Boundedness of fractional integrals on grand weighted Herz spaces with variable exponent

1.   Introduction

During the previous decade, there was a vast boom of research in the so-called variable exponent spaces because they can be used to model electrorheological fluids, image restoration, and continuum medium mechanics see, for instance, ^{[9,10,11,12,13,14,17,18,21,31,32]}. Consequently, over hundred scholars have contributed to the study of function spaces and related differential equations, so the theory of function spaces advanced quickly. For the time being, the theory of such variable exponent Lebesgue, Orlicz, Sobolev and Lorentz function spaces is widely developed, cf. ^{[3,4,7,8,19,20,29,33,34]}. The first generalization of variable exponent Herz spaces was established in ^[14]. The most general results were obtained in ^[1], where $\alpha$ was a variable. In ^[28], variable parameters were used to define continual Herz spaces, and the boundedness of sublinear operators (including the maximal function and Calderón-Zygmund singular operators)was proved in these spaces.

Boundedness of other operators, such as Riesz potential operators and the Marcinkiewicz integral, as proved in ^[22,27]. The concept of Morrey spaces $L^{ p, \lambda}$ was introduced by C. Morrey in 1938 (see ^[23]) in order to study regularity questions which appear in the calculus of variations.They describe local regularity more precisely than Lebesgue spaces and are widely used in not only harmonic analysis but also partial differential equations. In ^[22], Meskhi introduced the idea of grand Morrey spaces $L ^{r), \theta, \lambda}$ and derived the boundedness of a class of integral operators (Hardy-Littlewood maximal functions, Calderón-Zygmund singular integrals and potentials) in these spaces. Muckenhoupt ^[24] has established the theory on weights called the Muckenhoupt $A_p$ theory in the study of weighted function spaces and greatly developed real analysis. Recently, a generalization of the Muceknhoupt weights in terms of variable exponents has been studied in ^[2,5]. Weighted norm inequalities for the maximal operator on variable Lebesgue spaces were proved in ^[5]. Boundedness of the fractional integrals on variable weighted Lebesgue spaces by using the extrapolation theorem can be checked in ^[6,35].

Rafeiro et al. ^[25,26] established the idea of grand variable Herz spaces $\dot{K}^{\alpha(\cdot), p), \theta}_{q(\cdot)}(\mathbb{R}^n)$ and derived the boundedness of the sublinear operators and the Marcinkiewicz integral on $\dot{K}^{\alpha(\cdot), p), \theta}_{q(\cdot)}(\mathbb{R}^n)$. Inspired by the concept, in this article we introduce the concept of grand weighted Herz spaces with variable exponent and prove the boundedness of the fractional integral operator in these spaces. There are four sections in this article. The first section is dedicated to the introduction, and the second section contains some basic definitions and lemmas. In the third section, we introduce the concept of grand weighted Herz spaces with variable exponent, and the boundedness of the fractional integral operator on grand weighted Herz spaces with variable exponent is proved in the last section.

2.   Preliminaries

For this section, we refer to ^{[8,13,15,25,27,28,30]}.

2.1.   Lebesgue space with variable exponent

Assume that $G \subseteq \mathbb{R}^n$ is an open set, and $p(\cdot): G \rightarrow [1, \infty)$ is a real-valued measurable function. Let the following condition holds:

where

ⅰ) $p_-: = \operatorname{ess} \inf \limits _{g\in G}\; p(g)$,

ⅱ) $p_+: = \operatorname{ess} \sup \limits _{g\in G}\; p(g)$.

Lebesgue space $L^{p(\cdot)}(G)$ is the space of measurable functions $f$ on $G$ such that,

and the norm is defined as

This is the Banach function space and $p'(g) = \frac{p(g)}{p(g)-1}$ denotes the conjugate exponent of $p(g)$.

Next, we will define the space $L_{\mathrm{loc}}^{p(\cdot)}(G)$ as,

Now, to define the $\log$-condition,

where $C = C(\eta) > 0$ does not depend on $z_1, z_2$.

