Bessel–Riesz Operator in Variable Lebesgue Spaces L^p(·)(${\mathbb{R}}_{+}$)

Abstract

This paper investigates the Bessel–Riesz operator within the framework of variable Lebesgue spaces. We extend existing results by establishing boundedness under more general conditions. The analysis is based on the Hardy–Littlewood maximal function, Hölder’s inequality, and dyadic decomposition techniques. For a given domain space, we construct a suitable range space such that the operator remains bounded. Conversely, for a prescribed range space, we identify a corresponding domain space that guarantees boundedness. Illustrative examples are included to demonstrate the construction of such spaces. The main results hold when the essential infimum of the exponent function exceeds one, and we also establish weak-type estimates in the limiting case.

1. Introduction

Harmonic analysis is one of the most influential branches of mathematics, wherein the properties of various operators play a pivotal role. A substantial body of work in this field focuses on the analysis of integral operators acting on different function spaces. Among these, the Hardy–Littlewood maximal operator is of particular significance due to its capability to control singular integrals, thereby serving as a fundamental tool in harmonic analysis. Moreover, the study of the properties of operators in variable Lebesgue spaces has emerged as a central topic in recent developments.

In recent years, significant attention has been devoted to a particular class of integral operators known as the Bessel–Riesz operators. These operators arise naturally in the study of various problems in harmonic analysis and partial differential equations. In particular, they are useful in estimating solutions to the Schrödinger equation, as the Bessel–Riesz kernel shares structural similarities with kernels appearing in fundamental solutions and potential-type estimates. The Schrödinger equation, which serves as the quantum mechanical analogue of Newton’s second law in classical physics, governs the evolution of the wave function of a quantum system. Given the analytical significance of Bessel–Riesz operators in this context, as well as their broader applications in mathematical analysis and mathematical physics, a detailed investigation of their properties is both relevant and timely.

In this context, Kurata et al. [1] studied the boundedness of an operator formed by the product of an operator W and the Bessel–Riesz operator in generalized Morrey spaces. Subsequently, Idris et al. [2] examined the boundedness of generalized Bessel–Riesz operators on Morrey spaces defined over Euclidean domains. Furthermore, Ruzhansky et al. [3] investigated the boundedness of Bessel–Riesz operators and generalized Bessel–Riesz operators in generalized local (central) Morrey spaces on homogeneous groups. This line of research was further developed by Mehmood et al. [4], who studied the boundedness of Bessel–Riesz operators in the framework of Lebesgue and Morrey spaces over metric measure spaces.

It is well known that the solution of the Schrödinger operator can be estimated using the Bessel–Riesz kernel. The Schrödinger equation, which serves as the quantum mechanical counterpart of Newton’s law in classical physics, is used to describe the wave function or state function of a quantum system. Therefore, the involvement of the Bessel–Riesz operator becomes particularly significant for further study. This motivates us to investigate the properties of the Bessel–Riesz operator due to its importance in both mathematics and physics.

However, all the discussed results concerning Bessel–Riesz operators have been restricted to constant exponent spaces. Motivated by the seminal work of David Cruz-Uribe et al. [5,6], who extended the boundedness of the Hardy–Littlewood maximal operator to variable Lebesgue spaces, Nasir et al. [7] initiated the study of the Bessel–Riesz operator in the framework of variable Lebesgue spaces. Their work focused on establishing the boundedness of the Bessel–Riesz operator in this non-classical setting.

Unlike the classical Lebesgue spaces, where the boundedness of convolution-type operators such as the Bessel–Riesz operator is often derived from Young’s inequality, the variable Lebesgue spaces lack translation invariance, a property essential for the validity of Young’s inequality. Consequently, Young’s inequality does not hold in these spaces. To overcome this challenge, Nasir et al. [7] developed results on the boundedness of Bessel–Riesz operators in variable Lebesgue spaces, providing an analogue to the classical results obtained via Young’s inequality. They established the boundedness of the Bessel–Riesz operators under conditions inspired by Young’s inequality, wherein the norm of the Bessel–Riesz operator is controlled by the norm of the Bessel–Riesz kernel.

In contrast, this manuscript investigates the boundedness of the Bessel–Riesz operator without relying on the norm of its kernel, as dependence on the norm of the Bessel–Riesz kernel leads to additional assumptions regarding the membership of the kernel in variable exponent spaces, making the results more precise but also more restrictive. Thus, our approach yields more general conditions that do not require any membership assumptions on the Bessel–Riesz kernel. Further, two key scenarios are discussed: constructing a suitable range space for a given domain space and conversely, identifying a corresponding domain space for a prescribed range space such that the Bessel–Riesz operator maps the domain space into the range space. To support our theoretical findings, we provide carefully constructed examples within the framework of variable Lebesgue spaces. As the boundedness of the Hardy–Littlewood maximal operator is a crucial component in our analysis, and since this boundedness is not established when the essential infimum $p_{-}=1$, our results do not cover this limiting case. Therefore, we also investigate the weak-type $(1,q(·\left)\right)$ boundedness of the Bessel–Riesz operators in this context. Finally, we present corresponding formulations of these results in the classical Lebesgue spaces, many of which are novel.

This paper develops the results within a more general framework than those previously established in [7]. We investigate several fundamental properties of the Bessel–Riesz operators in the setting of variable Lebesgue spaces. The paper is organized into four sections. In Section 1, we provide a literature review, highlighting the significance of the Bessel–Riesz operators in harmonic analysis and discussing the contributions of various authors in this area. Section 2 covers essential preliminary results related to variable Lebesgue spaces, the Hardy–Littlewood maximal operators, and the Bessel–Riesz operators. Section 3 presents the main findings of the paper, including five core theorems, illustrative examples, remarks, and corollaries.

Theorem 4 addresses the construction of a range space $L^{q(·)}\left({\mathbb{R}}_{+}\right)$ for a given domain space $L^{p(·)}\left({\mathbb{R}}_{+}\right)$, ensuring that the Bessel–Riesz operator $I_{\alpha ,\gamma }$ maps $L^{p(·)}\left({\mathbb{R}}_{+}\right)$ into $L^{q(·)}\left({\mathbb{R}}_{+}\right)$ under suitable conditions. Conversely, Theorem 6 establishes that for a prescribed range space $L^{q(·)}\left({\mathbb{R}}_{+}\right)$, there exists a corresponding domain space $L^{p(·)}\left({\mathbb{R}}_{+}\right)$ such that $I_{\alpha ,\gamma }$ maps $L^{p(·)}\left({\mathbb{R}}_{+}\right)$ into $L^{q(·)}\left({\mathbb{R}}_{+}\right)$. Corollary 1 and Corollary 3 provide the analogous results in the setting of classical Lebesgue spaces corresponding to Theorem 4 and Theorem 6, respectively.

Example 1 supports Theorem 4 and Corollary 1 by constructing range spaces for given domain spaces, while Example 2 supports Theorem 6 and Corollary 3 by constructing domain spaces for given range spaces. Theorems 5 and 7 extend the boundedness results to more general scenarios not covered by Theorems 4 and 6. Furthermore, Theorem 8 establishes the weak-type $(1,q(·\left)\right)$ boundedness of the Bessel–Riesz operator. Corresponding classical results and various remarks regarding their interrelations are also discussed. Finally, Section 4 concludes the manuscript by summarizing the main contributions.

2. Preliminaries

In the following, we present some key concepts of variable Lebesgue spaces from [6] that are essential for a better understanding of this article.

