Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds

Mohammed Ali; Hussain Al-Qassem

doi:10.1515/math-2022-0603

Article Open Access

Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds

Mohammed Ali and Hussain Al-Qassem

Published/Copyright: July 21, 2023

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$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

We establish certain delicate L p bounds for a class of generalized Marcinkiewicz integral operators along submanifolds with rough kernels. These bounds allow us to use Yano’s extrapolation method to prove the L p boundedness of the aforementioned integral operators under very weak assumptions on the kernels. Our results in this article improve and generalize many previously known results.

Keywords: generalized Marcinkiewicz; Triebel-Lizorkin space; rough kernels; extrapolation

MSC 2010: 42B15; 42B20; 42B25; 42B35

1 Introduction

For κ ≥ 2 , let R κ be the κ -dimensional Euclidean space and S κ − 1 be the unit sphere in R κ , which is equipped with the normalized Lebesgue surface measure d σ ≡ d σ κ ( ⋅ ) . Also, let y ′ = y ∕ ∣ y ∣ for y ∈ R κ \ { 0 } .

For τ = τ 1 + i τ 2 ( τ 1 , τ 2 ∈ R with τ 1 > 0 ), let K Λ , g ( y ) = Λ ( y ) g ( ∣ y ∣ ) ∣ y ∣ κ − τ , where g is a measurable function on R + ≡ [ 0 , ∞ ) and Λ is a homogeneous function of degree zero on R κ , integrable over S κ − 1 , and satisfies the following:

(1.1) ∫ S κ − 1 Λ ( υ ) d σ ( υ ) = 0 .

For appropriate mappings ϕ , φ : R + → R , we consider the generalized Marcinkiewicz integral M Λ , ϕ , φ , g ( μ ) on R κ + 1 by:

M Λ , ϕ , φ , g ( μ ) ( f ) ( x ˜ ) = ∫ 0 ∞ 1 l τ ∫ ∣ y ∣ ≤ l f ( x − ϕ ( ∣ y ∣ ) y ′ , x κ + 1 − φ ( ∣ y ∣ ) ) K Λ , g ( y ) d y μ d l l 1 ∕ μ ,

where f ∈ S ( R κ + 1 ) , x ˜ = ( x , x κ + 1 ) ∈ R κ + 1 and μ > 1 .

When ϕ ( l ) = l , φ ≡ 0 , g ≡ 1 , and μ = 2 , we denote M Λ , ϕ , φ , g ( μ ) by M Λ , τ ; when τ = 1 , we denote M Λ , τ by M Λ . In this case, the operator M Λ is essentially the classical Marcinkiewicz integral operator that was introduced by Stein in [1] in which he established the L p ( 1 < p ≤ 2 ) boundedness of M Λ under the condition Λ ∈ Lip α ( S κ − 1 ) for some α ∈ ( 0 , 1 ] . Thereafter, the operator M Λ and its generalizations have been studied extensively by many authors. For a sample of past studies, the readers are referred to [2–5], among others. Let us now recall some pertinent results to our study. Walsh in [3] proved the L 2 ( R κ ) boundedness of M Λ if Λ ∈ L ( log L ) 1 ∕ 2 ( S κ − 1 ) , and he showed that the condition Λ ∈ L ( log L ) 1 ∕ 2 ( S κ − 1 ) is optimal in the sense that for any ε ∈ ( 0 , 1 ∕ 2 ) , there exists a Λ that belongs to L ( log L ) ε ( S κ − 1 ) such that M Λ is not bounded on L 2 ( R κ ) . In [4], the authors proved the L p ( R κ ) boundedness of M Λ for all p ∈ ( 1 , ∞ ) . On the other hand, the authors of [5] found that the operator M Λ is bounded on L p ( R κ ) for all p ∈ ( 1 , ∞ ) whenever the kernel Λ is in the block space B q ( 0 , − 1 ∕ 2 ) ( S κ − 1 ) for some q > 1 . Moreover, they proved that the condition Λ ∈ B q ( 0 , − 1 ∕ 2 ) ( S κ − 1 ) is optimal in the sense that M Λ will lose the L 2 boundedness if the assumption Λ ∈ B q ( 0 , − 1 ∕ 2 ) ( S κ − 1 ) is replaced by Λ ∈ B q ( 0 , ν ) ( S κ − 1 ) for any − 1 < ν < − 1 ∕ 2 .

On the other hand, the study of the L p boundedness of the parametric Marcinkiewicz operator was initiated by Hörmander in [6] and then continued by many authors. Readers are referred to [1,4,7–13] and references therein. Let us now recall some of the relevant results to our study in this article.

In [11], the authors proved that M Λ , ϕ , 0 , g ( 2 ) is bounded on L 2 ( R κ ) if Λ ∈ L ( log L ) ( S κ − 1 ) , ϕ ( l ) = l , and g ∈ ∇ λ ( R + ) for some λ > 1 , where ∇ λ ( R + ) is the collection of all measurable functions g : R + → C such that

‖ h ‖ ∇ λ ( R + ) = sup j ∈ Z ∫ 2 j 2 j + 1 ∣ g ( l ) ∣ λ d l l 1 ∕ λ < ∞ .

The discussion of the L p mapping properties of rough integral operators along surfaces as well as surfaces of revolution was initiated in [14] and then studied by many authors (see, for instance, [10,15–19] and references therein).

We point out that when τ = 1 , Λ ∈ L ( log L ) ( S κ − 1 ) , g ∈ ∇ λ ( R + ) for some λ > 1 , ϕ ( l ) = l , and φ ∈ C 2 ( R + ) , convex and increasing function with φ ( 0 ) = 0 , then the L p boundedness of M Λ , ϕ , φ , g ( 2 ) was obtained in [20] for all ∣ 1 ∕ p − 1 ∕ 2 ∣ < min { 1 ∕ λ ′ , 1 ∕ 2 } . In addition, whenever τ = 1 , Λ ∈ L ( log L ) 1 ∕ 2 ( S κ − 1 ) , g ∈ ∇ λ ( R + ) with λ > 1 , and ϕ ∈ C 2 ( [ 0 , ∞ ) ) , convex and increasing function with ϕ ( 0 ) = 0 , then the L p boundedness of M Λ , ϕ , 0 , g ( 2 ) was proved in [21] for all ∣ 1 ∕ p − 1 ∕ 2 ∣ < min { 1 ∕ λ ′ , 1 ∕ 2 } . Later on, the authors of [22] studied the L p boundedness of M Λ , ϕ , φ , g ( 2 ) under the assumptions Λ ∈ L ( log L ) α ( S κ − 1 ) (for certain values of α ), g ∈ ∇ λ ( R + ) for some λ > 1 and ϕ , φ ∈ C 2 ( [ 0 , ∞ ) ) , convex and increasing functions with ϕ ( 0 ) = φ ( 0 ) = 0 .

