A boundedness result for Marcinkiewicz integral operator

Laith Hawawsheh; Mohammad Abudayah

doi:10.1515/math-2020-0046

Open Access Published by De Gruyter Open Access July 29, 2020

A boundedness result for Marcinkiewicz integral operator

Laith Hawawsheh and Mohammad Abudayah

From the journal Open Mathematics

https://doi.org/10.1515/math-2020-0046

Abstract

We extend a boundedness result for Marcinkiewicz integral operator. We find a new space of radial functions for which this class of singular integral operators remains L p -bounded when its kernel satisfies only the sole integrability condition.

Keywords: Marcinkiewicz integral operator; L p -bounded integral operator; maximal integral operator

MSC 2010: 42B15; 42B20; 42B25

1 Introduction

For n ≥ 2 , let ℝ n be the n-dimensional Euclidean space and S n − 1 denote the unit sphere in ℝ n equipped with the normalized surface measure d σ . For any nonzero y ∈ ℝ n , let y ′ be the projection of y on S n − 1 , that is, y ′ = y | y | . For α > 0 , the Zygmund class L ( log L ) α ( S n − 1 ) consists of all measurable functions Ω on S n − 1 which satisfy

∫ S n − 1 | Ω ( x ) | log α ( e + | Ω ( x ) | ) d σ ( x ) < ∞ .

The following inclusions among Zygmund classes and Lebesgue spaces hold and are proper. For q > 1 and 0 < α < β , we have

L q ( S n − 1 ) ⊂ L ( log L ) ( S n − 1 ) ⊂ L 1 ( S n − 1 ) , L ( log L ) β ( S n − 1 ) ⊂ L ( log L ) α ( S n − 1 ) .

Let Ω ∈ L 1 ( S n − 1 ) be a homogeneous function of degree zero and satisfy

(1.1) ∫ S n − 1 Ω ( y ) d σ ( y ) = 0 .

Condition (1.1) is referred to as the cancellation property. It was shown that Ω must satisfy (1.1) for the existence of the singular integrals in our work, see [9]. For ρ > 0 and a measurable radial function h, consider the integral operator

ℳ Ω h ρ f ( x ) = ∫ 0 ∞ t − ρ ∫ | y | ≤ t f ( x − y ) K Ω, h ( y ) d y 2 d t t 1 / 2 ,

where

K Ω , h , ρ ( y ) = Ω ( y ) | y | n − ρ h ( | y | ) .

When n = 1 , ρ = 1 , h ≡ 1 and Ω ( y ) = sign ( y ) , we denote ℳ Ω , h ρ by ℳ . In 1938, Marcinkiewicz introduced the operator ℳ to give an analogue of certain Littlewood-Paley g-functions [7]. In [8], Stein introduced the integral operator ℳ Ω, 1 1 as a higher dimensional version of ℳ . Since then there was much work on the L p -boundedness of the operator ℳ Ω , 1 1 whenever Ω belongs to a variety of function spaces. For instance, Stein proved that ℳ Ω , 1 1 is bounded on L p ( ℝ n ) for all 1 < p ≤ 2 whenever Ω ∈ Lip α ( S n − 1 ) , 0 < α ≤ 1 . In [10], Walsh proved the boundedness of ℳ Ω , 1 1 when p = 2 and Ω ∈ L ( log L ) 1 / 2 ( S d − 1 ) . Moreover, he showed that we may not attain the boundedness of ℳ Ω , 1 1 on L 2 if we reduce the power 1/2 by a smaller one in the Zygmund class L ( log L ) 1 / 2 ( S d − 1 ) . In a recent study, Al-Salman et al. [3] proved that ℳ Ω , 1 1 is bounded for p ∈ ( 1 , ∞ ) and Ω ∈ L ( log L ) 1 / 2 ( S d − 1 ) . We refer the readers to [1,2,3,4,11,12,13,14] for more background information and related results.