Let $⅁\subseteq {\mathbb{R}}^n$ be a measurable set, and let $\mathfrak{P}(⅁)$ denote the collection of all Lebesgue measurable functions $p(·):⅁\to [1,\infty ]$. For any $p(·)\in \mathfrak{P}(⅁)$ and measurable subset $\Omega \subseteq ⅁$, we define $p_{-}\left(\Omega \right)=\mathrm{ess}{inf}_{\xi \in \Omega }p\left(\xi \right)$ and $p_{+}\left(\Omega \right)=\mathrm{ess}{sup}_{\xi \in \Omega }p\left(\xi \right)$, when there is no ambiguity in the domain, we simply write $p_{-}$ and $p_{+}$. The conjugate exponent function $p^{\prime }(·)$ is defined pointwise by ${\displaystyle \frac{1}{p\left(\xi \right)}}+{\displaystyle \frac{1}{p^{\prime }\left(\xi \right)}}=1$, and it holds that ${\left(p^{\prime }\right)}_{+}={\left(p_{-}\right)}^{\prime }$ and ${\left(p^{\prime }\right)}_{-}={\left(p_{+}\right)}^{\prime }$. Given an exponent function $p(·)\in \mathfrak{P}(⅁)$ with $p_{+}<\infty$ and a measurable function $\mathcal{F}$, the associated modular functional is defined as follows:

$${\Phi }_{p(·)}\left(\mathcal{F}\right)={\int }_{⅁}{\left|\mathcal{F}\left(\xi \right)\right|}^{p\left(\xi \right)}d\xi .$$

Using the modular functional, the variable Lebesgue space $L^{p(·)}(⅁)$ is defined as follows:

$$L^{p(·)}(⅁)=\left\{\mathcal{F}:{\Phi }_{p(·)}(\mathcal{F}/\lambda )<\infty ,\mathrm{for}\mathrm{some}\lambda >0\right\}.$$

This space is equipped with the Luxemburg norm defined as follows:

$${\parallel \mathcal{F}\parallel }_{L^{p(·)}(⅁)}=inf\left\{\lambda >0:{\Phi }_{p(·)}(\mathcal{F}/\lambda )\le 1\right\},$$

which makes $L^{p(·)}(⅁)$ a normed vector space. For any $\mathcal{F}\in L^{p(·)}(⅁)$ with ${\parallel \mathcal{F}\parallel }_{L^{p(·)}(⅁)}>0$, we have ${\Phi }_{p(·)}{(\mathcal{F}/\parallel \mathcal{F}\parallel }_{L^{p(·)}(⅁)})\le 1$, and if $p_{+}<\infty$, then equality holds, i.e., ${\Phi }_{p(·)}{(\mathcal{F}/\parallel \mathcal{F}\parallel }_{L^{p(·)}(⅁)})=1$ for all non-trivial $\mathcal{F}$. Moreover, if $p_{+}<\infty$, then for ${\parallel \mathcal{F}\parallel }_{L^{p(·)}(⅁)}>1$, we obtain

$${\Phi }_{p(·)}{\left(\mathcal{F}\right)}^{1/p_{+}}\le {\parallel \mathcal{F}\parallel }_{L^{p(·)}(⅁)}\le {\Phi }_{p(·)}{\left(\mathcal{F}\right)}^{1/p_{-}},$$

and if ${0<\parallel \mathcal{F}\parallel }_{L^{p(·)}(⅁)}\le 1$, then

$${\Phi }_{p(·)}{\left(\mathcal{F}\right)}^{1/p_{-}}\le {\parallel \mathcal{F}\parallel }_{L^{p(·)}(⅁)}\le {\Phi }_{p(·)}{\left(\mathcal{F}\right)}^{1/p_{+}}.$$

In the special case when $p(·)$ is a constant function, i.e., $p_{-}=p_{+}=p$, these inequalities reduce to the identity

$${\parallel \mathcal{F}\parallel }_{L^p(⅁)}={\left({\int }_{⅁}{\left|\mathcal{F}\left(\xi \right)\right|}^{p}d\xi \right)}^{1/p},$$

which coincides with the classical Lebesgue norm. A version of Hölder’s inequality also holds: for ${\mathcal{F}}_1\in L^{p(·)}(⅁)$ and ${\mathcal{F}}_2\in L^{p^{\prime }(·)}(⅁)$, one has ${\mathcal{F}}_1{\mathcal{F}}_2\in L^1(⅁)$ and

$${\int }_{⅁}|{\mathcal{F}}_1\left(\xi \right){\mathcal{F}}_2\left(\xi \right)|d\xi \le K_{p(·)}\parallel {\mathcal{F}}_1{\parallel }_{p(·)}{\parallel {\mathcal{F}}_2\parallel }_{p^{\prime }(·)},$$

where $K_{p(·)}$ is a constant depending on $p(·)$. The regularity of the exponent function $p(·)$ is crucial in applications, as the boundedness of certain operators, such as the Hardy–Littlewood maximal operator, depends on it. In particular, if the exponent function is log-Hölder continuous, then the maximal operator is bounded on the corresponding variable Lebesgue space. We say that $p(·)\in L{H}_0(⅁)$ if it is locally log-Hölder continuous, meaning there exists a constant $C_0$ such that

$$|p\left({\xi }_1\right)-p\left({\xi }_2\right)|\le {\displaystyle \frac{C_0}{-log\left(\right|{\xi }_1-{\xi }_2\left|\right)}},\mathrm{whenever}|{\xi }_1-{\xi }_2|<\frac{1}{2},{\xi }_1,{\xi }_2\in ⅁.$$

Moreover, $p(·)\in L{H}_{\infty }(⅁)$ if it is log-Hölder continuous at infinity, that is, there exist constants $C_{\infty }$ and $p_{\infty }$ such that

$$|p\left(\xi \right)-p_{\infty }|\le {\displaystyle \frac{C_{\infty }}{log(e+|\xi \left|\right)}},\forall \xi \in ⅁.$$

If both conditions are satisfied, we write $p(·)\in LH(⅁)$. Further, if $p_{+}<\infty$, then the hypothesis ${\displaystyle \frac{1}{p(·)}}\in LH(⅁)$ is equivalent to assume $p(·)\in LH(⅁)$. Finally, given a function $\mathcal{F}\in L_{\mathrm{loc}}^1\left({\mathbb{R}}^n\right)$, the Hardy–Littlewood maximal operator $\mathcal{M}$ is defined as

$$\mathcal{M}\mathcal{F}\left(\xi \right)=\underset{Q\ni \xi }{sup}{\displaystyle \frac{1}{\left|Q\right|}}{\int }_Q\left|\mathcal{F}\left(v\right)\right|dv,\xi \in {\mathbb{R}}^n,$$

where the supremum is taken over all cubes $Q\subset {\mathbb{R}}^n$ with sides parallel to the coordinate axes and containing the point $\xi$.

The boundedness of the Hardy–Littlewood maximal operator $\mathcal{M}$ on weighted Lebesgue spaces $L_w^p$ was characterized by Muckenhoupt [8] and later extended by Sawyer in [9,10]. Following these developments, the boundedness of $\mathcal{M}$ on variable Lebesgue spaces was studied in [5,6].

