Oscillatory hyper-Hilbert transform on Wiener amalgam spaces

Wei Sun; Ru-Long Xie; Liang-Yu Xu

doi:10.1515/math-2021-0106

Artikel Open Access

Oscillatory hyper-Hilbert transform on Wiener amalgam spaces

Wei Sun , Ru-Long Xie und Liang-Yu Xu

Veröffentlicht/Copyright: 31. Dezember 2021

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Abstract

We study the boundedness of the oscillatory integral

T α , β f ( x , y ) = ∫ Q 2 f ( x − γ 1 ( t ) , y − γ 2 ( s ) ) e − 2 π i t − β 1 s − β 2 t − α 1 − 1 s − α 2 − 1 d t d s

on Wiener amalgam spaces, where Q 2 = [ 0 , 1 ] × [ 0 , 1 ] is the unit square in two dimensions, ( x , y ) ∈ R n × R m , γ 1 ( t ) = ( t p 1 , t p 2 , … , t p n ) , γ 2 ( s ) = ( s q 1 , s q 2 , … , s q m ) are homogeneous curves on R n and R m .

Keywords: oscillatory integral; Wiener amalgam spaces; curves; product domain

MSC 2010: 42B20; 42B25

1 Introduction

Time-frequency analysis is an important branch of modern harmonic analysis, with relevant applications in wireless communication and signal analysis. In order to measure the time and frequency of a function or distribution, Feichtinger has introduced the Wiener amalgam spaces [1,2] (see Section 2 for details), which describes local and global properties of a function or distribution. In the last decade, Wiener amalgam spaces have been widely used in time-frequency and phase-space analyses, and also employed to study boundedness properties of pseudo-differential operators, Fourier integral, Fourier multiplier operators and well-posedness of solutions to partial differential equations. More details may be found in [3,4, 5,6,7, 8,9].

In harmonic analysis, there is a large interest in the study of the properties of singular integral operators. Our starting point is the well-known fact that the Hilbert transform

H γ f ( x ) = p.v. ∫ − 1 1 f ( x − γ ( t ) ) d t t ,

where γ : ( − 1 , 1 ) → R n is a continuous curve in R n . The study of these operators was initiated by Fabes and Rivière [10]. Stein and Wainger [11] proved that H γ is bounded on L p ( 1 < p < ∞ ) if γ is well-curved in R n . Here, we say that γ is well-curved, if γ is smooth with γ ( 0 ) = 0 and a segment of the curve containing the origin lies in a subspace of R n spanned by

d ( k ) γ ( t ) d t t = 0 , k = 1 , 2 , … .

One may force the singularity of operator H γ at the origin by considering the operator

H γ , α f ( x , y ) = p.v. ∫ − 1 1 f ( x − t , y − γ ( t ) ) d t t ∣ t ∣ α , α > 0 ,

where γ ( t ) = ∣ t ∣ k or γ ( t ) = ∣ t ∣ k sgn ( t ) and k ≥ 2 . In this case, it is not difficult to see that the operator is not bounded even on L 2 , since its Fourier multiplier is not uniformly bounded (see, e.g., [12]). In order to balance the singularity of operator H γ , α at the origin, we may introduce an additional oscillatory term e − 2 π i ∣ t ∣ − β and define the operator

H α , β f ( x , y ) = p.v. ∫ − 1 1 f ( x − t , y − γ ( t ) ) e − 2 π i ∣ t ∣ − β t ∣ t ∣ α d t ; α , β > 0 .

These operators have been first studied back in 1985 by Zielinski [13], who proved that the operator H α , β is bounded on L 2 ( R 2 ) if and only if β ≥ 3 α , where γ ( t ) = t 2 . In 1996, this result was improved by Chandarana [12], who considered the general homogeneous curves γ ( t ) = ∣ t ∣ k or γ ( t ) = ∣ t ∣ k sgn ( t ) for k ≥ 2 , and proved that if β > 3 α > 0 , then the operator H α , β is bounded on L p ( R 2 ) for

1 + 1 + 3 α ( β + 1 ) β ( β + 1 ) + ( β − 3 α ) < p < β ( β + 1 ) + ( β − 3 α ) 1 + 3 α ( β + 1 ) + 1 .