In the following, we discuss some results that are relevant to our work. For 1 ≤ γ < ∞ , let

l ∞ ( L γ ) ( ℝ + ) = h : ℝ n → ℂ : ∥ h ∥ l ∞ ( L γ ) ( ℝ + ) ≔ sup j ∈ Z ∫ 2 j − 1 2 j | h ( r ) | γ d r r 1 / γ < ∞ .

If γ = ∞ , we set

l ∞ ( L γ ) ( ℝ + ) = L ∞ ( ℝ + ) .

For 1 < γ < β < ∞ and t > 0 , the spaces l ∞ ( L γ ) ( ℝ + ) have the following interesting properties:

(1.2) L ∞ ( ℝ + ) ⊂ l ∞ ( L β ) ( ℝ + ) ⊂ l ∞ ( L γ ) ( ℝ + ) ⊂ l ∞ ( L 1 ) ( ℝ + ) ,

(1.3) ∫ 2 j − 1 2 j | h ( t r ) | γ d r r 1 / γ = ∫ 2 j − 1 t 2 j t | h ( r ) | γ d r r 1 / γ ≤ 2 ∥ h ∥ l ∞ ( L γ ) ( ℝ + ) .

Ding, Lu and Yabuta [4] studied the L 2 -boundedness of ℳ Ω , h ρ when h ∈ l ∞ ( L γ ) ( ℝ + ) and proved that ℳ Ω , h ρ is bounded on L 2 ( ℝ n ) for Ω ∈ L ( log L ) ( S n − 1 ) and h ∈ l ∞ ( L γ ) ( ℝ + ) , 1 < γ ≤ ∞ . In 2007, Al-Salman and Al-Qassem [2] improved this result and proved that ℳ Ω , h ρ is bounded on L p ( ℝ n ) for 1 < p < ∞ . In a very recent work, Hawawsheh et al. [5] introduced a new subspace D γ ( ℝ + ) of l ∞ ( L γ ) ( ℝ + ) . More precisely, D γ ( ℝ + ) consists of all measurable radial functions on ℝ n , which satisfy the following conditions:

(1.4) h ( r ) r 1 / γ ′ ∈ l ∞ ( L γ ) ( ℝ + ) ,

(1.5) ∑ j = 1 ∞ ∫ 2 j − 1 2 j | h ( r ) | γ d r r 1 / γ < ∞ .

The space D γ ( ℝ + ) has many remarkable properties. For instance, D γ ⊄ L ∞ ( ℝ + ) and D β ( ℝ + ) ⊂ D γ ( ℝ + ) for 1 < γ < β < ∞ . Hawawsheh and Al-Salman [6] proved that taking h ∈ D γ ( ℝ + ) allows ℳ Ω , h ρ to be bounded whenever Ω ∈ L 1 ( S n − 1 ) , which is known as the sole integrability condition. In fact, they obtained the following result.

Theorem 1.1

If h ∈ D γ for some γ > 1 and Ω ∈ L 1 ( S n − 1 ) is a homogeneous function of degree zero on ℝ n , then ℳ Ω, h ρ is bounded on L p ( ℝ n ) for 2 γ 2 γ − 1 < p < ∞ .

In view of Theorem 1.1, the natural question that arises is: what is the largest subspace of l ∞ ( L γ ) ( ℝ + ) for which the result of Theorem 1.1 holds? In this work, we aim to improve the result of the previous theorem by finding a subspace which contains D γ .

2 Lemmas

This section prepares for the proof of our main result by using a variation of the technique that was used to prove Theorem 1.1.

Let D γ ∗ ( ℝ + ) consist of all measurable radial functions on ℝ n , which satisfy the following conditions:

(2.1) r 1 / γ ′ h ( r ) ∈ l ∞ ( L γ ) ( ℝ + ) ,

(2.2) ∑ j = 0 ∞ ∫ 2 − j − 1 2 − j | h ( r ) | γ d r r 1 / γ < ∞ .