For the decay condition: Let $\eta_\infty \in (1, \infty)$, such that

Equation (2.4) holds for $\eta_0 \in (1, \infty)$ in the case of homogenous Herz spaces. We adopted the following notations in this paper:

(ⅰ) The Hardy-Littlewood maximal operator $M$ for $f \in L_{\mathrm{loc}}^1(G)$ is defined as

where $D(g, t): = \{y \in G : |g-y| < t\}$.

(ⅱ) The set $\mathcal{P}(G)$ is the collection of all $p(\cdot)$ satisfying $p_{-} > 1$ and $p_+ < \infty.$

(ⅲ) A weight is a locally integrable and positive function which is defined on $\mathbb{R}^n$, and it can be written as $\omega(G): = \int _G \omega(g) dg$ for a weight $w$ and measurable set $G$.

(ⅳ) The set of $p(\cdot)$ satisfying (2.3) and (2.4) is represented by $LH(\mathbb{R}^n)$.

$C$ is a constant which is independent of the main parameters involved, and its value varies from line to line.

Lemma 2.1 (Generalized Hölder's inequality). Assume that $G$ is a measurable subset of $\mathbb{R}^n$, and $1 \leq p_-(G) \leq p_+(G)\leq \infty$. Then,

holds, where $f \in {L^{p(\cdot)}(G)}$, $g \in {L^{q(\cdot)}(G)},$ and $\frac{1}{r(z)} = \frac{1}{p(z)}+\frac{1}{q(z)}$ for every $z \in G$.

2.2.   Herz spaces with variable exponent

We adopted the following notations in this subsection:

(a) $\chi _ k = \chi_{R_k}$.

(b) $R_k = D_k \setminus D_{k-1}$.

(c) $D_k = D(0, 2^k) = \{x \in \mathbb{R}^n : |x| < 2^k\}$ for all $k \in \mathbb{Z}.$

(d) $R_{t, \tau}: = D(0, \tau)\backslash D(0, t)$.

Definition 2.2. Let $r \in [1, \infty)$, $\alpha \in \mathbb{R}$ and $s(\cdot) \in \mathcal{P}(\mathbb{R}^n)$. The homogenous Herz space $\dot{K}^{\alpha, r}_{s(\cdot)}(\mathbb{R}^n)$ is defined by

where

Definition 2.3. Let $r\in [1, \infty)$, $\alpha \in \mathbb{R}$ and $s(\cdot)\in \mathcal{P}(\mathbb{R}^n)$. The non-homogenous Herz space $K^{\alpha, r}_{s(\cdot)}(\mathbb{R}^n)$ is defined by

where

2.3.   The variable exponent Muckenhoupt weights

Let $r(\cdot) \in \mathcal{P}(\mathbb{R}^n),$ and $w$ is a weight. The weighted Lebesgue space $L^{r(\cdot)}$ is the set of all complex-valued measurable functions $f$ such that $f w^{\frac{1}{r(\cdot)}} \in L^{r(\cdot)}(\mathbb{R}^n)$. $L^{r(\cdot)}(w)$ is a Banach space its norm is given by

where $r'(\cdot)$ is the conjugate exponent of $r(\cdot)$ given by $\frac{1}{r(\cdot)} + \frac{1}{r'(\cdot)} = 1$. Next we will define Muckenhoupt classes by starting with classical Muckenhoupt weights.

Definition 2.4. Suppose $r(\cdot) \in \mathcal{P}(\mathbb{R}^n).$ A weight $w$ is called an $A_{r(\cdot)}$ weight if

The set $A_{r(\cdot)}$ consists of all $A_{r(\cdot)}$ weights.

Now, we shall give the definitions of the Muckenhoupt classes $A_r$ with $r = 1, \infty.$

Definition 2.5. (ⅰ) A weight $w$ is called a Muckenhoupt $A_1$ weight if $Mw(z) \leq w(z)$ holds for almost every $z \in \mathbb{R}^n$. The set $A_1$ consists of all Muckenhoupt $A_1$ weights. For every $w\in A_1,$

Then a finite value of $[w]_{A_1}$ is called an $A_1$ constant.