Theorem 1

([5,6]). Given $p(.)\in \mathfrak{P}\left({\mathbb{R}}^n\right),$ if $1/p(·)\in LH\left({\mathbb{R}}^n\right)$, then there exist $C>0$ such that for all $\beta >0$ and $f\in L^{p(·)}\left({\mathbb{R}}^n\right),$ we have

$$\parallel \beta {\chi }_{\{\xi :\mathcal{M}\mathcal{F}(\xi )>\beta \}}{\parallel }_{L^{p(·)}\left({\mathbb{R}}^n\right)}\le C{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}^n\right)}.$$

Furthermore, if $p_{-}>1$, then there exist $C>0$ such that for all $f\in L^{p(·)}\left({\mathbb{R}}^n\right),$ we have

$${\parallel \mathcal{M}\mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}^n\right)}\le C{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}^n\right)}.$$

On the other hand, similar to maximal operators, Bessel–Riesz operators have attracted significant research interest. We now turn our attention to the study of Bessel–Riesz operators. In the following, we provide some definitions, theorems, and lemmas regarding the Bessel–Riesz operators that are useful for a better understanding of the article.

Definition 1.

Let $0<\gamma <\infty$ and $0<\alpha <n$. The Bessel–Riesz operator, denoted by $I_{\alpha ,\gamma }$, is defined as the convolution of a function $\mathcal{F}\in L_{\mathrm{loc}}^{p(·)}\left({\mathbb{R}}^n\right)$ with the Bessel–Riesz kernel:

$$\begin{array}{c}\hfill I_{\alpha ,\gamma }*\mathcal{F}\left(\xi \right)={\int }_{{\mathbb{R}}^n}{K}_{\alpha ,\gamma }(\xi -v)\mathcal{F}\left(v\right)dv,\end{array}$$

where

$$K_{\alpha ,\gamma }\left(\xi \right)=\left\{\begin{array}{cc}{\displaystyle \frac{{\left|\xi \right|}^{\alpha -n}}{{(1+|\xi \left|\right)}^{\gamma }}},\hfill & \xi \in {\mathbb{R}}^n\setminus \left\{0\right\},\hfill \\ 0,\hfill & \xi =0.\hfill \end{array}\right.$$

Note that the kernel $K_{\alpha ,\gamma }$ can be expressed as the product of two distinct kernels:

$$K_{\alpha ,\gamma }\left(\xi \right)=K_{\gamma }\left(\xi \right)K_{\alpha }\left(\xi \right),\forall \xi \in {\mathbb{R}}^n,$$

where $K_{\gamma }\left(\xi \right)={\displaystyle \frac{1}{{(1+|\xi \left|\right)}^{\gamma }}}$ and $K_{\alpha }\left(\xi \right)={\left|\xi \right|}^{\alpha -n}$ are known as the Bessel and Riesz kernels, respectively [11]. Thus, $K_{\alpha ,\gamma }$ is referred to as the Bessel–Riesz kernel, and the associated operator $I_{\alpha ,\gamma }$ is known as the Bessel–Riesz operator. Before proceeding with further discussion, we first state Young’s inequality for convolution in classical Lebesgue spaces.

Lemma 1

([7]). Suppose $p,\sigma ,q\in [1,\infty ]$ and ${\displaystyle \frac{1}{q}}={\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{\sigma }}-1$. If ${\mathcal{F}}_1\in L^p\left(\mathbb{R}\right)$ and ${\mathcal{F}}_2\in L^{\sigma }\left(\mathbb{R}\right)$, then ${\mathcal{F}}_1*{\mathcal{F}}_2\in L^q\left(\mathbb{R}\right)$, and

$$\begin{array}{c}\hfill \parallel {\mathcal{F}}_1*{\mathcal{F}}_2{\parallel }_{L^q\left(\mathbb{R}\right)}\le \parallel {\mathcal{F}}_1{\parallel }_{L^p\left(\mathbb{R}\right)}{\parallel {\mathcal{F}}_2\parallel }_{L^{\sigma }\left(\mathbb{R}\right)}.\end{array}$$

In classical Lebesgue spaces, the boundedness of Bessel–Riesz operators can be directly established using Young’s inequality. However, in variable Lebesgue spaces, Young’s inequality for convolutions generally fails due to the lack of translation invariance, as noted in [5,6]. Consequently, Nasir et al. [7] developed alternative results analogous to those obtained in classical Lebesgue spaces using Young’s inequality.

We now present the major results of their work in [7]. In the following lemma, they investigate the membership of the Bessel–Riesz kernel in variable Lebesgue spaces.

Lemma 2

([7]). Let $0<\alpha <1$ and $0<\gamma <\infty$. Then $K_{\alpha ,\gamma }\in L^{\sigma (·)}\left({\mathbb{R}}_{+}\right)$ for

$${\displaystyle \frac{1}{\gamma +1-\alpha }}<{\sigma }_{-}\le {\sigma }_{+}<{\displaystyle \frac{1}{1-\alpha }}.$$

In the next lemma, an estimate is established using the maximal operator, the dyadic decomposition technique, and Hölder’s inequality. This estimate will also play a crucial role in our results in this manuscript.

Lemma 3

([7]). For any $\xi \in {\mathbb{R}}_{+}$ and $\mathcal{F}\in L^{p(·)}\left({\mathbb{R}}_{+}\right)$, there exist a constant $C^{*}>0$ and an integer $N_r$ depending on $"r"$ such that, the following estimate holds for the Bessel–Riesz operator:

$$|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|\le C^{*}\left(\mathcal{M}\mathcal{F}\left(\xi \right)\sum _{k=-\infty }^{N_r-1}{\displaystyle \frac{{\left(2^{k}r\right)}^{\alpha }}{{(1+2^{k}r)}^{\gamma }}}+{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\sum _{k=N_r}^{\infty }{\displaystyle \frac{{\left(2^{k}r\right)}^{\alpha -\frac{1}{p_{+}}}}{{(1+2^{k}r)}^{\gamma }}}\right).$$

(1)

In the following theorem, they proved the boundedness of $I_{\alpha ,\gamma }$ from $L^{p(·)}\left({\mathbb{R}}_{+}\right)$ to $L^{q(·)}\left({\mathbb{R}}_{+}\right)$, without imposing any conditions on the maximal function.

Theorem 2

([7]). Let $p(·),q(·),\sigma (·)\in \mathfrak{P}\left({\mathbb{R}}_{+}\right)$, $0<\alpha <1$, and $0<\gamma <\infty$, with

$$\mathrm{Range}\left(\sigma (·)\right)\subset \left({\displaystyle \frac{1}{\gamma +1-\alpha }},{\displaystyle \frac{1}{1-\alpha }}\right).$$

Suppose there exists a constant $c>0$ such that

$${\displaystyle \frac{1}{q_{+}}}\le c\left({\displaystyle \frac{1}{p_{+}}}+{\displaystyle \frac{1}{{\sigma }_{+}}}-1\right),\mathrm{and}{\displaystyle \frac{q(·)}{p(·)}}=1+{\displaystyle \frac{c{q}_{+}(\alpha ({\sigma }_{+}-{\sigma }_{-})+{\sigma }_{-}-1)}{{\sigma }_{+}}}\mathrm{a}.\mathrm{e}.$$

with $p(·)\in LH\left({\mathbb{R}}_{+}\right)$. Then, the Bessel–Riesz operator

$$I_{\alpha ,\gamma }:L^{p(·)}\left({\mathbb{R}}_{+}\right)\to L^{q(·)}\left({\mathbb{R}}_{+}\right)$$

is bounded. Furthermore, for any $\mathcal{F}\in L^{p(·)}\left({\mathbb{R}}_{+}\right)$, there exists a constant $C>0$ such that