More recently, Cheng et al. have studied the mapping properties of the operator H α , β on Wiener amalgam spaces in [14]. They proved that if β ≥ 3 α , then the operator H α , β is bounded on W ( ℱ L p , L q ) ( R 2 ) for 1 ≤ p < ∞ , 1 ≤ q ≤ ∞ and γ ( t ) = ∣ t ∣ k or γ ( t ) = ∣ t ∣ k sgn ( t ) for k ≥ 2 .

Let us first consider the following oscillatory integral along curves in the two-dimensional product space R × R

(1.1) T f ( x , y ) = ∫ Q 2 f ( x − γ 1 ( t ) , y − γ 2 ( s ) ) t 1 + α 1 s 1 + α 2 e − 2 π i t − β 1 s − β 2 d t d s ; ( x , y ) ∈ R × R ,

where Q 2 = [ 0 , 1 ] × [ 0 , 1 ] is the unit square in two dimensions, and γ 1 ( t ) = t k , γ 2 ( s ) = s j .

Our first theorem may be stated as follows.

Theorem 1.1

Let 1 ≤ p < ∞ , 1 ≤ q ≤ ∞ , and operator T is defined by (1.1). Then we have

Suppose β 1 > α 1 ≥ 0 , β 2 > α 2 ≥ 0 , if k = 0 , j ≥ 0 or j = 0 , k ≥ 0 , then the operator T is bounded on W ( ℱ L p , L q ) ( R × R ) ;
Suppose β 1 > 2 α 1 ≥ 0 , β 2 > 2 α 2 ≥ 0 , if k ≥ 1 , j ≥ 0 or j ≥ 1 , k ≥ 0 , then the operator T is bounded on W ( ℱ L p , L q ) ( R × R ) .

In 2008, Chen et al. extended Chandarana’s results to n -dimensional space [15]. In particular, they have shown that if β > ( n + 1 ) α and Γ ( t ) = ( t p 1 , t p 2 , … , t p n ) , then the operator defined by

H n , α , β f ( x ) = p.v. ∫ − 1 1 f ( x − Γ ( t ) ) e − 2 π i ∣ t ∣ − β d t t ∣ t ∣ α

is bounded on L p ( R n ) , where

2 β 2 β − ( n + 1 ) α < p < 2 β ( n + 1 ) α .

Soon after [16], they investigated the mapping properties of the oscillatory integral along certain curves in high-dimensional product domain on Lebesgue space and defined the operator T α , β in R n × R m by

(1.2) T α , β f ( x , y ) = ∫ Q 2 f ( x − γ 1 ( t ) , y − γ 2 ( s ) ) t 1 + α 1 s 1 + α 2 e − 2 π i t − β 1 s − β 2 d t d s , ( x , y ) ∈ R n × R m

where γ 1 ( t ) = ( t p 1 , t p 2 , … , t p n ) , γ 2 ( s ) = ( s q 1 , s q 2 , … , s q m ) and 0 < p 1 < p 2 < ⋯ < p n . They proved that if β 1 > ( n + 1 ) α 1 , β 2 > ( n + 1 ) α 2 and σ = min { β 1 − ( n + 1 ) α 1 , β 2 − ( n + 1 ) α 2 } , if ∣ 2 − 1 − p − 1 ∣ < σ , then the operator T α , β f ( x , y ) is bounded on L p ( R n × R m ) .

For the operator of T α , β , we prove the following result.

Theorem 1.2

Let β 1 > ( n + 1 ) α 1 ≥ 0 and β 2 > ( n + 1 ) α 2 ≥ 0 , and assume that p 1 , p 2 , … , p n and q 1 , q 2 , … , q m are two groups of distinct positive real numbers. Then the operator T α , β , which is defined by (1.2), is bounded on W ( ℱ L p , L q ) ( R n × R m ) , where 1 ≤ p < ∞ , 1 ≤ q ≤ ∞ .

Throughout this paper, we always denote by C any positive constant that may vary at each occurrence, but it is independent on the essential variable.

This paper is organized as follows. In Section 2, we recall the definitions of Wiener amalgam spaces and modulation space. Some basic properties of Wiener amalgam spaces are also reviewed. In Section 3, we prove our main results. Finally, we discuss two extensions in Section 4.