(ⅱ) A weight is called a Muckenhoupt $A_\infty$ weight if the weight belongs to the following set:

Definition 2.6. Suppose $r(\cdot) \in \mathcal{P}(\mathbb{R}^n).$ A weight is called an ${A'}_{r(\cdot)}$ weight if

where $P_D: = (\frac{1}{|D|} \int_{D} \frac{1}{r(z)}dz)^{-1}$ is the harmonic average of $r(\cdot)$ over $D$. The set $A' _{r(\cdot)}$ consists of all $A' _{r(\cdot)}$ weights.

Definition 2.7. Let $0 < \alpha < n$ and $r_1(\cdot), r_2(\cdot) \in \mathcal{P}(\mathbb{R}^n)$ such that $\frac{1}{r_2(z)}\equiv \frac{1}{r_1(z)}- \frac{\alpha}{n}$. A weight $w$ is called an $A(r_1(\cdot), r_2(\cdot))$ weight if

holds for all balls $D \subset \mathbb{R}^n$.

Lemma 2.8. Assume that $G$ is a Banach function space, and the Hardy-Littlewood maximal operator $M$ is weakly bounded on $G$, that is,

is true for all $g \in G$ and all $\lambda > 0.$ Then, we have

Lemma 2.9. ^[16] Let $X$ be a Banach function space, and $M$ is bounded on the associate space $X'$. Then, there exists a constant $\delta \in(0, 1)$ such that for all measurable sets $E \subset D$ and for all balls $D \subset \mathbb{R}^n$,

Let ${r_2(\cdot)} \in \mathcal{P} (\mathbb{R}^n) \cap LH(\mathbb{R}^n)$, $w^{r_2(\cdot)} \in A_{r_2(\cdot)}$, $w^{-r'_2(\cdot)} \in A_{r'_2(\cdot)}$ and $\delta _1, \delta_2 \in (0, 1).$

For more details, see ^[16].

3.   Grand weighted Herz spaces with variable exponent

In this section, first we will define grand Herz spaces and then introduce the concept of grand weighted Herz spaces with variable exponent.

Definition 3.1. ^[25] Let $\alpha(\cdot) \in L^\infty(\mathbb{R}^n)$, $r \in [1, \infty)$, $s :\mathbb{R}^n \rightarrow [1, \infty)$, $\theta > 0$. A grand Herz spaces with variable exponent ${\dot{K} ^{\alpha(\cdot), r), \theta}_{ s(\cdot)}(\mathbb{R}^n)}$ is defined by

where

Now, we will define a variable exponent weighted Lebesgue space.

Definition 3.2. ^[16] Let $\Omega \subset \mathbb{R}^n$ be a measurable set, and $w$ is a positive and locally integrable function on $\Omega$. $L^{r(\cdot)} _{\mathrm{loc}} (\Omega, w)$ is the class of all functions $f$ which satisfy the following condition: For all compact sets $E \subset \Omega,$ there is a constant $\lambda > 0$ such that

Definition 3.3. Let $q(\cdot) \in \mathcal{P}(\mathbb{R}^n)$, $0 < r < \infty$, $\alpha \in \mathbb{R}$, $\theta > 0$. The homogeneous grand weighted Herz spaces with variable exponent ${\dot{K} ^{\alpha, r), \theta}_{ q(\cdot)}(w)}$ is the collection of $L^{q(\cdot)} _{\mathrm{loc}}(\mathbb{R}^n/(\{0\}, w)$ is such that,

where

Non-homogeneous grand weighted Herz spaces can be defined in a similar way.

4.   Boundedness of the fractional integrals

Definition 4.1. Fractional integrals are given by the following.