Our focus in this article will be on studying the L p of the generalized parametric Marcinkiewicz integral operator M Λ , ϕ , φ , g ( μ ) . In fact, the study of the operator M Λ , ϕ , φ , g ( μ ) was started in [23], where the authors showed that if Λ ∈ L q ( S κ − 1 ) with q > 1 , ϕ ( l ) = l , and 1 < μ < ∞ , then

(1.2) ∥ M Λ , ϕ , 0 , 1 ( μ ) ∥ L p ( R κ ) ≤ C ∥ f ∥ F . p 0 , μ ( R κ )

for all p ∈ ( 1 , ∞ ) . Later, Le in [24] proved that the inequality (1.2) holds under the weaker condition g ∈ ∇ max { μ ′ , 2 } ( R + ) and Λ ∈ L ( log L ) ( S κ − 1 ) .

These results were improved and extended in [25], where the authors showed that if g ∈ ∇ λ ( R + ) with λ > 2 , Λ ∈ L ( log L ) 1 ∕ μ ( S κ − 1 ) ∪ B q ( 0 , 1 μ − 1 ) ( S κ − 1 ) with q > 1 and ϕ ( l ) = l , then the L p boundedness of M Λ , ϕ , 0 , 1 ( μ ) holds for all 1 λ ′ . For recent advances on the study of such operators and their developments, we refer the readers to see the articles [25–29], among others.

Let us now recall the definition of the Triebel-Lizorkin spaces F . p s , μ ( R κ ) . For s ∈ R and p , μ ∈ ( 1 , ∞ ) , the homogeneous Triebel-Lizorkin space F . p s , μ ( R κ ) is defined by:

F . p s , μ ( R κ ) = f ∈ S ′ ( R κ ) : ∥ f ∥ F . p s , μ ( R κ ) = ∑ m ∈ Z 2 m s μ ∣ T m * f ∣ μ 1 ∕ μ L p ( R κ ) < ∞ ,

where S ′ refers to the tempered distribution class on R κ , T m ^ ( η ) = H ( 2 − m η ) for m ∈ Z and H ∈ C 0 ∞ ( R κ ) is a radial function that satisfies the following properties:

H ∈ [ 0 , 1 ] ,
supp ( H ) ⊂ { η : ∣ η ∣ ∈ [ 1 ∕ 2 , 2 ] } ,
H ( η ) ≥ C > 0 if ∣ η ∣ ∈ 3 5 , 5 3 ,
∑ m ∈ Z H ( 2 − m η ) = 1 with η ≠ 0 .

It is well known that the space F . p s , μ ( R κ ) satisfies the following properties:

The Schwartz space S ( R κ ) is dense in F . p s , μ ( R κ ) ,
F . p 0 , 2 ( R κ ) = L p ( R κ ) for 1 < p < ∞ ,
F . p s , μ 1 ( R κ ) ⊆ F . p s , μ 2 ( R κ ) if μ 1 ≤ μ 2 ,
( F . p s , μ ( R κ ) ) ∗ = F . p ′ − s , μ ′ ( R κ ) .

In view of the results in [22] on parametric Marcinkiewicz integrals and the results in [25] on the generalized parametric Marcinkiewicz integrals, a question that arises naturally is the following:

Question: Does the L p boundedness of the generalized parametric Marcinkiewicz integral M Λ , ϕ , φ , g ( μ ) hold under the same assumptions as in Theorem 1.1 in [22]?

We shall provide answer to the aforementioned question in the affirmative. The main results of this article can be formulated as follows:

Theorem 1.1

Let Λ ∈ L q ( S κ − 1 ) for some q ∈ ( 1 , 2 ] and satisfy Condition (1.1). Suppose that g ∈ ∇ λ ( R + ) for some λ ∈ ( 1 , 2 ] and ϕ , φ ∈ C 2 ( [ 0 , ∞ ) ) , increasing and convex functions with ϕ ( 0 ) = 0 = φ ( 0 ) . Then, for any function f that belongs to the space F . p 0 , μ ( R κ + 1 ) , there exists a positive constant C p , Λ , g (independent of ϕ , φ , μ , and τ ) such that

∥ M Λ , ϕ , φ , g ( μ ) ( f ) ∥ L p ( R κ + 1 ) ≤ C p , Λ , g 1 ( q − 1 ) ( λ − 1 ) 1 ∕ μ ∥ f ∥ F . p 0 , μ ( R κ + 1 ) i f μ ≤ p ≤ λ μ ′ μ ′ − λ , ∥ M Λ , ϕ , φ , g ( μ ) ( f ) ∥ L p ( R κ + 1 ) ≤ C p , Λ , g 1 ( q − 1 ) ( λ − 1 ) λ μ − μ + λ λ μ ∥ f ∥ F . p 0 , μ ( R κ + 1 ) if λ μ λ μ − μ + λ < p < μ ,

and

∥ M Λ , ϕ , φ , g ( μ ) ( f ) ∥ L p ( R κ + 1 ) ≤ C p , Λ , g 1 ( q − 1 ) ( λ − 1 ) λ μ − μ + 1 μ λ ∥ f ∥ F . p 0 , μ ( R κ + 1 ) i f μ λ λ μ − μ + 1 < p < μ ,

where C p , Λ , g = C p ∥ Λ ∥ L q ( S κ − 1 ) ∥ g ∥ ∇ λ ( R + ) .

Theorem 1.2

Let ϕ , φ , and Λ be given as in Theorem 1.1, and g ∈ ∇ λ ( R + ) with 2 < λ < ∞ . Then, there is a constant C p , Λ , g > 0 such that

∥ M Λ , ϕ , φ , g ( μ ) ( f ) ∥ L p ( R κ + 1 ) ≤ C p , Λ , g 1 q − 1 1 ∕ λ ′ ∥ f ∥ F . p 0 , μ ( R κ + 1 )

for λ ′ λ ′ , and

∥ M Λ , ϕ , φ , g ( μ ) ( f ) ∥ L p ( R κ + 1 ) ≤ C p , Λ , g 1 q − 1 ∥ f ∥ F . p 0 , μ ( R κ + 1 )

for 1 < p < μ and μ ≤ λ ′ .

The conclusions of Theorems 1.1 and 1.2 together with Yano’s extrapolation argument (see also [22,30]) give the following results.

Theorem 1.3

Assume that ϕ and φ are given as in Theorem 1.1 and that g ∈ ∇ λ ( R + ) for some λ ∈ ( 1 , 2 ] .