It can be readily seen that D γ ∗ ( ℝ + ) ⊂ l ∞ ( L γ ) ( ℝ + ) and D γ ∗ ( ℝ + ) ⊄ D γ ( ℝ + ) . In fact, one can consider the function h ( r ) = 1 r 1 / 2 γ ′ ( 0 , 1 ) ( r ) .

Lemma 2.1

Let h ∈ D γ ∗ ( ℝ + ) for some γ > 1 and Ω ∈ L 1 ( S n − 1 ) . Then,

∫ t / 2 < | y | ≤ t | K Ω , h ( y ) | d y ≤ 2 1 + 1 / γ ′ ( log 2 ) 1 / γ ′ t ρ − 1 / γ ′ ∥ Ω ∥ 1 ∥ r 1 / γ ′ h ∥ l ∞ ( L γ ) ( ℝ + ) .

Proof

By switching to polar coordinates, Hölder’s inequality and using (1.3), we have

∫ t / 2 < | y | ≤ t | K Ω, h ( y ) | d y = ∫ t / 2 t r ρ | h ( r ) | ∫ S n − 1 | Ω ( y ′ ) | d σ ( y ′ ) d r r ≤ 2 1 / γ ′ t ρ − 1 / γ ′ ∥ Ω ∥ 1 ∫ t / 2 t | r 1 / γ ′ h ( r ) | d r r ≤ 2 1 + 1 / γ ′ ( log 2 ) 1 / γ ′ t ρ − 1 / γ ′ ∥ Ω ∥ 1 ∥ r 1 / γ ′ h ∥ l ∞ ( L γ ) ( ℝ + ) .

□

Lemma 2.2

Let Ω and h be as in Lemma 2.1. If

F ( x , t ) = t − ρ ∫ t / 2 < | y | ≤ t f ( x − y ) K Ω , h ( y ) d y ,

then

∥ F ( x , t ) ∥ p ≤ 2 1 + 1 / γ ′ ( log 2 ) 1 / γ ′ t − 1 / γ ′ ∥ Ω ∥ 1 ∥ r 1 / γ ′ h ∥ l ∞ ( L γ ) ( ℝ + ) ∥ f ∥ p .

Proof

An application of Minkowski’s inequality for integrals yields

∥ F ( x , t ) ∥ p = ∫ ℝ n | F ( x , t ) | p d x 1 / p = ∫ ℝ n t − ρ ∫ t / 2 < | y | ≤ t f ( x − y ) K Ω , h ( y ) d y p d x 1 / p ≤ t − ρ ∫ t / 2 < | y | ≤ t K Ω, h ( y ) ∫ ℝ n | f ( x − y ) | p d x 1 / p d y .

Now, applying Lemma 2.1 and recalling that the Lebesgue integral is invariant under translation we have the desired result.□

Definition 2.3

For a function u defined on ℝ n , the maximal integral operator of u is defined as

σ ∗ ( u ) ( x ) = sup 1 < t < ∞ t − ρ ∫ t / 2 < | y | ≤ t | u ( x − y ) | | Ω ( y ) | | y | n − ρ | h ( | y | ) | d y .

The following lemma constitutes the boundedness of the maximal integral operator σ ∗ ( u ) .

Lemma 2.4

Let Ω and h be as in Lemma 2.1 and γ > 1 . Then,

∥ σ ∗ ( u ) ∥ q ≤ C ∥ u ∥ q

for q > γ ′ .