2 Preliminaries and important lemmas

Let S ( R n ) be the Schwartz space of all complex-valued, rapidly decreasing, infinitely differentiable functions and S ′ ( R n ) be the topological dual of S ( R n ) . For a function f ∈ S ( R n ) , its Fourier transform is defined by f ˆ ( ω ) = ∫ f ( t ) e − 2 π i ω ⋅ t d t , and the inverse Fourier transform is f ˇ ( t ) = f ˆ ( − t ) . The translation and modulation operators are defined by

T x f ( t ) = f ( t − x ) , M ω f ( t ) = e 2 π i ω ⋅ t f ( t ) ,

for every x , ω ∈ R n .

The short-time Fourier transform (STFT) is defined by

V g f ( x , ω ) = ⟨ f , M ω T x g ⟩ = ∫ e − 2 π i ω ⋅ y f ( y ) g ( y − x ) ¯ d y ,

i.e., the Fourier transform applied to f T x g ¯ .

Starting from STFT, let g be a nonzero Schwartz function and 1 ≤ p , q ≤ ∞ , then the modulation space M p , q ( R n ) is defined as the closure of the Schwartz class with respect to the norm

‖ f ‖ M p , q = ∫ R n ∫ R n ∣ V g f ( x , ω ) ∣ p d x q p d ω 1 q ,

with the general modifications for p or q = ∞ .

For 1 ≤ p ≤ ∞ , let ℱ L p ( R n ) be the space of tempered distributions with Fourier transforms in L p ( R n ) , that is,

ℱ L p ( R n ) = { f ∈ S ′ ( R n ) ∣ f ˆ ∈ L p }

with norm ‖ f ‖ ℱ L p = ‖ f ˆ ‖ L p .

Using ℱ L p ( R n ) , Feichtinger [1] introduced Wiener amalgam spaces as follows:

Definition 2.1

For 1 ≤ p , q ≤ ∞ , a tempered distribution f is in the Wiener amalgam spaces W ( ℱ L p , L q ) ( R n ) , if f is locally in ℱ L p ( R n ) and globally in L q ( R n ) , that is, for every non-zero g ∈ C 0 ∞ ( R n ) , ℱ ( f T x g ) ∈ ℱ L p ( R n ) and

‖ f ‖ W ( ℱ L p , L q ) = ∫ R n ∫ R n ∣ ℱ ( f T x g ) ( y ) ∣ p d y q p d x 1 q

is finite, with the general modifications for p or q = ∞ .

To prove our main results, the following lemmas play an important role.

Lemma 2.1

Let us consider 1 ≤ p 1 , p 2 , q 2 , r 1 , r 2 ≤ ∞ , 1 ≤ q 1 < ∞ which satisfy the conditions 1 q 1 = 1 p 1 + 1 r 1 , 1 + 1 q 2 = 1 p 2 + 1 r 2 , then

W ( ℱ L p 1 , L p 2 ) ∗ W ( ℱ L r 1 , L r 2 ) ↪ W ( ℱ L q 1 , L q 2 ) .

Remark

In particular, if r 1 = ∞ and r 2 = 1 , then p 1 = q 1 ≜ p , p 2 = q 2 ≜ q , and the statement of Lemma 2.1 may be written as

W ( ℱ L p , L q ) ∗ W ( ℱ L ∞ , L 1 ) ↪ W ( ℱ L p , L q ) .

This conclusion provides us with a method to prove that singular convolution operators are bounded on Wiener amalgam spaces. Suppose that the operator T is defined by T f ( x ) = ( k ∗ f ) ( x ) , where the kernel function of operator T is k ( x ) . If we can prove that the kernel k ∈ W ( ℱ L ∞ , L 1 ) , then the operator is bounded on Wiener amalgam spaces W ( ℱ L p , L q ) .

Lemma 2.2

f ∈ W ( ℱ L ∞ , L 1 ) if and only if f ˆ ∈ M ∞ , 1 .

Lemma 2.3

Let K = 2 n 2 + 1 , and G K defined as follows:

G K = f ∈ C K ( R n ) ∑ ∣ l ∣ ≤ K ‖ ∂ l f ‖ ∞ ≤ C ,

then G K ⊂ M ∞ , 1 .

The proofs of Lemmas 2.1–2.3 can be found in [1,2], and the following lemma is the so-called Van der Corput Lemma, which can be found in [17].

