by the FWO Odysseus 1 grant G.0H94.18N:
Analysis and Partial Differential Equations and by the Methusalem programme of the
Ghent University Special Research Fund (BOF) (Grant number 01M01021) and also
supported by EPSRC grant EP/R003025/2.
Data availability. The manuscript has no associated data.
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OSCILLATORY INTEGRALS FOR MITTAG-LEFFLER FUNCTIONS
11
Isroil A. Ikromov
V.I. Romanovsky Institute of Mathematics of the Academy of Sciences of Uzbekistan
Olmazor district, University 46, Tashkent, Uzbekistan
Samarkand State University
Department of Mathematics, 15 University Boulevard
Samarkand, 140104, Uzbekistan
Email address: [email protected]
Michael Ruzhansky
Department of Mathematics: Analysis, Logic and Discrete Mathematics
Ghent University,
Krijgslaan 281, Ghent, Belgium,
School of Mathematical Sciences, Queen Mary University of London,
United Kingdom
Email address: [email protected]
Akbar R.Safarov
Uzbek-Finnish Pedagogical Institute
Spitamenshox 166, Samarkand, Uzbekistan
Samarkand State University
Department of Mathematics, 15 University Boulevard
Samarkand, 140104, Uzbekistan
Email address: [email protected]