( i ) If Λ ∈ L ( log L ) 1 ∕ μ ( S κ − 1 ) , then for p ∈ μ , λ μ ′ μ ′ − λ ,

∥ M Λ , ϕ , φ , g ( μ ) ( f ) ∥ L p ( R κ + 1 ) ≤ C p ∥ g ∥ ∇ λ ( R + ) ( ∥ Λ ∥ L ( log L ) 1 ∕ μ ( S κ − 1 ) + 1 ) ∥ f ∥ F . p 0 , μ ( R κ + 1 ) .

( i i ) If Λ ∈ L ( log L ) λ μ − μ + λ λ μ ( S κ − 1 ) , then for p ∈ λ μ λ μ − μ + λ , μ ,

∥ M Λ , ϕ , φ , g ( μ ) ( f ) ∥ L p ( R κ + 1 ) ≤ C p ∥ g ∥ ∇ λ ( R + ) ∥ Λ ∥ L ( log L ) λ μ − μ + λ λ μ ( S κ − 1 ) + 1 ∥ f ∥ F . p 0 , μ ( R κ + 1 ) .

( i i i ) If Λ ∈ L ( log L ) λ μ − μ + 1 μ λ ( S κ − 1 ) , then for p ∈ μ λ μ − λ μ + 1 , μ ,

∥ M Λ , ϕ , φ , g ( μ ) ( f ) ∥ L p ( R κ + 1 ) ≤ C p ∥ g ∥ ∇ λ ( R + ) ∥ Λ ∥ L ( log L ) λ μ − μ + 1 λ μ ( S s − 1 ) + 1 ∥ f ∥ F . p 0 , μ ( R κ + 1 ) .

( i v ) If Λ ∈ B q ( 0 , − 1 ∕ μ ′ ) ( S κ − 1 ) for some q > 1 , then for p ∈ μ , λ μ ′ μ ′ − λ ,

∥ M Λ , ϕ , φ , g ( μ ) ( f ) ∥ L p ( R κ + 1 ) ≤ C p ∥ g ∥ ∇ λ ( R + ) ( ∥ Λ ∥ q ( 0 , − 1 ∕ μ ′ ) ( S s − 1 ) + 1 ) ∥ f ∥ F . p 0 , μ ( R κ + 1 ) .

( v ) If Λ ∈ B q ( 0 , λ − μ λ μ ) ( S κ − 1 ) for some q > 1 , then for p ∈ λ μ λ μ − μ + λ , μ ,

∥ M Λ , ϕ , φ , g ( μ ) ( f ) ∥ L p ( R κ + 1 ) ≤ C p ∥ g ∥ ∇ λ ( R + ) ∥ Λ ∥ B q ( 0 , λ − μ λ μ ) ( S s − 1 ) + 1 ∥ f ∥ F . p 0 , μ ( R κ + 1 ) .

( v i ) If Λ ∈ B q ( 0 , 1 − μ μ λ ) ( S κ − 1 ) for some q > 1 , then for p ∈ μ λ μ − μ λ + 1 , μ ,

∥ M Λ , ϕ , φ , g ( μ ) ( f ) ∥ L p ( R κ + 1 ) ≤ C p ∥ g ∥ ∇ λ ( R + ) ∥ Λ ∥ B q ( 0 , 1 − μ μ λ ) ( S s − 1 ) + 1 ∥ f ∥ F . p 0 , μ ( R κ + 1 ) .

Theorem 1.4

Let ϕ and φ be given as in Theorem 1.2, and let g ∈ ∇ λ ( R + ) for some 2 < λ < ∞ .

If Λ ∈ L ( log L ) 1 ∕ λ ′ ( S κ − 1 ) , then
∥ M Λ , ϕ , φ , g ( μ ) ( f ) ∥ L p ( R κ + 1 ) ≤ C p ∥ g ∥ ∇ λ ( R + ) ( ∥ Λ ∥ L ( log L ) 1 ∕ λ ′ ( S s − 1 ) + 1 ) ∥ f ∥ F . p 0 , μ ( R κ + 1 )
for λ ′ λ ′ .
If Λ ∈ L ( log L ) ( S κ − 1 ) , then we have
∥ M Λ , ϕ , φ , g ( μ ) ( f ) ∥ L p ( R κ + 1 ) ≤ C p ∥ g ∥ ∇ λ ( R + ) ( ∥ Λ ∥ L ( log L ) ( S s − 1 ) + 1 ) ∥ f ∥ F . p 0 , μ ( R κ + 1 ) ,
for 1 < p < μ and μ ≤ λ ′ .
If Λ ∈ B q ( 0 , − 1 ∕ λ ) ( S κ − 1 ) for some q > 1 , then
∥ M Λ , ϕ , φ , g ( μ ) ( f ) ∥ L p ( R κ + 1 ) ≤ C p ∥ g ∥ ∇ λ ( R + ) ( ∥ Λ ∥ q ( 0 , − 1 ∕ λ ) ( S s − 1 ) + 1 ) ∥ f ∥ F . p 0 , μ ( R κ + 1 )
for λ ′ 1 , then
∥ M Λ , ϕ , φ , g ( μ ) ( f ) ∥ L p ( R κ + 1 ) ≤ C p ∥ g ∥ ∇ λ ( R + ) ( ∥ Λ ∥ q ( 0 , 0 ) ( S s − 1 ) + 1 ) ∥ f ∥ F . p 0 , μ ( R κ + 1 )
for all 1 < p < μ and μ ≤ λ ′ .

Remark

For any 0 < α ≤ 1 , s > 0 , and q > 1 , the following inclusions hold and are proper:
C 1 ( S κ − 1 ) ⊂ Lip α ( S κ − 1 ) ⊂ L q ( S κ − 1 ) ⊂ L ( log L ) s ( S κ − 1 ) ⊂ L 1 ( S κ − 1 ) ,

⋃ r > 1 L r ( S κ − 1 ) ⊂ B q ( 0 , v ) ( S κ − 1 ) ⊂ L 1 ( S κ − 1 ) for any v > − 1 ,