Proof

With the aid of Hölder’s inequality we get

(2.3) σ ∗ ( u ) ( x ) = sup 1 < t < ∞ t − ρ ∫ t / 2 < | y | ≤ t | u ( x − y ) | | Ω ( y ) | | y | n − ρ + 1 / γ ′ | | y | 1 / γ ′ h ( | y | ) | d y ≤ sup 1 < t < ∞ t − ρ ∫ t / 2 < | y | ≤ t | u ( x − y ) | γ ′ | Ω ( y ) | | y | n − ρ + 1 d y 1 / γ ′ t − ρ ∫ t / 2 < | y | ≤ t | Ω ( y ) | | y | n − ρ | | y | 1 / γ ′ h ( | y | ) | γ d y 1 / γ ≤ ∥ Ω ∥ 1 1 / γ ∥ r 1 / γ ′ ⋅ h ∥ l ∞ ( L γ ) ( ℝ + ) × sup 1 < t < ∞ t − ρ ∫ t / 2 < | y | ≤ t | u ( x − y ) | γ ′ | Ω ( y ) | | y | n − ρ + 1 d y 1 / γ ′ ≤ C sup 1 < t < ∞ ∫ t / 2 t ∫ S n − 1 | u ( x − r y ′ ) | γ ′ | Ω ( y ′ ) | d σ ( y ′ ) d r r 2 1 / γ ′ ≤ C ∫ 1 / 2 ∞ ∫ S n − 1 | u ( x − r y ′ ) | γ ′ | Ω ( y ′ ) | d σ ( y ′ ) d r r 2 1 / γ ′ .

Finally, by using (2.3) and Minkowski’s inequality for integrals we obtain

□

Now, we are in a position to study the Dyadic operators

S Ω , h ρ , j g ( x ) = ∫ 2 j 2 j + 1 t − ρ ∫ t / 2 < | y | ≤ t K Ω , h ( y ) g ( x + y , t ) d y 2 d t t 1 / 2 .

Lemma 2.5

Let 2 γ 2 γ − 1 < p < 2 . If g ∈ Z ≡ L p ′ d x , L 2 [ 2 j , 2 j + 1 ] , d t t for j = 0 , 1 , 2 , … , then

∥ S Ω , h ρ , j g ∥ p ′ 2 ≤ C 2 − j / γ ′ ∥ g ∥ Z .

Proof

Since p ′ > 2 , there exists a non-negative function u ∈ L ( p ′ / 2 ) ′ ( ℝ n ) with ∥ u ∥ L ( p ′ / 2 ) ′ ( ℝ n ) = 1 such that

(2.5) S Ω , h ρ , j g ( x ) p ′ 2 = ∫ ℝ n ∫ 2 j 2 j + 1 t − ρ ∫ t / 2 < | y | ≤ t K Ω , h ( y ) g ( x + y , t ) d y 2 d t t u ( x ) d x .

Let G ( x , t ) = t − ρ ∫ t / 2 < | y | ≤ t K Ω , h ( y ) g ( x + y , t ) d y . Then, by applying the Cauchy-Schwartz inequality we have

(2.6) G ( x , t ) 2 ≤ t − ρ ∫ t / 2 < | y | ≤ t | K Ω , h ( y ) | | g ( x + y , t ) | d y 2 ≤ t − ρ ∫ t / 2 < | y | ≤ t | K Ω , h ( y ) | d y t − ρ ∫ t / 2 < | y | ≤ t | K Ω , h ( y ) | | g ( x + y , t ) | 2 d y .

Thus, by Lemma 2.1 and (2.6) we get

(2.7) G ( x , t ) 2 ≤ C t − 1 / γ ′ t − ρ ∫ t / 2 < | y | ≤ t | K Ω , h ( y ) | | g ( x + y , t ) | 2 d y .

Now, combining (2.5) and (2.7) gives

(2.8) ∥ S Ω , h ρ , j g ( x ) ∥ p ′ 2 ≤ C 2 − j / γ ′ ∫ ℝ n ∫ 2 j 2 j + 1 ∫ t / 2 < | y | ≤ t t − ρ | K Ω , h ( y ) | | g ( x + y , t ) | 2 u ( x ) d y d t t d x = C 2 − j / γ ′ ∫ 2 j 2 j + 1 ∫ t / 2 < | y | ≤ t t − ρ | K Ω , h ( y ) | ∫ ℝ n | g ( x + y , t ) | 2 u ( x ) d x d y d t t = C 2 − j / γ ′ ∫ 2 j 2 j + 1 ∫ t / 2 < | y | ≤ t t − ρ | K Ω , h ( y ) | ∫ ℝ n | g ( x , t ) | 2 u ( x − y ) d x d y d t t = C 2 − j / γ ′ ∫ ℝ n ∫ 2 j 2 j + 1 | g ( x , t ) | 2 ∫ t / 2 < | y | ≤ t t − ρ | K Ω , h ( y ) | u ( x − y ) d y d t t d x ≤ C 2 − j / γ ′ ∫ ℝ n ∫ 2 j 2 j + 1 | g ( x , t ) | 2 d t t σ ∗ ( u ) ( x ) d x .