Lemma 2.4

I ( a , b ) = ∫ a b e i h ( t ) d t ,

then we have

If ∣ h ′ ( t ) ∣ ≥ λ > 0 and h ′ is monotonic on ( a , b ) , then
∣ I ( a , b ) ∣ ≤ C λ − 1 ;
If k ≥ 2 , h ∈ C k ( [ a , b ] ) and ∣ h ( k ) ( t ) ∣ ≥ λ > 0 , then
∣ I ( a , b ) ∣ ≤ C λ − 1 / k ,

where the constant C is independent of a , b .

Lemma 2.5

Let p ( t ) and q ( t ) be real valued smooth functions on the interval ( a , b ) and k ∈ N . If p ( t ) satisfies ∣ p ( k ) ( t ) ∣ ≥ λ for all t ∈ ( a , b ) and (1) k = 1 , q ′ ( t ) is monotonic on ( a , b ) , or (2) k ≥ 2 , then we have

∫ a b e i p ( t ) q ( t ) d t ≤ C k λ − 1 k ‖ q ‖ L ∞ ( a , b ) + ∫ a b ∣ q ′ ( t ) ∣ d t .

It is not difficult to see that Lemma 2.5 is a generalization of Lemma 2.4, which may also be found in [17].

3 Proofs of main results

Before proving the main results, we first estimate two oscillatory integrals.

Let

P s ( ξ ) = ∫ 0 1 t k l 1 − α 1 − 1 e − 2 π i ( ξ t k + t − β 1 s − β 2 ) d t , s ∈ ( 0 , 1 ) ,

where ξ ∈ R and k , l 1 , α 1 , β 1 , β 2 ≥ 0 , then we have the following estimates for P s ( ξ ) :

Lemma 3.1

Existence of a positive constant C such that

If k = 0 and β 1 > α 1 , then ∣ P s ( ξ ) ∣ ≤ C s β 2 ;
If k ≥ 1 and β 1 > 2 α 1 , then ∣ P s ( ξ ) ∣ ≤ C s β 2 2 ,

where the constant C is independent of ξ .

Proof

Let ϕ s ( u ) = ξ u k + u − β 1 s − β 2 and G ( t ) = ∫ 0 t e − 2 π i ϕ s ( u ) d u .

If k = 0 , differentiating ϕ s ( u ) by first order, we get
ϕ s ′ ( u ) = − β 1 u − β 1 − 1 s − β 2 < 0 ,
which implies that
∣ ϕ s ′ ( u ) ∣ ≥ C t − β 1 − 1 s − β 2
for u ∈ ( 0 , t ) . Lemma 2.4 indicates that
∣ G ( t ) ∣ ≤ C t β 1 + 1 s β 2 .
Integrating by parts for P s ( ξ ) , we have
P s ( ξ ) = ∫ 0 1 t − α 1 − 1 G ′ ( t ) d t = [ t − α 1 − 1 G ( t ) ] 0 1 − ( − α 1 − 1 ) ∫ 0 1 t − α 1 − 2 G ( t ) d t ,
and
∣ P s ( ξ ) ∣ ≤ C s β 2 [ t β 1 − α 1 ] 0 1 + ∣ − α 1 − 1 ∣ ∫ 0 1 t β 1 − α 1 − 1 d t ≤ C s β 2
for β 1 > α 1 .
If k ≥ 1 , we consider two cases ξ ≤ 0 and ξ > 0 . For ξ ≤ 0 , differentiating ϕ s ( u ) by first order, then
ϕ s ′ ( u ) = k ξ u k − 1 − β 1 u − β 1 − 1 s − β 2 ≤ − β 1 u − β 1 − 1 s − β 2 < 0 ,
which implies that
∣ ϕ s ′ ( u ) ∣ ≥ C t − β 1 − 1 s − β 2
for u ∈ [ 0 , t ] . Repeating the proof of (i), we get
∣ P s ( ξ ) ∣ ≤ C s β 2 ≤ C s β 2 2 .
For ξ > 0 , differentiating ϕ s ( u ) by second order, we now have
ϕ s ″ ( u ) = k ( k − 1 ) ξ u k − 2 + β 1 ( β 1 + 1 ) u − β 1 − 2 s − β 2 ≥ C t − β 1 − 2 s − β 2
for u ∈ [ 0 , t ] . Lemma 2.4 gives that
∣ G ( t ) ∣ ≤ C t β 1 + 2 2 s β 2 2 .
Integrating by parts for P s ( ξ ) , we get
P s ( ξ ) = ∫ 0 1 t k l 1 − α 1 − 1 G ′ ( t ) d t = [ t k l 1 − α 1 − 1 G ( t ) ] 0 1 − ( k l 1 − α 1 − 1 ) ∫ 0 1 t k l 1 − α 1 − 2 G ( t ) d t ,
thus
∣ P s ( ξ ) ∣ ≤ C s β 2 2 t β 1 − 2 α 1 2 + k l 1 0 1 + ∣ k l 1 − α 1 − 1 ∣ ∫ 0 1 t β 1 − 2 α 1 2 + k l 1 − 1 d t ≤ C s β 2 2
for β 1 > 2 α 1 .□