• If $0<\parallel K_{\alpha ,\gamma }{\parallel }_{L^{\sigma (·)}\left({\mathbb{R}}_{+}\right)}<1$, then

  $$\parallel I_{\alpha ,\gamma }{\mathcal{F}\parallel }_{L^{q(·)}\left({\mathbb{R}}_{+}\right)}\le C\parallel K_{\alpha ,\gamma }{\parallel }_{L^{\sigma (·)}\left({\mathbb{R}}_{+}\right)}^{1/{\sigma }_{+}}{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}.$$
• If $\parallel K_{\alpha ,\gamma }{\parallel }_{L^{\sigma (·)}\left({\mathbb{R}}_{+}\right)}\ge 1$, then

  $$\parallel I_{\alpha ,\gamma }{\mathcal{F}\parallel }_{L^{q(·)}\left({\mathbb{R}}_{+}\right)}\le C\parallel K_{\alpha ,\gamma }{\parallel }_{L^{\sigma (·)}\left({\mathbb{R}}_{+}\right)}{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}.$$

In the following, Nasir et al. [7] extend the result to a broader setting not covered by Theorem 2, by investigating the boundedness of the Bessel–Riesz operator from $L^{p(·)}\left({\mathbb{R}}_{+}\right)$ to $L^{q(·)}\left({\mathbb{R}}_{+}\right)$ under certain constraints on the maximal function.

Definition 2.

Let $0<\alpha <1$. The pair of exponent functions $\left(q\right(·),p(·\left)\right)$ is said to satisfy the boundedness property if the following conditions hold:

• For any $\xi \in {⅁}_1=\left\{\xi \in {\mathbb{R}}_{+}:\mathcal{M}\mathcal{F}\left(\xi \right)\le {\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\right\}$, the exponents satisfy

  $${\displaystyle \frac{q\left(\xi \right)}{p\left(\xi \right)}}\ge 1+{\displaystyle \frac{q_{+}(\alpha ({\sigma }_{+}-{\sigma }_{-})+{\sigma }_{-}-1)}{{\sigma }_{+}}}\mathrm{a}.\mathrm{e}.$$
• For any $\xi \in {⅁}_2=\left\{\xi \in {\mathbb{R}}_{+}:\mathcal{M}\mathcal{F}\left(\xi \right)>{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\right\}$, the exponents satisfy

  $${\displaystyle \frac{q\left(\xi \right)}{p\left(\xi \right)}}<1+{\displaystyle \frac{q_{+}(\alpha ({\sigma }_{+}-{\sigma }_{-})+{\sigma }_{-}-1)}{{\sigma }_{+}}}\mathrm{a}.\mathrm{e}.$$

Theorem 3

([7]). Let $0<\gamma <\infty$, and suppose the pair $\left(q\right(·),p(·\left)\right)$ satisfies the boundedness property. Assume

$${\displaystyle \frac{1}{\gamma +1-\alpha }}<{\sigma }_{-}\le {\sigma }_{+}<{\displaystyle \frac{1}{1-\alpha }},\mathrm{and}{\displaystyle \frac{1}{q_{+}}}={\displaystyle \frac{1}{p_{+}}}+{\displaystyle \frac{1}{{\sigma }_{+}}}-1,$$

with $p(·)\in LH\left({\mathbb{R}}_{+}\right)$. Then, the Bessel–Riesz operator

$$I_{\alpha ,\gamma }:L^{p(·)}\left({\mathbb{R}}_{+}\right)\to L^{q(·)}\left({\mathbb{R}}_{+}\right)$$

is bounded. Furthermore, for any $\mathcal{F}\in L^{p(·)}\left({\mathbb{R}}_{+}\right)$, there exists a constant $C>0$ such that

• If $0<\parallel K_{\alpha ,\gamma }{\parallel }_{L^{\sigma (·)}\left({\mathbb{R}}_{+}\right)}<1$, then

  $$\parallel I_{\alpha ,\gamma }{\mathcal{F}\parallel }_{L^{q(·)}\left({\mathbb{R}}_{+}\right)}\le C\parallel K_{\alpha ,\gamma }{\parallel }_{L^{\sigma (·)}\left({\mathbb{R}}_{+}\right)}^{1/{\sigma }_{+}}{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}.$$
• If $\parallel K_{\alpha ,\gamma }{\parallel }_{L^{\sigma (·)}\left({\mathbb{R}}_{+}\right)}\ge 1$, then

  $$\parallel I_{\alpha ,\gamma }{\mathcal{F}\parallel }_{L^{q(·)}\left({\mathbb{R}}_{+}\right)}\le C\parallel K_{\alpha ,\gamma }{\parallel }_{L^{\sigma (·)}\left({\mathbb{R}}_{+}\right)}{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}.$$

In the above theorem, the condition

$${\displaystyle \frac{1}{\gamma +1-\alpha }}<{\sigma }_{-}\le {\sigma }_{+}<{\displaystyle \frac{1}{1-\alpha }}$$

is required to ensure that the Bessel–Riesz kernel belongs to the variable Lebesgue space $L^{\sigma (·)}\left({\mathbb{R}}_{+}\right)$. Additionally, the conditions ${\displaystyle \frac{1}{q_{+}}}={\displaystyle \frac{1}{p_{+}}}+{\displaystyle \frac{1}{{\sigma }_{+}}}-1\mathrm{and}p(·)\in LH\left({\mathbb{R}}_{+}\right)$ are essential for the proof.

3. Main Results

Suppose $\mathcal{F}\in L^{p(.)}\left(\mathbb{R}\right)$ then for $0<\alpha <1$ and $0<\gamma <\infty ,$ the Bessel–Riesz operator is defined as

$$\begin{array}{c}\hfill I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right):=(K_{\alpha ,\gamma }*\mathcal{F})\left(\xi \right)={\int }_{\mathbb{R}}{\displaystyle \frac{{|\xi -v|}^{\alpha -1}\mathcal{F}\left(v\right)}{{[1+|\xi -v\left|\right]}^{\gamma }}}dv;\xi \in \mathbb{R}.\end{array}$$

Theorem 4.

Let $p(·)\in \mathfrak{P}\left({\mathbb{R}}_{+}\right),$ and $1<p_{-}\le p_{+}<\infty$ with $p(.)\in LH\left({\mathbb{R}}_{+}\right)$. If $\alpha -{\displaystyle \frac{1}{p_{+}}}<\gamma$ then there exist $q(·)\in \mathfrak{P}\left({\mathbb{R}}_{+}\right),$ satisfying ${\displaystyle \frac{1}{q(·)}}={\displaystyle \frac{1}{1+\gamma p_{+}}}\left({\displaystyle \frac{1}{p(·)}}-(\alpha -\gamma ){\displaystyle \frac{p_{+}}{p(·)}}\right)$, a.e. such that $I_{\alpha ,\gamma }:L^{p(·)}\left({\mathbb{R}}_{+}\right)\to L^{q(·)}\left({\mathbb{R}}_{+}\right)$ is continuous, or the Bessel–Riesz operator maps $L^{p(·)}\left({\mathbb{R}}_{+}\right)$ into $L^{q(·)}\left({\mathbb{R}}_{+}\right).$

Proof. 