An application of Hölder’s inequality yields

(2.9) S Ω , h ρ , j g ( x ) p ′ 2 ≤ C ∥ g ∥ Z ∥ σ ∗ ( u ) ∥ L ( p ′ / 2 ) ′ ( ℝ n ) .

Finally, by the observation that ( p ′ / 2 ) ′ > γ ′ whenever 2 γ 2 γ − 1 < p < 2 and Lemma 2.4 we get the required estimate.□

3 Main result

Theorem 3.1

If h ∈ D γ + D γ ∗ for some γ > 1 and Ω ∈ L 1 ( S n − 1 ) is a homogeneous function of degree zero on ℝ n , then ℳ Ω , h ρ is bounded on L p ( ℝ n ) for 2 γ 2 γ − 1 < p < ∞ .

Proof

Based on the observation that the operator ℳ Ω , h ρ is sublinear in h and Theorem 1.1, we need only to prove Theorem 3.1 for h ∈ D γ ∗ . First, note that

(3.1) ℳ Ω , h ρ f ( x ) = ∫ 0 ∞ t − ρ ∫ | y | ≤ t f ( x − y ) K Ω , h ( y ) d y 2 d t t 1 / 2 = ∫ 0 ∞ ∑ k = 0 ∞ t − ρ ∫ t 2 − k − 1 < | y | ≤ t 2 k f ( x − y ) K Ω , h ( y ) d y 2 d t t 1 / 2 ≤ ∑ k = 0 ∞ ∫ 0 ∞ t − ρ ∫ t 2 − k − 1 < | y | ≤ t 2 k f ( x − y ) K Ω , h ( y ) d y 2 d t t 1 / 2 = ∑ k = 0 ∞ 2 − k ρ ∫ 0 ∞ t − ρ ∫ t / 2 < | y | ≤ t f ( x − y ) K Ω , h ( y ) d y 2 d t t 1 / 2 ≔ C ρ ∫ 0 ∞ t − ρ ∫ t / 2 < | y | ≤ t f ( x − y ) K Ω , h ( y ) d y 2 d t t 1 / 2 ≤ C ρ ℛ Ω , h ρ f ( x ) + S Ω , h ρ f ( x ) ,

where

ℛ Ω , h ρ f ( x ) = ∫ 0 1 t − ρ ∫ t / 2 < | y | ≤ t f ( x − y ) K Ω , h ( y ) d y 2 d t t 1 / 2 = ∫ 0 1 F ( x , t ) 2 d t t 1 / 2 , S Ω , h ρ f ( x ) = ∫ 1 ∞ t − ρ ∫ t / 2 < | y | ≤ t f ( x − y ) K Ω , h ( y ) d y 2 d t t 1 / 2 = ∫ 1 ∞ F ( x , t ) 2 d t t 1 / 2 .

By using the homogeneity of L p -norm and the classical Minkowski inequality we have

(3.2) ∥ ℳ Ω , h ρ f ∥ p ≤ C ρ ( ∥ ℛ Ω , h ρ f ∥ p + ∥ S Ω , h ρ f ∥ p ) .