Let function Φ ( t ) ∈ C 0 ∞ ( R ) satisfying supp Φ ∈ 1 2 , 2 , ( ξ , η ) ∈ ( R n × R m ) . For v = 0 , 1 , 2 , … , define the oscillatory integrals as

I v , s ( ξ , η ) = ∫ 1 2 2 Φ ( t ) t p 1 l 1 + p 2 l 2 + ⋯ p n l n − α 1 − 1 e − 2 π i ϕ ( 2 − v t , s ) d t , s ∈ ( 0 , 1 ) ,

where the ϕ ( t , s ) = ξ 1 t p 1 + ξ 2 t p 2 + ⋯ + ξ n t p n + η 1 s q 1 + ⋯ + η m s q m + t − β 1 s − β 2 , the range of index p 1 , p 2 , … , p n , q 1 , q 2 , … , q m and α 1 , β 1 , β 2 are the same as that in Theorem 1.2. Then we have the following estimates for I v , s ( ξ , η ) :

Lemma 3.2

Existence of a positive constant C such that

∣ I v , s ( ξ , η ) ∣ ≤ C 2 − v β 1 n + 1 s β 2 n + 1 ,

where the constant C is independent of ( ξ , η ) .

Proof

Differentiate ϕ ( t , s ) for t by m order, then for m = 1 , 2 , 3 , … , n + 1 , we have

∂ m ϕ ∂ t m = − β 1 ( − β 1 − 1 ) ⋯ ( − β 1 − m + 1 ) t − β 1 − m s − β 2 + ∑ j = 1 n p j ( p j − 1 ) ⋯ ( p j − m + 1 ) t p j − m ξ j .

Let Δ j m = ∏ i = 1 m ( p j − i − 1 ) , p 0 = − β 1 , ξ 0 = s − β 2 , it is easy to see that

t β 1 + m s β 2 ∂ m ϕ ( t , s ) ∂ t m = ∑ j = 0 n Δ j m t p j + β 1 s β 2 ξ j .

Denote vector ω = ( ω 1 , ω 2 , … , ω n + 1 ) , μ = ( μ 0 , μ 1 , … , μ n ) , where ω m = t β 1 + m s β 2 ∂ m ϕ ∂ t m , μ j = t p j + β 1 s β 2 ξ j , then ω = D μ and D is a ( n + 1 ) -order Vandermonde matrix with

D = Δ 0 1 Δ 1 1 ⋯ Δ n 1 Δ 0 2 Δ 1 2 ⋯ Δ n 2 ⋮ ⋮ ⋱ ⋮ Δ 0 n + 1 Δ 1 n + 1 ⋯ Δ n n + 1 .

It is not difficult to check that det D = ∏ j = 0 n p j ∏ 0 ≤ i < j ≤ n ( p j − p i ) , for distinct positive real numbers p 0 , p 1 , … , p n , we know ∣ det D ∣ ≠ 0 . On the other hand, it is easy to see that the norm of u is larger than 1 2 for every t ∈ 1 2 , 2 . Thus, there is a positive C independent of s , t and ( ξ , η ) ∈ R n × R m such that

t β 1 + m s β 2 ∂ m ϕ ( t , s ) ∂ t m ≥ C

for at least one m ∈ { 1 , 2 , … , n + 1 } . Therefore, for any 1 2 ≤ t ≤ 2 , we have

∂ m ϕ ( 2 − v t , s ) ∂ t m ≥ C 2 v β 1 s − β 2 .