Since $1<p_{-}\le p_{+}<\infty$ and $p(·)\in LH\left({\mathbb{R}}_{+}\right),$ so by boundedness of maximal operator, there exist $C_0>0,$ such that for any $f\in L^{p(·)}\left({\mathbb{R}}_{+}\right),$ we have

$${\parallel \mathcal{M}\mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\le C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}.$$

Further, definition of norm in variable Lebesgue spaces, we have

$${\parallel \mathcal{M}\mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}=inf\{\lambda >0:{\Phi }_{p(·)}\left(\mathcal{M}\mathcal{F}/\lambda \right)\le 1\}.$$

Thus, $C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\in \left\{\lambda >0:{\Phi }_{p(·)}\left({\displaystyle \frac{\mathcal{M}\mathcal{F}}{\lambda }}\right)\le 1\right\}$. Now, for any $\xi \in {\mathbb{R}}_{+}$ and $\mathcal{F}\in L^{p(·)}\left({\mathbb{R}}_{+}\right)$, with $\mathcal{F}\ne 0$ (since the case $\mathcal{F}=0$ is trivial), and using the maximal function, dyadic decomposition technique, and Hölder’s inequality, there exists an integer $N_r$ such that, by following (1), we have

$$|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|\le C^{**}\left(r^{\alpha }\mathcal{M}\mathcal{F}\left(\xi \right)+r^{\alpha -{\displaystyle \frac{1}{p_{+}}}-\gamma }{\parallel \mathcal{F}\parallel }_{L^{p(.)}\left({\mathbb{R}}_{+}\right)}\right).$$

(2)

We choose $r={\left({\displaystyle \frac{\mathcal{M}\mathcal{F}\left(\xi \right)}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)}^{{\displaystyle \frac{p_{+}p\left(\xi \right)}{((\alpha -\gamma )p_{+}-1)q\left(\xi \right)}}},$ where $(\alpha -\gamma )p_{+}-1<0$; thus, using the fact that $q\left(\xi \right)={\displaystyle \frac{1+\gamma p_{+}}{1-(\alpha -\gamma )p_{+}}}p\left(\xi \right),$ a.e. for any $\xi \in {\mathbb{R}}_{+},$ we get

$$\begin{array}{cc}\hfill A& =r^{\alpha }\mathcal{M}\mathcal{F}\left(\xi \right)+r^{\alpha -{\displaystyle \frac{1}{p_{+}}}-\gamma }{\parallel \mathcal{F}\parallel }_{L^{p(.)}\left({\mathbb{R}}_{+}\right)}\hfill \\ \hfill & =\left(C_0+1\right){\left({\displaystyle \frac{\mathcal{M}\mathcal{F}\left(\xi \right)}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)}^{{\displaystyle \frac{p\left(\xi \right)}{q\left(\xi \right)}}}{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}.\hfill \end{array}$$

(3)

Thus, by following (2) and (3), and choosing $C_1=(C_0+1)C^{**},$ we get

$$|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|\le C_1{\left({\displaystyle \frac{\mathcal{M}\mathcal{F}\left(\xi \right)}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)}^{{\displaystyle \frac{p\left(\xi \right)}{q\left(\xi \right)}}}{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)},$$

thus,

$${\left({\displaystyle \frac{|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|}{C_1{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)}^{q\left(\xi \right)}\le {\left({\displaystyle \frac{\mathcal{M}\mathcal{F}\left(\xi \right)}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)}^{p\left(\xi \right)}.$$

From the definition of modular, we get

$${\Phi }_{q(·)}\left({\displaystyle \frac{I_{\alpha ,\gamma }\mathcal{F}}{C_1{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)\le {\Phi }_{p(·)}\left({\displaystyle \frac{\mathcal{M}\mathcal{F}}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right).$$

Since the boundedness of the Hardy–Littlewood maximal operator ensures that the modular on the right-hand side is less than or equal to one, we obtain the following as a consequence.

$$\parallel I_{\alpha ,\gamma }{\mathcal{F}\parallel }_{L^{q(·)}\left({\mathbb{R}}_{+}\right)}\le C_1{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}.$$

□

Corollary 1.

Let $1<p<\infty$ and $\alpha -{\displaystyle \frac{1}{p}}<\gamma$; then, there exist $1<q<\infty ,$ satisfying ${\displaystyle \frac{1}{q}}={\displaystyle \frac{1}{1+\gamma p}}\left({\displaystyle \frac{1}{p}}-(\alpha -\gamma )\right)$ a.e., such that $I_{\alpha ,\gamma }:L^p\left({\mathbb{R}}_{+}\right)\to L^q\left({\mathbb{R}}_{+}\right)$ is continuous, or the Bessel–Riesz operator maps $L^p\left({\mathbb{R}}_{+}\right)$ into $L^q\left({\mathbb{R}}_{+}\right).$

Proof. 

Since p is constant, we have $p_{-}=p_{+}=p$, and $\alpha -{\displaystyle \frac{1}{p}}<\gamma$. Therefore, all the assumptions of Theorem 4 are satisfied, and the proof follows by substituting $p(·)=p$ into the proof of Theorem 4. □

Example 1.

Let

$$p\left(\xi \right)=\left\{\begin{array}{c}\hfill \xi +2\mathrm{if}\xi \in [0,1),\\ \hfill {\displaystyle \frac{1}{\xi }}+2\mathrm{if}\xi \in [1,\infty ),\end{array}\right.$$

Then, $p_{-}=2,p_{+}=3,1<p_{-}\le p_{+}<\infty ,$ and for $\alpha =1/2,\gamma =2,$ we have $\alpha -{\displaystyle \frac{1}{p_{+}}}<\gamma .$ Further, since $p(·)\in LH\left({\mathbb{R}}_{+}\right)),$ by following Theorem 4, there exist

$$q\left(\xi \right)=\left\{\begin{array}{c}\hfill {\displaystyle \frac{14\xi }{11}}+{\displaystyle \frac{28}{11}}\mathrm{if}\xi \in [0,1),\\ \hfill {\displaystyle \frac{14}{11\xi }}+{\displaystyle \frac{28}{11}}\mathrm{if}\xi \in [1,\infty ),\end{array}\right.$$

satisfying ${\displaystyle \frac{1}{q(·)}}={\displaystyle \frac{1}{1+\gamma p_{+}}}\left({\displaystyle \frac{1}{p(·)}}-(\alpha -\gamma ){\displaystyle \frac{p_{+}}{p(·)}}\right)$ a.e., such that $I_{\alpha ,\gamma }:L^{p(·)}\left({\mathbb{R}}_{+}\right)\to L^{q(·)}\left({\mathbb{R}}_{+}\right)$ is continuous, or the Bessel–Riesz operator maps $L^{p(·)}\left({\mathbb{R}}_{+}\right)$ into $L^{q(·)}\left({\mathbb{R}}_{+}\right).$ Further, for the constant case, if we take $p=5/2,\alpha =1/2$ and $\gamma =2,$ we have $\alpha -{\displaystyle \frac{1}{p}}<\gamma$; thus, by following Corollary 1, there exists $q=60/19$ such that $I_{\alpha ,\gamma }:L^p\left({\mathbb{R}}_{+}\right)\to L^q\left({\mathbb{R}}_{+}\right)$ is continuous, or the Bessel–Riesz operator maps $L^p\left({\mathbb{R}}_{+}\right)$ into $L^q\left({\mathbb{R}}_{+}\right).$

Definition 3.

