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A boundedness result for Marcinkiewicz integral operator

  • Laith Hawawsheh EMAIL logo and Mohammad Abudayah
Published/Copyright: July 29, 2020
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Open Mathematics
From the journal Open Mathematics Volume 18 Issue 1

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

(2.4) ∥ σ ∗ ( u ) ∥ q γ ′ ≤ C ∫ ℝ n ∫ 1 / 2 ∞ ∫ S n − 1 | u ( x − r y ′ ) | γ ′ | Ω ( y ′ ) | d σ ( y ′ ) d r r 2 q / γ ′ d x γ ′ / q ≤ C ∫ 1 / 2 ∞ ∫ ℝ n ∫ S n − 1 | u ( x − r y ′ ) | γ ′ | Ω ( y ′ ) | d σ ( y ′ ) q / γ ′ d x γ ′ / q d r r 2 ≤ C ∫ 1 / 2 ∞ ∫ S n − 1 ∫ ℝ n | u ( x − r y ′ ) | q d x γ ′ / q | Ω ( y ′ ) | d σ ( y ′ ) d r r 2 ≤ C ∥ u ∥ q γ ′ .

□

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

© 2020 Laith Hawawsheh and Mohammad Abudayah, published by De Gruyter

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

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  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
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  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
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  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
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  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
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  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
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  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
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  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
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  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
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  102. Estimates of entropy numbers in probabilistic setting
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  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
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  123. Special Issue on Graph Theory (GWGT 2019), Part II
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  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
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  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
https://doi.org/10.1515/math-2020-0046
Keywords for this article
Marcinkiewicz integral operator; L p -bounded integral operator; maximal integral operator
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  152. Three classes of decomposable distributions
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