If the first order of ϕ satisfies the above inequality, we can split the interval t ∈ 1 2 , 2 into finite number of subintervals such that ϕ ′ is monotonic on these subintervals, and the number of subintervals depends only on n . So by Lemma 2.5 and s ∈ [ 0 , 1 ] , we get

∣ I v , s ( ξ , η ) ∣ ≤ C 2 − v β 1 m s β 2 m [ ‖ Φ ( t ) t p 1 l 1 + p 2 l 2 + ⋯ p n l n − α 1 − 1 ‖ L ∞ 1 2 , 2 + ∫ 1 2 2 ∣ ( Φ ( t ) t p 1 l 1 + p 2 l 2 + ⋯ p n l n − α 1 − 1 ) ′ ∣ d t ] ≤ C 2 − v β 1 m s β 2 m ≤ C 2 − v β 1 n + 1 s β 2 n + 1 . □

Now, we begin with the proof of Theorem 1.1.

Proof

We take the Fourier transform on T and write T as

T f ^ ( ξ 1 , ξ 2 ) = m ( ξ 1 , ξ 2 ) f ˆ ( ξ 1 , ξ 2 ) ,

where

m ( ξ 1 , ξ 2 ) = ∫ Q 2 e − 2 π i [ ξ 1 ⋅ t k + ξ 2 ⋅ s j + t − β 1 s − β 2 ] t − 1 − α 1 s − 1 − α 2 d t d s .

By Lemmas 2.1–2.3, we only need to prove ‖ ∂ l m ‖ ∞ ≤ C for every ∣ l ∣ ≤ 4 . Denote that l = ( l 1 , l 2 ) ∈ N 2 and ∣ l ∣ = l 1 + l 2 , then

∂ ξ 1 l 1 ∂ ξ 2 l 2 m ( ξ 1 , ξ 2 ) = ∫ Q 2 ( − 2 π i t k ) l 1 ( − 2 π i s j ) l 2 e − 2 π i ( ξ 1 t k + ξ 2 s j + t − β 1 s − β 2 ) t − 1 − α 1 s − 1 − α 2 d t d s = ( − 2 π i ) ∣ l ∣ ∫ 0 1 ∫ 0 1 t k l 1 − α 1 − 1 e − 2 π i ( ξ 1 t k + t − β 1 s − β 2 ) d t e − 2 π i ξ 2 s j s j l 2 − α 2 − 1 d s = ( − 2 π i ) ∣ l ∣ ∫ 0 1 P s ( ξ 1 ) e − 2 π i ξ 2 s j s j l 2 − α 2 − 1 d s .

k = 0 , j ≥ 0 , by Lemma 3.1(i), we have
‖ ∂ ξ 1 l 1 ∂ ξ 2 l 2 m ‖ ∞ ≤ C ∫ 0 1 s β 2 − α 2 + j l 2 − 1 d s ≤ C
for β 2 > α 2 .
k ≥ 1 , j ≥ 0 , by Lemma 3.1(ii), we get
‖ ∂ ξ 1 l 1 ∂ ξ 2 l 2 m ‖ ∞ ≤ C ∫ 0 1 s β 2 − 2 α 2 2 + j l 2 − 1 d s ≤ C
for β 2 > 2 α 2 . Combining the proofs of (i) and (ii), we completed the proof of Theorem 1.1.□

Next, we will give the proof of Theorem 1.2.

Proof

Using Fourier transform for operator T α , β , write T α , β as

T α , β f ∧ ( ξ , η ) = m ( ξ , η ) f ^ ( ξ , η ) ,

where ( ξ , η ) ∈ R n × R m and

m ( ξ , η ) = ∫ Q 2 e − 2 π i [ γ 1 ( t ) ξ + γ 2 ( s ) η + t − β 1 s − β 2 ] t − 1 − α 1 s − 1 − α 2 d t d s .

Note that L = ( L n , L m ) , and L n = ( l 1 , l 2 , … , l n ) ∈ N n , L m = ( l 1 ′ , l 2 ′ , … , l m ′ ) ∈ N m . By Lemmas 2.1–2.3, if we can prove ∂ ξ L n ∂ η L m m ∈ L ∞ for any ∣ L ∣ = ∣ L n ∣ + ∣ L m ∣ ≤ 2 n + m 2 + 1 , then we can complete the proof of Theorem 1.2.