For p ≥ 2 , by Lemma 2.2 we have

(3.3) ∥ S Ω , h ρ f ∥ p 2 = ∫ ℝ n ∫ 1 ∞ F ( x , t ) 2 d t t p / 2 d x 2 / p ≤ ∫ 1 ∞ ∥ F ( x , t ) ∥ p 2 d t t ≤ C ∥ f ∥ p 2 ∫ 1 ∞ 1 t 1 + 2 γ ′ d t = C ∥ f ∥ p 2 .

Next, we consider the case where 1 < p < 2 . By duality, there exists a function g ( x , t ) defined on ℝ n × ℝ + in Z = L p ′ d x , L 2 [ 1 , ∞ ] , d t t with ∥ g ∥ Z ≤ 1 such that

(3.4) ∥ S Ω , h ρ f ∥ p = ∫ ℝ n ∫ 1 ∞ t − ρ ∫ t / 2 < | y | ≤ t f ( x − y ) K Ω , h ( y ) d y g ( x , t ) d t t d x = ∫ 1 ∞ ∫ t / 2 < | y | ≤ t ∫ ℝ n f ( x − y ) g ( x , t ) d x t − ρ K Ω , h ( y ) d y d t t = ∫ ℝ n ∑ j = 0 ∞ ∫ 2 j 2 j + 1 t − ρ ∫ t / 2 < | y | ≤ t K Ω , h ( y ) g ( x + y , t ) d y d t t f ( x ) d x ≤ ( ln 2 ) 1 / 2 ∫ ℝ n ∑ j = 0 ∞ ∫ 2 j 2 j + 1 t − ρ ∫ t / 2 < | y | ≤ t K Ω , h ( y ) g ( x + y , t ) d y 2 d t t 1 / 2 f ( x ) d x ≔ ( ln 2 ) 1 / 2 ∫ ℝ n ∑ j = 0 ∞ S Ω , h ρ , j g ( x ) f ( x ) d x .

By Hölder’s inequality and Lemma 2.5, we get

(3.5) ∥ S Ω , h ρ f ∥ p ≤ ( ln 2 ) 1 / 2 ∥ f ∥ p ∑ j = 0 ∞ ∥ ( S Ω , h ρ , j g ( x ) ∥ p ′ ≤ C ∥ f ∥ p .

On the other hand, to prove the boundedness of the operator ℛ Ω , h ρ , consider

(3.6) ℛ Ω , h ρ f ( x ) = ∫ 0 1 t − ρ ∫ t / 2 < | y | ≤ t f ( x − y ) K Ω , h ( y ) d y 2 d t t 1 / 2 ≤ ∑ j = 0 ∞ ∫ 2 − j − 1 2 − j t − ρ ∫ t / 2 < | y | ≤ t f ( x − y ) K Ω , h ( y ) d y 2 d t t 1 / 2 ≔ ∑ j = 1 ∞ ℛ Ω , h ρ , j f ( x ) .

Therefore, by Minkowski’s inequality we have

(3.7) ∥ ℛ Ω , h ρ f ( x ) ∥ p ≤ ∑ j = 0 ∞ ∥ ℛ Ω , h ρ , j f ∥ p

for 1 < p < ∞ . Thus, we need to estimate ∥ ℛ Ω , h ρ , j f ∥ p so that the series on the right-hand side of (3.7) converges. The proof follows the same lines of the one in [6]. In fact, by using (2.2) in the proofs of Lemmas 2.4 and 2.5 we achieve our goal.□

Acknowledgment

This article is dedicated to our mentor, role model and dear friend, the late Prof. Hasan, who has never failed to show support and kindness, and share his invaluable knowledge with others. Rest in Peace Prof. Hassan Alezeh.

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Received: 2020-04-02

Revised: 2020-06-08

Accepted: 2020-06-12

Published Online: 2020-07-29

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

A boundedness result for Marcinkiewicz integral operator

Abstract

1 Introduction

Theorem 1.1

2 Lemmas

Lemma 2.1

Proof

Lemma 2.2

Proof

Definition 2.3

Lemma 2.4

Proof

Lemma 2.5

Proof

3 Main result

Theorem 3.1

Proof

Acknowledgment

References

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