L ( log L ) s 1 ( S κ − 1 ) ⊂ L ( log L ) s 2 ( S κ − 1 ) for 0 < s 2 < s 1 ,

B q ( 0 , v 1 ) ( S κ − 1 ) ⊂ B q ( 0 , v 2 ) ( S κ − 1 ) for − 1 < v 2 < v 1 .
For the special case ϕ ( l ) = l , φ ≡ 0 , g ≡ 1 , μ = 2 , and τ = 1 , the author of [1] proved the L p ( 1 < p ≤ 2 ) boundedness of M Λ , ϕ , φ , g ( μ ) whenever Λ ∈ Lip α ( S κ − 1 ) for some α ∈ ( 0 , 1 ] . Hence, by Remark (1), our results extend and improve the results in [1].
For the special case ϕ ( l ) = l , φ ≡ 0 , μ = 2 , τ = 1 , and g ∈ ∇ λ ( R + ) , the authors of [11] proved only the L 2 ( R κ ) boundedness of the operator M Λ , ϕ , φ , g ( μ ) whenever Λ ∈ L ( log L ) ( S κ − 1 ) . Therefore, our results are fundamental improvement and generalization of the results in [11].
For the case μ = 2 and λ ∈ ( 1 , 2 ] , the range of p in Theorem 1.3 is better than the range of the results obtained in [22], in which the authors proved the L p boundedness of M Λ , ϕ , φ , g ( 2 ) only for p ∈ 2 λ ′ λ ′ − 2 , 2 λ 2 − λ .
When ϕ ( l ) = l and φ ≡ 0 , the operator was studied in [23] only under the condition Λ ∈ L q ( S κ − 1 ) , which is much stronger than the conditions assumed on Λ in Theorems 1.3 and 1.4.
In Theorem 1.3, we have that L ( log L ) λ μ − μ + λ λ μ ( S κ − 1 ) in (iii) is subspace of L ( log L ) λ μ − μ + 1 μ λ ( S κ − 1 ) in (ii) while the range of p in (iii) is better than the range of p in (ii). Similarly from (v) and (vi), we note that B q ( 0 , λ − μ λ μ ) ( S κ − 1 ) ⊂ B q ( 0 , 1 − μ μ λ ) ( S κ − 1 ) , while the range of p in (vi) is better than that in (v).
As L ( log L ) ( S κ − 1 ) ⊂ L ( log L ) 1 ∕ λ ′ ( S κ − 1 ) and B q ( 0 , 0 ) ( S κ − 1 ) ⊂ B q ( 0 , − 1 ∕ λ ) ( S κ − 1 ) , then the spaces in Theorems 1.4 ( i ) and ( i i i ) are better than the spaces in ( i i ) and ( i v ) .

Throughout this article, the letter C stands for a positive constant that may vary at each occurrence, but it is independent of the essential variables.

2 Some lemmas

This section is devoted to giving some auxiliary lemmas that will play key roles in the proof of our main results in this article.

Let ω ≥ 2 . For suitable mappings ϕ , φ : R + → R , Λ : S κ − 1 → R and measurable g : R + → C , we define the family of measures { σ Λ , ϕ , ϕ , g , l ≔ σ g , l : l ∈ R + } and their corresponding maximal operator σ Λ , g ∗ and M Λ , g , ω on R κ + 1 by:

∫ R κ + 1 f d σ g , l = 1 l τ ∫ l ∕ 2 ≤ ∣ y ∣ ≤ l f ( ϕ ( ∣ y ∣ ) y ′ , φ ( ∣ y ∣ ) ) Λ ( y ) g ( ∣ y ∣ ) ∣ y ∣ κ − τ d y , σ Λ , g ∗ f ( x ˜ ) = sup l ∈ R + ∣ ∣ σ g , l ∣ * f ( x ˜ ) ∣ ,

and

M Λ , g , ω f ( x ˜ ) = sup j ∈ Z ∫ ω j ω j + 1 ∣ ∣ σ g , l ∣ ∗ f ( x ˜ ) ∣ d l l ,

where ∣ σ g , l ∣ is defined in the same way as σ g , l with replacing g Λ by ∣ g Λ ∣ .

The next two lemmas can be obtained by the following similar arguments (with minor modifications) as those used in [22].

Lemma 2.1

Let ω ≥ 2 , Λ ∈ L q ( S κ − 1 ) , and g ∈ ∇ λ ( R + ) for some q , λ > 1 . Suppose that ϕ is given as in Theorem 1.1 and φ is an arbitrary mapping on R + . Then, there exist constants ε and C such that

∫ ω j ω j + 1 ∣ σ ˆ g , l ( ζ , ξ ) ∣ 2 d l l ≤ C ( ln ω ) , ∫ ω j ω j + 1 ∣ σ ˆ g , l ( ζ , ξ ) ∣ 2 d l l ≤ C ( ln ω ) ∥ Λ ∥ L q ( S κ − 1 ) 2 ∥ h ∥ ∇ λ ( R + ) 2 min ∣ ζ ϕ ( ω j − 1 ) ∣ − ε ln ω , ∣ ζ ϕ ( ω j + 1 ) ∣ ε ln ω .

Lemma 2.2

Let ϕ , φ , g , and Λ be given as in Theorem 1.1. Then, for all p > λ ′ , there exists a positive constant C p such that

(2.1) ‖ M Λ , g , ω ( f ) ‖ L p ( R κ + 1 ) ≤ C p ( ln ω ) ∥ Λ ∥ L q ( S κ − 1 ) ∥ g ∥ ∇ λ ( R + ) ‖ f ‖ L p ( R κ + 1 )

and

(2.2) ‖ σ Λ , g ∗ ( f ) ‖ L p ( R κ + 1 ) ≤ C p ( ln ω ) 1 ∕ λ ′ ∥ Λ ∥ L q ( S κ − 1 ) ∥ g ∥ ∇ λ ( R + ) ‖ f ‖ L p ( R κ + 1 )

hold for all f ∈ L p ( R κ + 1 ) .

By using similar arguments as that used in [25], we obtain the following:

Lemma 2.3

Let ω ≥ 2 , and let Λ , ϕ , φ , and μ be given as in Theorem 1.1. Assume that g ∈ ∇ λ ( R + ) with 2 < λ < ∞ . Then, there exists a constant C p , Λ , g such that

(a) If μ > λ ′ , we have

∑ j ∈ Z ∫ ω j ω j + 1 ∣ σ g , l ∗ A j ∣ μ d l l 1 μ L p ( R κ + 1 ) ≤ C p , Λ , g ( ln ω ) 1 ∕ λ ′ ∑ j ∈ Z ∣ A j ∣ μ 1 ∕ μ L p ( R κ + 1 ) for λ ′ < p < ∞ ,

(b) If μ ≤ λ ′ , we have

∑ j ∈ Z ∫ ω j ω j + 1 ∣ σ g , l ∗ A j ∣ μ d l l 1 μ L p ( R κ + 1 ) ≤ C p , Λ , g ( ln ω ) ∑ j ∈ Z ∣ A j ∣ μ 1 ∕ μ L p ( R κ + 1 ) for 1 < p < μ ,

where { A j ( ⋅ ) , j ∈ Z } is an arbitrary sequence of functions on R κ + 1 .