Let $p(·),q(·)\in \mathfrak{P}\left({\mathbb{R}}_{+}\right)$ and $\alpha -{\displaystyle \frac{1}{p_{+}}}<\gamma$; then, the pair $\left(p\right(·),q(·\left)\right)$ is said to satisfy the $(\alpha -{\displaystyle \frac{1}{p_{+}}}<\gamma )$-boundedness property if for any $\xi \in {⅁}_1=\{\xi \in {\mathbb{R}}_{+}:\mathcal{M}\mathcal{F}\left(\xi \right)\le C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\}$ the exponents satisfy ${\displaystyle \frac{1}{q(·)}}\ge {\displaystyle \frac{1}{1+\gamma p_{+}}}\left({\displaystyle \frac{1}{p(·)}}-(\alpha -\gamma ){\displaystyle \frac{p_{+}}{p(·)}}\right)$ a.e. and for any $\xi \in {⅁}_2=\{\xi \in {\mathbb{R}}_{+}:\mathcal{M}\mathcal{F}\left(\xi \right)>C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\}$ the exponents satisfy ${\displaystyle \frac{1}{q(·)}}<{\displaystyle \frac{1}{1+\gamma p_{+}}}\left({\displaystyle \frac{1}{p(·)}}-(\alpha -\gamma ){\displaystyle \frac{p_{+}}{p(·)}}\right)$ a.e.

Theorem 5.

Let $p(·),q(·)\in \mathfrak{P}\left({\mathbb{R}}_{+}\right),$ with $\alpha -{\displaystyle \frac{1}{p_{+}}}<\gamma$ and the pair $\left(p\right(·),q(·\left)\right)$ satisfies the $(\alpha -{\displaystyle \frac{1}{p_{+}}}<\gamma )$-boundedness property with $p(.)\in LH\left({\mathbb{R}}_{+}\right)$ and $1<p_{-}\le p_{+}<\infty ,q_{+}<\infty$; then, the Bessel–Riesz operator $I_{\alpha ,\gamma }:L^{p(·)}\left({\mathbb{R}}_{+}\right)\to L^{q(·)}\left({\mathbb{R}}_{+}\right)$ is bounded. Further, for any $\mathcal{F}\in L^{p(.)}\left({\mathbb{R}}_{+}\right)$ there exists a constant $C>0$, such that

$$\parallel I_{\alpha ,\gamma }{\mathcal{F}\parallel }_{L^{q(.)}\left({\mathbb{R}}_{+}\right)}\le C{\parallel \mathcal{F}\parallel }_{L^{p(.)}\left({\mathbb{R}}_{+}\right)}.$$

Proof. 

Since $1<p_{-}\le p_{+}<\infty$ and $p(·)\in LH\left({\mathbb{R}}_{+}\right),$ so by boundedness of maximal operator, there exist $C_0>0,$ such that for any $f\in L^{p(·)}\left({\mathbb{R}}^n\right),$ we have

$${\parallel \mathcal{M}\mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\le C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}.$$

Now, for any $\xi \in {\mathbb{R}}_{+}$ and $\mathcal{F}\in L^{p(·)}\left({\mathbb{R}}_{+}\right)$, with $\mathcal{F}\ne 0$ (since the case $\mathcal{F}=0$ is trivial), and using Lemma 3 along with the fact that $\alpha -{\displaystyle \frac{1}{p_{+}}}<\gamma ,$ we get

$$|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|\le C_2\left(r^{\alpha }\mathcal{M}\mathcal{F}\left(\xi \right)+r^{\alpha -{\displaystyle \frac{1}{p_{+}}}-\gamma }{\parallel \mathcal{F}\parallel }_{L^{p(.)}\left({\mathbb{R}}_{+}\right)}\right).$$

Further, by choosing $r={\left({\displaystyle \frac{\mathcal{M}\mathcal{F}\left(\xi \right)}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)}^{{\displaystyle \frac{p_{+}p\left(\xi \right)}{((\alpha -\gamma )p_{+}-1)q\left(\xi \right)}}},$ and then by simplifying, we get

$$|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|\le C_2{\displaystyle \frac{C_0{\left(C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\right)}^{\frac{\alpha p_{+}p\left(\xi \right)}{(1-(\alpha -\gamma )p_{+})q\left(\xi \right)}+\frac{p\left(\xi \right)}{q\left(\xi \right)}-1}+{\left(\mathcal{M}\mathcal{F}\left(\xi \right)\right)}^{\frac{\alpha p_{+}p\left(\xi \right)}{(1-(\alpha -\gamma )p_{+})q\left(\xi \right)}+\frac{p\left(\xi \right)}{q\left(\xi \right)}-1}}{C_0{\left(\mathcal{M}\mathcal{F}\left(\xi \right)\right)}^{\frac{\alpha p_{+}p\left(\xi \right)}{(1-(\alpha -\gamma )p_{+})q\left(\xi \right)}-1}{\left(C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\right)}^{\frac{p\left(\xi \right)}{q\left(\xi \right)}-1}}}.$$

For any $\xi \in {\mathbb{R}}_{+},$ we discuss following two possibilities. If $\xi \in {⅁}_1,$ then ${\displaystyle \frac{1}{q(·)}}\ge {\displaystyle \frac{1}{1+\gamma p_{+}}}\left({\displaystyle \frac{1}{p(·)}}-(\alpha -\gamma ){\displaystyle \frac{p_{+}}{p(·)}}\right)$ a.e., thus ${\displaystyle \frac{\alpha p_{+}p\left(\xi \right)}{(1-(\alpha -\gamma )p_{+})q\left(\xi \right)}}+{\displaystyle \frac{p\left(\xi \right)}{q\left(\xi \right)}}-1\le 0$ a.e., we get

$$\begin{array}{c}\hfill |I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|\le C_2(C_0+1){\left({\displaystyle \frac{\mathcal{M}\mathcal{F}\left(\xi \right)}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)}^{{\displaystyle \frac{p\left(\xi \right)}{q\left(\xi \right)}}}{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}.\end{array}$$

(4)

If $\xi \in {⅁}_2,$ then, ${\displaystyle \frac{1}{q(·)}}<{\displaystyle \frac{1}{1+\gamma p_{+}}}\left({\displaystyle \frac{1}{p(·)}}-(\alpha -\gamma ){\displaystyle \frac{p_{+}}{p(·)}}\right)$ a.e., thus ${\displaystyle \frac{\alpha p_{+}p\left(\xi \right)}{(1-(\alpha -\gamma )p_{+})q\left(\xi \right)}}+{\displaystyle \frac{p\left(\xi \right)}{q\left(\xi \right)}}-1>0$ a.e., we get

$$\begin{array}{c}\hfill |I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|\le C_2(C_0+1){\left({\displaystyle \frac{\mathcal{M}\mathcal{F}\left(\xi \right)}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)}^{{\displaystyle \frac{p\left(\xi \right)}{q\left(\xi \right)}}}{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}.\end{array}$$

(5)

By following (4) and (5), we get for any $\xi \in R_{+},$

$${\left({\displaystyle \frac{|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|}{C_3{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)}^{q\left(\xi \right)}\le {\left({\displaystyle \frac{\mathcal{M}\mathcal{F}\left(\xi \right)}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)}^{p\left(\xi \right)}.$$

From the definition of modular, we get

$${\Phi }_{q(·)}\left({\displaystyle \frac{I_{\alpha ,\gamma }\mathcal{F}}{C_3{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)\le {\Phi }_{p(·)}\left({\displaystyle \frac{\mathcal{M}\mathcal{F}}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right).$$

Thus by following the boundedness of the Hardy–Littlewood maximal operator and the definition of the norm in variable Lebesgue spaces, we have $C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\in \{\lambda >0:{\Phi }_{p(·)}\left(\mathcal{M}\mathcal{F}/\lambda \right)\le 1\},$ consequently, we get

$${\Phi }_{q(·)}\left({\displaystyle \frac{I_{\alpha ,\gamma }\mathcal{F}}{C_3{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)\le {\Phi }_{p(·)}\left({\displaystyle \frac{\mathcal{M}\mathcal{F}}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)\le 1.$$

Thus,

$$C_3{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\in \{\lambda >0:{\Phi }_{q(·)}\left(\left(I_{\alpha ,\gamma }\mathcal{F}\right)/\lambda \right)\le 1\}.$$

Hence,

$$\parallel I_{\alpha ,\gamma }{\mathcal{F}\parallel }_{L^{q(·)}\left({\mathbb{R}}_{+}\right)}\le C_3{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}.$$

□

Corollary 2.