Let $0 < \zeta < n$, and then the fractional integral operator $I^\zeta$ is defined by

Theorem 4.2. ^[16] Let ${r_1(\cdot)} \in \mathcal{P} (\mathbb{R}^n) \cap LH(\mathbb{R}^n)$, $0 < \zeta < n/r_1+$ and $\sigma : = (n/\zeta)'$. Define $r_2(\cdot)$ by $1/r_2(\cdot) = 1/r_1(\cdot)-\zeta /n$. Then, for all weights $w$ such that $(r_2(\cdot)/\sigma, w^\sigma)$ is an $M$-pair, $I^\zeta$ is bounded from $L^{r_1(\cdot)}(w^{r_1(\cdot)})$ to $L^{r_2(\cdot)}(w^{r_2(\cdot)}).$

Theorem 4.3. Let $1 < r < \infty$, $q_2(\cdot) \in \mathcal{P} (\mathbb{R}^n) \cap LH(\mathbb{R}^n)$, $w^{q_2(\cdot)} \in A_1$, $\delta _1, \delta_2 \in (0, 1)$ be the constants appearing in (2.11) and (2.12), respectively. $\alpha$ and $\zeta$ are such that

(i) $-n \delta_1 < \alpha < n \delta _2 -\zeta$.

(ii) $0 < \zeta < n(\delta_1+\delta_2)$.

Define $q_1(\cdot)$ by $1/q_2(\cdot) = 1/q_1(\cdot) -\zeta/n$. Then, the fractional integral operator $I^\zeta$ is a bounded operator from ${\dot{K} ^{\alpha, r), \theta}_{ q_2(\cdot)}(w^{q_2(\cdot)})}$ to ${\dot{K} ^{\alpha, r), \theta}_{ q_1(\cdot)}(w^{q_1(\cdot})}$.

Proof. Let $f \in {\dot{K} ^{\alpha, r), \theta}_{ q_2(\cdot)}(w^{ q_2(\cdot)})}$, and $f_j : = f \chi _j$ for any $j \in \mathbb{Z}.$ Then, $f = \sum \limits _{j = -\infty}^{\infty}f _j$, and we have

Operator $I ^\zeta$ is bounded on weighted Lebesgue space, so for $E_2$,

For $E_1$, by using the size condition and Hölder's inequality,

By using Lemma (2.8), we get

By using (2.12), we have

By the boundednesss of $I^\zeta:{L^{q_1(\cdot)}(w^{q_1(\cdot)})} \rightarrow {L^{q_2(\cdot)}(w^{q_2(\cdot)})}$, and the inequality $2^{j\zeta} \chi _{D_j} \leq (I^\zeta f_{D_j})(x)$, we have

By using Lemma (2.8) again, we obtain

By using the above inequalities, we get

It is known that $\zeta -n \delta _2 + \alpha < 0$, so we will consider two cases: $1 < r(1+\delta) < \infty$ and $0 < r(1+\delta) \leq 1$. By considering, the first case, $1 < r(1+\delta) < \infty,$ and by applying Hölder's inequality, we get

For $0 < r(1+\delta) \leq 1$, we get

Now, we will estimate $E_3$, by using the size condition, and Hölder's inequality, for $j, k \in \mathbb{Z}$ with $j \geq k+2,$ We have

By taking the $L^{p_2(\cdot)}(w^{p_2(\cdot)})$-norm, we have

By the definition of $A(p_1(\cdot), p_2(\cdot))$, we obtain

Hence, we have

Therefore, we get

For $\alpha + n \delta _1 > 0$, we will consider the two cases: $1 < r(1+\delta) < \infty$ and $0 < r(1+\delta) \leq 1$. Now, by considering the first case, $1 < r(1+\delta) < \infty,$ and by using Hölder's inequality, we have

For $0 < r(1+\delta) \leq 1$, we get

which completes the proof.

5.   Conclusions

In this work, we have introduced a new type of space called grand weighted Herz spaces with variable exponents, and we have proved the boundedness of the fractional integrals on those spaces. This spaces will open the door for many future research work in this field.

Acknowledgments

The authors F. Azmi and N. Mlaiki would like to thank Prince Sultan University for paying the publication fees for this work through TAS LAB.

Conflict of interest

The authors declare no conflicts of interest.