Proof

Let us first consider the case λ ′ μ ′ . By (2.2), it is easy to see that

∥ sup j ∈ Z sup l ∈ [ 1 , ω ] ∣ σ g , ω j l ∗ A j ∣ ∥ L p ( R κ + 1 ) ≤ ∥ σ Λ , g ∗ ( sup j ∈ Z ∣ A j ∣ ) ∥ L p ( R κ + 1 ) ≤ C p , Λ , g ln ( ω ) 1 ∕ λ ′ ∥ sup j ∈ Z ∣ A j ∣ ∥ L p ( R κ + 1 ) ,

and hence, we have

(2.3) ‖ σ g , ω j l ∗ A j ‖ L ∞ ( [ 1 , ω ] , d l l ) l ∞ ( Z ) L p ( R κ + 1 ) ≤ C p , Λ , g ln ( ω ) 1 ∕ λ ′ ∥ ∥ A j ∥ l ∞ ( Z ) ∥ L p ( R κ + 1 ) .

As p > λ ′ , then by duality, there exists Ω ∈ L ( p ∕ λ ′ ) ′ ( R κ + 1 ) satisfying ∥ Ω ∥ L ( p ∕ λ ′ ) ′ ( R κ ) ≤ 1 and

(2.4) ∑ j ∈ Z ∫ 1 ω ∣ σ g , ω j l ∗ A j ∣ λ ′ d l l 1 λ ′ L p ( R κ + 1 ) λ ′ = ∫ R κ + 1 ∑ j ∈ Z ∫ 1 ω ∣ σ g , ω j l ∗ A j ( y ˜ ) ∣ λ ′ d l l Ω ( y , y κ + 1 ) d y d y κ + 1 ≤ C ∥ Λ ∥ L 1 ( S κ − 1 ) ( λ ′ ∕ λ ) ∥ g ∥ ∇ λ ( R + ) λ ′ ∫ R κ + 1 ∑ j ∈ Z ∣ A j ( y , y κ + 1 ) ∣ λ ′ σ Λ , 1 ∗ Ω ¯ ( − y , − y κ + 1 ) d y d y κ + 1 ≤ C ∥ Λ ∥ L 1 ( S κ − 1 ) ( λ ′ ∕ λ ) ∥ g ∥ ∇ λ ( R + ) λ ′ ∑ j ∈ Z ∣ A j ∣ λ ′ L ( p ∕ λ ′ ) ( R κ + 1 ) ∥ σ Λ , 1 ∗ ( Ω ¯ ) ∥ L ( p ∕ λ ′ ) ′ ( R κ ) ≤ C ln ( ω ) ∥ Λ ∥ L q ( S κ − 1 ) ( λ ′ ∕ λ ) + 1 ∥ g ∥ ∇ λ ( R + ) λ ′ ∑ j ∈ Z ∣ A j ∣ λ ′ 1 λ ′ L p ( R κ + 1 ) λ ′ ,

where Ω ¯ ( y , y κ + 1 ) = Ω ( − y , − y κ + 1 ) . Hence,

(2.5) ∑ j ∈ Z ∫ 1 ω ∣ σ g , ω j l ∗ A j ∣ λ ′ d l l 1 λ ′ L p ( R κ + 1 ) ≤ C p , Λ , g ln ( ω ) 1 ∕ λ ′ ∑ j ∈ Z ∣ A j ∣ λ ′ 1 λ ′ L p ( R κ + 1 ) .

Let T be the linear operator defined on any function A = A j ( y , y κ + 1 ) by T ( A ) = σ g , ω j l ∗ A j ( y , y κ + 1 ) . By interpolation between (2.3) and (2.5), we obtain

(2.6) ∑ j ∈ Z ∫ ω j ω j + 1 ∣ σ g , l ∗ A j ∣ μ d l l 1 μ L p ( R κ + 1 ) ≤ ∑ j ∈ Z ∫ 1 ω ∣ σ g , ω j l ∗ A j ∣ μ d l l 1 ∕ μ L p ( R κ + 1 ) ≤ C p , Λ , g ln ( ω ) 1 ∕ λ ′ ∑ j ∈ Z ∣ A j ∣ μ ′ 1 μ ′ L p ( R κ + 1 )

for all λ ′ 2 and μ > λ ′ . Now, let us consider the case 1 < p < μ with μ ≤ λ ′ . By duality, there are functions θ j ( y ˜ , l ) defined on R κ + 1 × R + such that ‖ θ j ‖ L μ ′ ( [ ω j , ω j + 1 ] , d l l ) l μ ′ L p ′ ( R κ + 1 ) ≤ 1 and

(2.7) ∑ j ∈ Z ∫ ω j ω j ∣ σ g , l ∗ A j ∣ μ d l l 1 ∕ μ L p ( R κ + 1 ) = ∫ R κ + 1 ∑ j ∈ Z ∫ ω j ω j + 1 ( σ g , l ∗ A j ( x ) ) θ j ( y ˜ , l ) d l l d y ˜ ≤ C p ln ( ω ) 1 ∕ μ ∥ ( Z ( θ j ) ) 1 ∕ μ ′ ∥ L p ′ ( R κ + 1 ) ∑ j ∈ Z ∣ A j ∣ μ 1 ∕ μ L p ( R κ + 1 ) ,

where

Z ( θ j ) ( y ˜ ) = ∑ j ∈ Z ∫ ω j ω j ∣ σ g , l ∗ θ j ( y ˜ , l ) ∣ μ ′ d l l .

Since μ ≤ λ ′ ≤ 2 ≤ λ , by Hölder’s inequality, we have that

(2.8) ∣ σ g , l ∗ θ j ( y ˜ , l ) ∣ μ ′ ≤ C ∥ Λ ∥ L 1 ( S κ − 1 ) ( μ ′ ∕ μ ) ∥ g ∥ ∇ λ ( R + ) μ ′ ∫ ω j ω j + 1 ∫ S κ − 1 ∣ Λ ( υ ) ∣ ∣ θ j ( y − ϕ ( l ) υ , y κ + 1 − φ ( l ) , l ) ∣ μ ′ d σ ( υ ) d l l .

Again, since p ′ > μ ′ , we deduce that there is a function ψ ∈ L ( p ′ ∕ μ ′ ) ′ ( R κ + 1 ) that satisfies

∥ ( Z ( θ j ) ) 1 ∕ μ ′ ∥ L p ′ ( R κ + 1 ) μ ′ = ∑ j ∈ Z ∫ R κ + 1 ∫ ω j ω j + 1 ∣ σ g , l ∗ θ j ( y ˜ , l ) ∣ μ ′ d l l ψ ( y ˜ ) d y ˜ .