Let $1<p<\infty ,1<q<\infty ,$$\alpha -{\displaystyle \frac{1}{p}}<\gamma$, and the pair of exponents $(p,q)$ satisfies the $(\alpha -{\displaystyle \frac{1}{p}}<\gamma )$-boundedness property; then, the Bessel–Riesz operator $I_{\alpha ,\gamma }:L^p\left({\mathbb{R}}_{+}\right)\to L^q\left({\mathbb{R}}_{+}\right)$ is bounded.

Proof. 

The proof follows from the proof of Theorem 5, using the exponent functions constant such that $p(·)=p$ and $q(·)=q.$ □

Remark 1.

Since Theorem 4 describes the boundedness of the Bessel–Riesz operators under the condition

$${\displaystyle \frac{1}{q(·)}}={\displaystyle \frac{1}{1+\gamma p_{+}}}\left({\displaystyle \frac{1}{p(·)}}-(\alpha -\gamma ){\displaystyle \frac{p_{+}}{p(·)}}\right)\mathrm{a}.\mathrm{e}.,$$

whenever the exponent functions do not satisfy this condition, Theorem 5 provides an alternative result concerning the boundedness of the Bessel–Riesz operators. Moreover, Theorem 5 does not cover Theorem 4, as Theorem 5 requires an additional condition involving the Hardy–Littlewood maximal function, which is not needed in Theorem 4.

Theorem 6.

Let $q(·)\in \mathfrak{P}\left({\mathbb{R}}_{+}\right),$$q_{+}<\infty$ and ${\displaystyle \frac{1}{q_{-}}}+{\displaystyle \frac{\alpha q_{+}}{q_{-}}}<1$ with $q(·)\in LH\left({\mathbb{R}}_{+}\right)$; then, there exist $p(·)\in \mathfrak{P}\left({\mathbb{R}}_{+}\right),$ satisfying ${\displaystyle \frac{1}{p(·)}}={\displaystyle \frac{1}{q(·)}}+{\displaystyle \frac{\alpha q_{+}}{q(·)}}$ a.e. such that $I_{\alpha ,\gamma }:L^{p(·)}\left({\mathbb{R}}_{+}\right)\to L^{q(·)}\left({\mathbb{R}}_{+}\right)$ is continuous, or the Bessel–Riesz operator maps $L^{p(·)}\left({\mathbb{R}}_{+}\right)$ into $L^{q(·)}\left({\mathbb{R}}_{+}\right).$

Proof. 

Since $q_{+}<\infty$ and ${\displaystyle \frac{1}{q_{-}}}+{\displaystyle \frac{\alpha q_{+}}{q_{-}}}<1$, along with the relation ${\displaystyle \frac{1}{p\left(\xi \right)}}={\displaystyle \frac{1}{q\left(\xi \right)}}+{\displaystyle \frac{\alpha q_{+}}{q\left(\xi \right)}}$ almost everywhere, it follows that $1<p_{-}\le p_{+}<\infty$. Moreover, $q_{+}<\infty$, $q(·)\in LH\left({\mathbb{R}}_{+}\right)$, and $p(·)={\displaystyle \frac{q(·)}{1+\alpha q_{+}}}$ guarantee the log-Hölder continuity of $p(·)$. The only modification is that if $C_0$ and $C_{\infty }$ are the constants corresponding to the local and the log-Hölder continuity at infinity for $q(·)$, then ${\displaystyle \frac{C_0}{1+\alpha q_{+}}}$ and ${\displaystyle \frac{C_{\infty }}{1+\alpha q_{+}}}$ serve as the respective constants for $p(·)$. Consequently, $p(·)\in LH\left({\mathbb{R}}_{+}\right),$ so by boundedness of maximal operator, there exist $C_0>0,$ such that for any $\mathcal{F}\in L^{p(·)}\left({\mathbb{R}}_{+}\right),$ we have

$${\parallel \mathcal{M}\mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\le C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}.$$

Now, for any $\xi \in {\mathbb{R}}_{+}$ and any nonzero function $\mathcal{F}\in L^{p(·)}\left({\mathbb{R}}_{+}\right)$ (the case $\mathcal{F}=0$ being trivial), it follows from Lemma 3 that

$$|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|\le C^{*}\left(\mathcal{M}\mathcal{F}\left(\xi \right)\sum _{k=-\infty }^{N_r-1}{\left(2^{k}r\right)}^{\alpha }+{\parallel \mathcal{F}\parallel }_{L^{p(.)}\left({\mathbb{R}}_{+}\right)}\sum _{k=N_r}^{\infty }{\left(2^{k}r\right)}^{-{\displaystyle \frac{1}{q_{+}}}}\right),$$

since the series on the right are convergent, we have

$$|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|\le C_4\left(r^{\alpha }\mathcal{M}\mathcal{F}\left(\xi \right)+r^{-{\displaystyle \frac{1}{q_{+}}}}{\parallel \mathcal{F}\parallel }_{L^{p(.)}\left({\mathbb{R}}_{+}\right)}\right).$$

We choose $r={\left({\displaystyle \frac{\mathcal{M}\mathcal{F}\left(\xi \right)}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)}^{-{\displaystyle \frac{q_{+}p\left(\xi \right)}{q\left(\xi \right)}}}$; then, using the fact that ${\displaystyle \frac{1}{p(·)}}={\displaystyle \frac{1}{q(·)}}+{\displaystyle \frac{\alpha q_{+}}{q(·)}}$ a.e. on ${\mathbb{R}}_{+},$ we get

$$|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|\le C_5{\left({\displaystyle \frac{\mathcal{M}\mathcal{F}\left(\xi \right)}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)}^{{\displaystyle \frac{p\left(\xi \right)}{q\left(\xi \right)}}}{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}.$$

From the definition of modular, we get

$${\Phi }_{q(·)}\left({\displaystyle \frac{I_{\alpha ,\gamma }\mathcal{F}}{C_5{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)\le {\Phi }_{p(·)}\left({\displaystyle \frac{\mathcal{M}\mathcal{F}}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right).$$

Now, using the boundedness of the Hardy–Littlewood maximal operator and the definition of the norm in variable Lebesgue spaces, we obtain the desired boundedness of the Bessel–Riesz operator. □

Corollary 3.

If $1<q<\infty$ and ${\displaystyle \frac{1}{q}}+\alpha <1$, then there exist $1<p<\infty$ satisfying ${\displaystyle \frac{1}{p}}={\displaystyle \frac{1}{q}}+\alpha$ a.e. such that $I_{\alpha ,\gamma }:L^p\left({\mathbb{R}}_{+}\right)\to L^q\left({\mathbb{R}}_{+}\right)$ is continuous, or the Bessel–Riesz operator maps $L^p\left({\mathbb{R}}_{+}\right)$ into $L^q\left({\mathbb{R}}_{+}\right).$

Proof. 