∂ ξ L n ∂ η L m m ( ξ , η ) = ∂ ξ 1 l 1 ∂ ξ 2 l 2 ⋯ ∂ ξ n l n ∂ η 1 l 1 ′ ∂ η 2 l 2 ′ ⋯ ∂ η m l m ′ m ( ξ 1 , ξ 2 , … , ξ n , η 1 , η 2 , … , η m ) = ∫ Q 2 ( − 2 π i t p 1 ) l 1 ( − 2 π i t p 2 ) l 2 ⋯ ( − 2 π i t p n ) l n ( − 2 π i s q 1 ) l 1 ′ ( − 2 π i s q 2 ) l 2 ′ ⋯ × ( − 2 π i s q m ) l m ′ e − 2 π i ϕ ( t , s ) t − 1 − α 1 s − 1 − α 2 d t d s = ( − 2 π i ) ∣ L ∣ ∫ 0 1 ∫ 0 1 t p 1 l 1 + p 2 l 2 + ⋯ p n l n − α 1 − 1 e − 2 π i ϕ ( t , s ) d t s q 1 l 1 ′ + q 2 l 2 ′ ⋯ + q m l m ′ − α 2 − 1 d s ,

where ϕ ( t , s ) = ξ 1 t p 1 + ξ 2 t p 2 + ⋯ + ξ n t p n + η 1 s q 1 + ⋯ + η m s q m + t − β 1 s − β 2 .

Let

I s ( ξ , η ) = ∫ 0 1 t p 1 l 1 + p 2 l 2 + ⋯ p n l n − α 1 − 1 e − 2 π i ϕ ( t , s ) d t ,

choose a test function Φ ( t ) ∈ C 0 ∞ ( R ) satisfying supp Φ ∈ 1 2 , 2 and Σ v = 0 ∞ Φ ( 2 v t ) = 1 . Then I s ( ξ , η ) = Σ v = 0 ∞ I v , s ( ξ , η ) , where

I v , s ( ξ , η ) = ∫ 0 1 Φ ( 2 v t ) t p 1 l 1 + p 2 l 2 + ⋯ p n l n − α 1 − 1 e − 2 π i ϕ ( t , s ) d t = 2 v α 1 2 − v ( p 1 l 1 + ⋯ + p n l n ) ∫ 1 2 2 Φ ( t ) t p 1 l 1 + p 2 l 2 + ⋯ p n l n − α 1 − 1 e − 2 π i ϕ ( 2 − v t , s ) d t .

By Lemma 3.2, we have

∣ I s ( ξ , η ) ∣ ≤ ∑ v = 0 ∞ ∣ I v , s ( ξ , η ) ∣ ≤ C ∑ v = 0 ∞ 2 − v ( β 1 n + 1 − α 1 + p 1 l 1 + ⋯ + p n l n ) s β 2 n + 1 ≤ C s β 2 n + 1

for β 1 > ( n + 1 ) α 1 . So we get

∣ ∂ ξ L n ∂ η L m m ( ξ , η ) ∣ ≤ C ∫ 0 1 s β 2 n + 1 − α 2 + q 1 l 1 ′ + ⋯ + q m l m ′ − 1 ≤ C

for β 2 > ( n + 1 ) α 2 . To summarize, we completed the proof of Theorem 1.2.□

Acknowledgements

The authors would like to thank reviewers for their valuable comments and suggestions.

Funding information: This work was supported by Natural Science Foundation of Anhui Province of China (Nos KJ2016A506 and KJ2017A454), Natural Science Foundation for Outstanding Youth of Anhui Province of China (Nos GXYQ2017070 and GXYQ2020049) and Natural Science Foundation of Chaohu University (No. XLY-201904).
Author contributions: All authors read and approved the final manuscript.
Conflict of interest: The authors declare that they have no competing interests.