Therefore, by a simple change of variables together with Hölder’s inequality and Inequalities (2.2) and (2.8), we end with

(2.9) ∥ ( Z ( θ j ) ) 1 ∕ μ ′ ∥ L p ′ ( R κ + 1 ) μ ′ ≤ C ∥ g ∥ ∇ λ ( R + ) μ ′ ∥ Λ ∥ L 1 ( S κ − 1 ) ( μ ′ ∕ μ ) ∥ σ ∗ ∣ Λ ∣ , 1 ( ψ ) ∥ L ( p ′ ∕ μ ′ ) ′ ( R κ + 1 ) ∑ j ∈ Z ∫ ω j ω j + 1 ∣ θ j ( ⋅ , l ) ∣ μ ′ d l l L ( p ′ ∕ μ ′ ) ( R κ + 1 ) ≤ C p ln ( ω ) ∥ Λ ∥ L q ( S μ − 1 ) ( μ ′ ∕ μ ) + 1 ∥ g ∥ ∇ λ ( R + ) μ ′ ∥ ψ ∥ L ( p ′ ∕ μ ′ ) ′ ( R κ + 1 ) .

Consequently, by (2.7) and (2.9), Lemma 2.3 is proved for any 1 < p < μ with μ ≤ λ ′ < 2 .□

Lemma 2.4

Let g ∈ ∇ λ ( R + ) and Λ ∈ L q ( S κ − 1 ) for some λ , q ∈ ( 1 , 2 ] . Suppose that ω ≥ 2 and ϕ , φ are given as in Theorem 1.2. Then, there exists a positive constant C p , Λ , g such that for any sequence of functions { A j ( ⋅ ) , j ∈ Z } on R κ + 1 , we have

(2.10) ∑ j ∈ Z ∫ ω j ω j + 1 ∣ σ g , l ∗ A j ∣ μ d l l 1 μ L p ( R κ + 1 ) ≤ C p , Λ , g ( ln ω ) 1 ∕ μ ∑ j ∈ Z ∣ A j ∣ μ 1 ∕ μ L p ( R κ + 1 )

for all p ∈ [ μ , λ μ ′ μ ′ − λ ] ,

(2.11) ∑ j ∈ Z ∫ ω j ω j + 1 ∣ σ g , l ∗ A j ∣ μ d l l 1 μ L p ( R κ + 1 ) ≤ C p , Λ , g ( ln ω ) λ μ − μ + λ λ μ ∑ j ∈ Z ∣ A j ∣ μ 1 ∕ μ L p ( R κ + 1 )

for all p ∈ ( λ μ λ μ − μ + λ , μ ) , and

(2.12) ∑ j ∈ Z ∫ ω j ω j + 1 ∣ σ g , l ∗ A j ∣ μ d l l 1 μ L p ( R κ + 1 ) ≤ C p , Λ , g ( ln ω ) λ μ − μ + 1 μ λ ∑ j ∈ Z ∣ A j ∣ μ 1 ∕ μ L p ( R κ + 1 )

for all p ∈ μ λ λ μ − μ + 1 , μ .

Proof

To prove (2.10), we follow the same ideas as those used in the proof of Theorem 3.7, which can be traced all the way back to [31]. Let us consider the case p > μ . By duality, there exists a non-negative function b ∈ L ( p ∕ μ ) ′ ( R κ + 1 ) such that ∥ b ∥ L ( p ∕ μ ) ′ ( R κ + 1 ) ≤ 1 and

(2.13) ∑ j ∈ Z ∫ ω j ω j + 1 ∣ σ g , l ∗ A j ∣ μ d l l 1 ∕ μ L p ( R κ + 1 ) μ = ∫ R κ + 1 ∑ j ∈ Z ∫ ω j ω j + 1 ∣ σ g , l ∗ A j ( y ˜ ) ∣ μ d l l b ( y ˜ ) d y ˜ .

Thanks to Hölder’s inequality, we have

(2.14) ∣ σ g , l ∗ A j ( y ˜ ) ∣ μ ≤ C ∥ g ∥ ∇ λ ( R + ) ( μ ∕ μ ′ ) ∥ Λ ∥ L 1 ( S κ − 1 ) ( μ ∕ μ ′ ) ∫ 1 2 l l ∫ S κ − 1 ∣ A j ( x − ϕ ( t ) υ , x κ + 1 − φ ( t ) ) ∣ μ ∣ Λ ( υ ) ∣ ∣ g ( t ) ∣ μ − μ λ μ ′ d σ ( υ ) d t t .

Thus, by a simple change of variables and Hölder’s inequality together with Lemma 2.2, we obtain

∑ j ∈ Z ∫ ω j ω j + 1 ∣ σ g , l ∗ A j ∣ μ d l l 1 ∕ μ L p ( R κ + 1 ) μ ≤ C ∥ g ∥ ∇ λ ( R + ) ( μ ∕ μ ′ ) ∥ Λ ∥ L 1 ( S κ − 1 ) ( μ ∕ μ ′ ) ∫ R κ + 1 ∑ j ∈ Z ∣ A j ( y ˜ ) ∣ μ M ∣ Λ ∣ , ∣ g ∣ μ − μ λ μ ′ , ω b ¯ ( − y ˜ ) d y ˜ ≤ C ∥ g ∥ ∇ λ ( R + ) ( μ ∕ μ ′ ) ∥ Λ ∥ L 1 ( S κ − 1 ) ( μ ∕ μ ′ ) ∑ j ∈ Z ∣ A j ∣ μ L ( p ∕ μ ) ( R κ + 1 ) M ∣ Λ ∣ , ∣ g ∣ μ ( μ ′ − λ ) μ ′ ( b ¯ ) L ( p ∕ μ ) ′ ( R κ + 1 ) ≤ C ln ( ω ) ∥ g ∥ ∇ λ ( R + ) ( μ ∕ μ ′ ) + 1 ∥ Λ ∥ L q ( S κ − 1 ) ( μ ∕ μ ′ ) + 1 ∑ j ∈ Z ∣ A j ∣ μ L ( p ∕ μ ) ( R κ + 1 ) ∥ b ¯ ∥ L ( p ∕ μ ) ′ ( R κ + 1 ) ,

where b ¯ ( − y ˜ ) = b ( y ˜ ) . The last inequality is true since ∣ g ∣ μ ( μ ′ − λ ) μ ′ ∈ ∇ λ μ ′ μ ( μ ′ − λ ) ( R + ) . Consequently, (2.10) holds for all μ < p ≤ λ μ ′ μ ′ − λ . For the case p = μ , by Hölder’s inequality, and (2.14) and (2.1), we obtain

(2.15) ∑ j ∈ Z ∫ ω j ω j + 1 ∣ σ g , l ∗ A j ∣ μ d l l 1 μ L p ( R κ + 1 ) μ ≤ C ∥ g ∥ ∇ λ ( R + ) ( μ ∕ μ ′ ) ∥ Λ ∥ L 1 ( S κ − 1 ) ( μ ∕ μ ′ ) ∑ j ∈ Z ∫ R κ + 1 ∫ ω j ω j + 1 ∫ 1 2 l l ∫ S κ − 1 ∣ A j ( x − ϕ ( t ) υ , x κ + 1 − φ ( t ) ) ∣ μ ∣ Λ ( υ ) ∣ ∣ g ( t ) ∣ μ − μ λ μ ′ d σ ( υ ) d t t d l l d x ˜ ≤ C ( ln ω ) ∥ g ∥ ∇ λ ( R + ) ( μ ∕ μ ′ ) + 1 ∥ Λ ∥ L 1 ( S κ − 1 ) ( μ ∕ μ ′ ) + 1 ∫ R κ + 1 ∑ j ∈ Z ∣ A j ( x ˜ ) ∣ μ d x ˜ p ∕ μ .

Now, let us consider the case λ μ λ μ − μ + λ < p < μ . Hence, we have μ ′ < p ′ . By duality, there is a set of functions { h j ( y ˜ , l ) } defined on R κ + 1 × R + such that ‖ h j ‖ L μ ′ ( [ ω j , ω j + 1 ] , d l l ) l μ ′ L p ′ ( R κ + 1 ) ≤ 1 and

(2.16) ∑ j ∈ Z ∫ ω j ω j ∣ σ g , l ∗ A j ∣ μ d l l 1 ∕ μ L p ( R κ + 1 ) = ∫ R κ + 1 ∑ j ∈ Z ∫ ω j ω j + 1 ( σ g , l ∗ A j ( x ) ) h j ( y ˜ , l ) d l l d y ˜ .

Let ℋ ( h j ) be given by:

ℋ ( h j ) ( y ˜ ) = ∑ j ∈ Z ∫ ω j ω j ∣ σ g , l ∗ h j ( y ˜ , l ) ∣ μ ′ d l l .

Thus, by duality, there exists a function ρ ∈ L ( p ′ ∕ μ ′ ) ′ ( R κ + 1 ) with norm 1 such that

∥ ( ℋ ( h j ) ) 1 ∕ μ ′ ∥ L p ′ ( R κ + 1 ) μ ′ = ∑ j ∈ Z ∫ R κ + 1 ∫ ω j ω j + 1 ∣ σ g , l ∗ h j ( y ˜ , l ) ∣ μ ′ ρ ( y ˜ ) d l l d y ˜ ≤ C ∥ g ∥ ∇ λ ( R + ) ( μ ′ ∕ μ ) ∥ Λ ∥ L 1 ( S κ − 1 ) ( μ ′ ∕ μ ) σ ∗ ∣ Λ ∣ , ∣ g ∣ μ ′ ( μ − λ ) μ ( ρ ) L ( p ′ ∕ μ ′ ) ′ ( R κ + 1 ) ∑ j ∈ Z ∫ ω j ω j + 1 ∣ h j ( ⋅ , l ) ∣ μ ′ d l l L ( p ′ ∕ μ ′ ) ( R κ + 1 ) ≤ C p ( ln ω ) 1 ∕ μ λ μ − λ ′ ∥ Λ ∥ L q ( S κ − 1 ) ( μ ′ ∕ μ ) + 1 ∥ g ∥ ∇ λ ( R + ) ( μ ′ ∕ μ ) + 1 ∥ ρ ∥ L ( p ′ ∕ μ ′ ) ′ ( R κ + 1 )

for all μ > p > μ λ μ − λ ′ . Consequently, by Hölder’s inequality and (2.16), we obtain

(2.17) ∑ j ∈ Z ∫ ω j ω j ∣ σ g , l ∗ A j ∣ μ d l l 1 ∕ μ L p ( R κ + 1 ) ≤ C p ( ln ω ) λ μ − μ + λ λ μ ∥ ( ℋ ( h j ) ) 1 ∕ μ ′ ∥ L p ′ ( R κ + 1 ) ∑ j ∈ Z ∣ A j ∣ μ 1 ∕ μ L p ( R κ + 1 ) ≤ C p ( ln ω ) λ μ − μ + λ λ μ ∑ j ∈ Z ∣ A j ∣ μ 1 ∕ μ L p ( R κ + 1 ) ,

holds for all p ∈ ( λ μ λ μ − μ + λ , μ ) .

Now, it is left to prove (2.12). To this end, let T be the linear operator T defined as in the proof of Lemma 2.3. It is easy to see that the following inequality holds:

(2.18) ∥ T ( A ) ∥ L 1 ( 1 , ω ) , d l l l 1 ( Z ) L 1 ( R κ + 1 ) ≤ C ln ( ω ) ∑ j ∈ Z ∣ A j ∣ L 1 ( R κ + 1 ) .

By interpolating between (2.18) and (2.3), we obtain (2.12). The proof of Lemma 2.4 is complete.□

3 Proof of the main results

Proof of Theorem 1.1

We shall use similar arguments as those used in [25] and [32]. Suppose that g ∈ ∇ λ ( R + ) with 1 < λ ≤ 2 , Λ ∈ L q ( S κ − 1 ) with 1 < q ≤ 2 and ϕ , φ are C 2 ( R + ) , convex and increasing functions with ϕ ( 0 ) = 0 = φ ( 0 ) . It is clear that by Minkowski’s inequality,

(3.1) M Λ , ϕ , φ , g ( μ ) ( f ) ( x ˜ ) ≤ ∑ j = 0 ∞ ∫ R + 1 l τ ∫ 2 − j − 1 l < ∣ y ∣ ≤ 2 − j l f ( x − ϕ ( ∣ y ∣ ) y ′ , x κ + 1 − φ ( ∣ y ∣ ) ) K Λ , g ( y ) d y μ d l l 1 ∕ μ = 2 τ 1 2 τ 1 − 1 ∫ R + ∣ σ g , l ∗ f ( x ˜ ) ∣ μ d l l 1 ∕ μ .

Let ω = 2 λ ′ q ′ . Hence, we have ln ( ω ) ≤ ln ( 8 ) ( λ − 1 ) ( q − 1 ) . For j ∈ Z , let { Φ j } − ∞ ∞ be a partition of unity in C ∞ ( 0 , ∞ ) such that

0 ≤ Φ j ≤ 1 , ∑ j Φ j ( l ) = 1 , supp Φ j ⊆ [ ϕ ( ω − j − 1 ) , ϕ ( ω − j + 1 ) ] ≡ I j , ω , and d m Φ j ( l ) d l m ≤ C m l m .