References

[1] H. G. Feichtinger , Banach spaces of distributions of Wiener’s type and interpolation , in: H. G. Feichtinger , Sz.-Nagy , and E. Gorlich (eds.), Functional Analysis and Approximation, ISNM 60: International Series of Numerical Mathematics , vol. 60, 1981, pp. 153–165, https://doi.org/10.1007/978-3-0348-9369-5_16. Suche in Google Scholar

[2] H. G. Feichtinger , Banach Convolution algebra of Wiener type , in: Proceedings of the Conference on Functions Series, Operators, Budapest 1980, Colloquia Mathematica Societatis János Bolyai , vol. 35, North-Holland, Amsterdam, 1983, pp. 509–524. Suche in Google Scholar

[3] A. Bényi and T. Oh , Modulation spaces, Wiener amalgam spaces and Brownian motion, Adv. Math. 228 (2011), no. 5, 2943–2981, https://doi.org/10.1016/j.aim.2011.07.023. Suche in Google Scholar

[4] E. Cordero and F. Nicola , Sharpness of some properties of Wiener amalgam and modulation spaces, Bull. Aust. Math. Soc. 80 (2009), no. 1, 105–116, https://doi.org/10.1017/S0004972709000070. Suche in Google Scholar

[5] E. Cordero and F. Nicola , Pseudodifferential operators on Lp , Wiener amalgam and modulation spaces, Int. Math. Res. Not. IMRN 2010 (2010), no. 10, 1860–1893, https://doi.org/10.1093/imrn/rnp190. Suche in Google Scholar

[6] W. Czaja , Boundedness of pseudodifferential operators on modulation spaces, Math. Anal. Appl. 284 (2003), no. 1, 389–396, https://doi.org/10.1016/S0022-247X(03)00364-0 . 10.1016/S0022-247X(03)00364-0Suche in Google Scholar

[7] M. Kobayashi , Multipliers on modulation spaces, SUT J. Math. 42 (2006), no. 2, 305–312. 10.55937/sut/1262445124Suche in Google Scholar

[8] H. G. Feichtinger and G. Narimani , Fourier multipliers of classical modulation spaces, Appl. Comput. Harmon. Anal. 21 (2006), no. 3, 349–359, https://doi.org/10.1016/j.acha.2006.04.010. Suche in Google Scholar

[9] B. Wang and H. Hudzik , The global Cauchy problem for the NLS and NLKG with small rough data, Differ. Equ. 232 (2007), no. 1, 36–73, https://doi.org/10.1016/j.jde.2006.09.004. Suche in Google Scholar

[10] E. B. Fabes and N. M. Riviére , Singular intervals with mixed homogeneity, Studia Math. 27 (1966), 19–38, https://doi.org/10.4064/sm-27-1-19-38. Suche in Google Scholar

[11] E. M. Stein and S. Wainger , Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. (N.S.) 84 (1978), no. 6, 1239–1295, https://doi.org/10.1090/S0002-9904-1978-14554-6. Suche in Google Scholar

[12] S. Chandarana , Lp -bounds for hypersingular integral operators along curves, Pacific J. Math. 175 (1996), no. 2, 389–416. 10.2140/pjm.1996.175.389Suche in Google Scholar

[13] M. Zielinski , Highly oscillatory singular integrals along curves, Ph.D. thesis, University of Wisconsin-Madison, Madison WI, 1985. Suche in Google Scholar

[14] M. Cheng , W. Sun , and L. Shu , Boundedness properties of certain oscillatory integrals on Wiener amalgam space, Chinese J. Contemp. Math. 39 (2018), no. 2, 113–126. Suche in Google Scholar

[15] J. C. Chen , D. S. Fan , M. Wang , and X. R. Zhu , Lp bounds for oscillatory hyper Hilbert transform along curves, Proc. Amer. Math. Soc. 139 (2008), no. 9, 3145–3153. 10.1090/S0002-9939-08-09325-8Suche in Google Scholar

[16] J. C. Chen , D. S. Fan , H. X. Wu , and X. R. Zhu , Oscillatory integrals on unit square along surfaces, Front. Math. China 6 (2011), no. 1, 49–59, https://doi.org/10.1007/s11464-010-0088-3. Suche in Google Scholar

[17] E. Stein , Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993. 10.1515/9781400883929Suche in Google Scholar

Received: 2020-11-20

Revised: 2021-09-06

Accepted: 2021-09-07

Published Online: 2021-12-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2021-0106

Schlagwörter für diesen Artikel

oscillatory integral; Wiener amalgam spaces; curves; product domain

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