Define the multiplier operator Ψ j f ^ ( ζ , ξ ) = Φ j ( ∣ ζ ∣ ) f ^ ( ζ , ξ ) , where ( ζ , ξ ) ∈ R κ × R . Thus, for any f ∈ S ( R κ + 1 ) , we have

(3.2) M Λ , ϕ , φ , g ( κ ) ( f ) ≤ 2 τ 1 2 τ 1 − 1 ∑ j ∈ Z G Λ , ϕ , φ , g , j , μ ( f ) ,

where

G Λ , ϕ , φ , g , j , μ ( f ) ( x ˜ ) = ∫ R + ∣ U Λ , ϕ , φ , g , j , ω ( x ˜ , l ) ∣ μ d l l 1 ∕ μ , U Λ , ϕ , φ , g , j , ω ( x ˜ , l ) = ∑ m ∈ Z ( Φ m + j ∗ σ g , l ∗ f ) ( x ˜ ) χ [ ω m , ω m + 1 ) ( l ) .

Hence, to prove Theorem 1.1, it suffices to prove that there exists ε ∈ ( 0 , 1 ) such that

(3.3) ∥ G Λ , ϕ , φ , g , j , μ ( f ) ∥ L p ( R κ + 1 ) ≤ C p , Λ , g 2 − ε ∣ j ∣ 1 ( q − 1 ) ( λ − 1 ) 1 ∕ μ ∥ f ∥ F . p 0 , μ ( R κ + 1 )

for all p ∈ [ μ , λ μ ′ μ ′ − λ ] ,

(3.4) ∥ G Λ , ϕ , φ , g , j , μ ( f ) ∥ L p ( R κ + 1 ) ≤ C p , Λ , g 2 − ε ∣ j ∣ 1 ( q − 1 ) ( λ − 1 ) λ μ − μ + λ λ μ ∥ f ∥ F . p 0 , μ ( R κ + 1 )

for all p ∈ λ μ λ μ − μ + λ , μ , and

(3.5) ∥ G Λ , ϕ , φ , g , j , μ ( f ) ∥ L p ( R κ + 1 ) ≤ C p , Λ , g 2 − ε ∣ j ∣ 1 ( q − 1 ) ( λ − 1 ) λ μ − μ + 1 λ μ ∥ f ∥ F . p 0 , μ ( R κ + 1 )

for all p ∈ λ μ λ μ − μ + 1 , μ .

Let us start proving (3.3). Consider the case p = μ = 2 . This gives that ∥ f ∥ F . 2 0 , 2 ( R κ + 1 ) = ∥ f ∥ L 2 ( R κ + 1 ) . Thus, by Plancherel’s theorem and Lemma 2.1, we obtain

∥ G Λ , ϕ , φ , g , j , 2 ( f ) ∥ L 2 ( R κ + 1 ) 2 ≤ ∑ m ∈ Z ∫ D m + j , ω ∫ ω m ω m + 1 ∣ σ ˆ g , l ( ζ , ξ ) ∣ 2 d l l ∣ f ˆ ( ζ , ξ ) ∣ 2 d ζ d ξ ≤ C 2 , q , λ 2 ( ln ω ) ∑ m ∈ Z ∫ D m + j , ω min ∣ ζ ϕ ( ω j − 1 ) ∣ − ε ln ω , ∣ ζ ϕ ( ω j + 1 ) ∣ ε ln ω ∣ f ˆ ( ζ , ξ ) ∣ 2 d ζ d ξ ≤ C 2 , q , λ 2 ( ln ω ) 2 − ε ∣ j ∣ ∑ m ∈ Z ∫ D m + j , ω ∣ f ˆ ( ζ , ξ ) ∣ 2 d ζ d ξ ≤ C 2 , q , λ 2 ( ln ω ) 2 − ε ∣ j ∣ ∥ f ∥ L 2 ( R κ + 1 ) 2 ,

where D m , ω = { ( ζ , ξ ) ∈ R κ × R : ∣ ( ζ , ξ ) ∣ ∈ I m , ω } and 0 < ε < 1 . Therefore, we have

(3.6) ∥ G Λ , ϕ , φ , g , j , 2 ( f ) ∥ L 2 ( R κ + 1 ) ≤ C 2 , q , λ 2 − ε 2 ∣ j ∣ ( λ − 1 ) − 1 ∕ 2 ( q − 1 ) − 1 ∕ 2 ∥ f ∥ F . 0 2 , 2 ( R κ + 1 ) .

On the other hand, by Lemma 2.4 and invoking Lemma 2.1 in [25], we obtain

(3.7) ∥ G Λ , ϕ , φ , g , j , μ ( f ) ∥ L p ( R κ + 1 ) ≤ C p , Λ , g [ ( q − 1 ) ( λ − 1 ) ] − 1 ∕ μ ∥ f ∥ F . p 0 , μ ( R κ + 1 )

for μ ≤ p ≤ λ μ ′ μ ′ − λ ,

(3.8) ∥ G Λ , ϕ , φ , g , j , μ ( f ) ∥ L p ( R κ + 1 ) ≤ C p , Λ , g [ ( q − 1 ) ( λ − 1 ) ] μ − λ μ − λ λ μ ∥ f ∥ F . p 0 , μ ( R κ + 1 )

for λ μ λ μ − μ + λ < p < μ , and

(3.9) ∥ G Λ , ϕ , φ , g , j , μ ( f ) ∥ L p ( R κ + 1 ) ≤ C p , Λ , g [ ( q − 1 ) ( λ − 1 ) ] μ − λ μ − 1 λ μ ∥ f ∥ F . p 0 , μ ( R κ + 1 )

for λ μ μ λ − μ + 1 < p < μ . By interpolating between (3.6) and (3.7)–(3.9), we obtain (3.3)–(3.5).□

Proof of Theorem 1.2

We can prove this theorem by following a similar argument as in the proof of Theorem 1.1, but we need to invoke Lemma 2.3 instead of Lemma 2.4 and we need to choose ω = 2 q ′ instead of ω = 2 λ ′ q ′ .□

4 Conclusion

In this work, we established certain L p estimates for rough generalized Marcinkiewicz integrals along submanifolds. By using these estimates together with Yano’s extrapolation argument, we proved the boundedness of the generalized Marcinkiewicz integrals under very weak conditions on the kernel functions. Our results improve and extend several known results on Marcinkiewicz integrals.

Acknowledgements

The authors would like to express their gratitude to the referees for their valuable comments and suggestions in improving writing this article. In addition, Open Access funding was provided by the Qatar National Library.

Funding information: Open Access funding was provided by the Qatar National Library.
Author contributions: Formal analysis and writing-original draft preparation: M.A. and H.A.-Q. All authors have read and agreed to the published version of the manuscript.
Conflict of interest: The authors declare that they have no conflict of interest.
Data availability statement: No data were used to support this study.

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Received: 2022-12-25

Revised: 2023-05-20

Accepted: 2023-06-14

Published Online: 2023-07-21

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

Articles in the same Issue

https://doi.org/10.1515/math-2022-0603

Keywords for this article

generalized Marcinkiewicz; Triebel-Lizorkin space; rough kernels; extrapolation

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