Since q is constant, we have $q_{-}=q_{+}=q$, and ${\displaystyle \frac{1}{q}}+\alpha <1,$. Therefore, all the assumptions of Theorem 6 are satisfied, and the proof follows by substituting $q(·)=q$ into the proof of Theorem 6. □

Example 2.

Let

$$q\left(\xi \right)=\left\{\begin{array}{c}\hfill \xi +5\mathrm{if}\xi \in [0,1),\\ \hfill {\displaystyle \frac{1}{\xi }}+5\mathrm{if}\xi \in [1,\infty ),\end{array}\right.$$

then, $q_{-}=5,q_{+}=6,1<q_{-}\le q_{+}<\infty ,$ and for $\alpha =1/2,$ we have ${\displaystyle \frac{1}{q_{-}}}+{\displaystyle \frac{\alpha q_{+}}{q_{-}}}<1.$ Further, since $q(·)\in LH\left({\mathbb{R}}_{+}\right),$ thus by following Theorem 6, there exist

$$p\left(\xi \right)=\left\{\begin{array}{c}\hfill {\displaystyle \frac{\xi }{4}}+{\displaystyle \frac{5}{4}}\mathrm{if}\xi \in [0,1),\\ \hfill {\displaystyle \frac{1}{4\xi }}+{\displaystyle \frac{5}{4}}\mathrm{if}\xi \in [1,\infty ),\end{array}\right.$$

satisfying ${\displaystyle \frac{1}{p(·)}}={\displaystyle \frac{1}{q(·)}}+{\displaystyle \frac{\alpha q_{+}}{q(·)}}$ a.e. such that $I_{\alpha ,\gamma }:L^{p(·)}\left({\mathbb{R}}_{+}\right)\to L^{q(·)}\left({\mathbb{R}}_{+}\right)$ is continuous, or the Bessel–Riesz operator maps $L^{p(·)}\left({\mathbb{R}}_{+}\right)$ into $L^{q(·)}\left({\mathbb{R}}_{+}\right).$ Further, for the constant case, if we take $q=11/2$ and $\alpha =1/2,$ we have ${\displaystyle \frac{1}{q}}+\alpha <1$; thus, by following Corollary 3, there exist $p=22/15$ such that $I_{\alpha ,\gamma }:L^p\left({\mathbb{R}}_{+}\right)\to L^q\left({\mathbb{R}}_{+}\right)$ is continuous, or the Bessel–Riesz operator maps $L^p\left({\mathbb{R}}_{+}\right)$ into $L^q\left({\mathbb{R}}_{+}\right).$

Definition 4.

Let $p(·),q(·)\in \mathfrak{P}\left({\mathbb{R}}_{+}\right)$ and ${\displaystyle \frac{1}{p_{+}}}-{\displaystyle \frac{1}{q_{+}}}=\alpha$ then the pair $\left(p\right(·),q(·\left)\right)$ is said to satisfy the $({\displaystyle \frac{1}{p_{+}}}-{\displaystyle \frac{1}{q_{+}}}=\alpha )$-boundedness property if for any $\xi \in {⅁}_1=\{\xi \in {\mathbb{R}}_{+}:\mathcal{M}\mathcal{F}\left(\xi \right)\le C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\}$ the exponents satisfy ${\displaystyle \frac{1}{p(·)}}\ge {\displaystyle \frac{1}{q(·)}}+{\displaystyle \frac{\alpha q_{+}}{q(·)}}$ a.e., and for any $\xi \in {⅁}_2=\{\xi \in {\mathbb{R}}_{+}:\mathcal{M}\mathcal{F}\left(\xi \right)>C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\}$ the exponents satisfy ${\displaystyle \frac{1}{p(·)}}<{\displaystyle \frac{1}{q(·)}}+{\displaystyle \frac{\alpha q_{+}}{q(·)}}$ a.e.

Theorem 7.

Let $p(·),q(·)\in \mathfrak{P}\left({\mathbb{R}}_{+}\right),$ with ${\displaystyle \frac{1}{p_{+}}}-{\displaystyle \frac{1}{q_{+}}}=\alpha$ and the pair $\left(p\right(·),q(·\left)\right)$ satisfies the $({\displaystyle \frac{1}{p_{+}}}-{\displaystyle \frac{1}{q_{+}}}=\alpha )$-boundedness property with $p(.)\in LH\left({\mathbb{R}}_{+}\right)$ and $1<p_{-}\le p_{+}<\infty$; then, the Bessel–Riesz operator $I_{\alpha ,\gamma }:L^{p(·)}\left({\mathbb{R}}_{+}\right)\to L^{q(·)}\left({\mathbb{R}}_{+}\right)$ is bounded. Further, for any $\mathcal{F}\in L^{p(.)}\left({\mathbb{R}}_{+}\right)$ there exists a constant $C>0$, such that

$$\parallel I_{\alpha ,\gamma }{\mathcal{F}\parallel }_{L^{q(.)}\left({\mathbb{R}}_{+}\right)}\le C{\parallel \mathcal{F}\parallel }_{L^{p(.)}\left({\mathbb{R}}_{+}\right)}.$$

Proof. 

Since $1<p_{-}\le p_{+}<\infty$ and $p(·)\in LH\left({\mathbb{R}}_{+}\right),$ by the boundedness of maximal operator there exist $C_0>0$ such that for any $\mathcal{F}\in L^{p(·)}\left({\mathbb{R}}_{+}\right)$ we have

$${\parallel \mathcal{M}\mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\le C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}.$$

Now, for any $\xi \in {\mathbb{R}}_{+}$ and $\mathcal{F}\in L^{p(·)}\left({\mathbb{R}}_{+}\right)$, with $\mathcal{F}\ne 0$ (since the case $\mathcal{F}=0$ is trivial), and using Lemma 3 along with the fact that ${\displaystyle \frac{1}{p_{+}}}-{\displaystyle \frac{1}{q_{+}}}=\alpha ,$ we get

$$|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|\le C_5^{*}\left(r^{\alpha }\mathcal{M}\mathcal{F}\left(\xi \right)+r^{-{\displaystyle \frac{1}{q_{+}}}}{\parallel \mathcal{F}\parallel }_{L^{p(.)}\left({\mathbb{R}}_{+}\right)}\right).$$

Further, by choosing $r={\left({\displaystyle \frac{\mathcal{M}\mathcal{F}\left(\xi \right)}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)}^{-{\displaystyle \frac{q_{+}p\left(\xi \right)}{q\left(\xi \right)}}}$ and then by simplifying, we get

$$|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|\le C_5^{*}{\displaystyle \frac{C_0{\left(C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\right)}^{\frac{\alpha q_{+}p\left(\xi \right)}{q\left(\xi \right)}+\frac{p\left(\xi \right)}{q\left(\xi \right)}-1}+{\left(\mathcal{M}\mathcal{F}\left(\xi \right)\right)}^{\frac{\alpha q_{+}p\left(\xi \right)}{q\left(\xi \right)}+\frac{p\left(\xi \right)}{q\left(\xi \right)}-1}}{C_0{\left(\mathcal{M}\mathcal{F}\left(\xi \right)\right)}^{\frac{\alpha q_{+}p\left(\xi \right)}{q\left(\xi \right)}-1}{\left(C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\right)}^{\frac{p\left(\xi \right)}{q\left(\xi \right)}-1}}}.$$

For any $\xi \in {\mathbb{R}}_{+},$ we discuss following two possibilities. If $\xi \in {⅁}_1,$ then ${\displaystyle \frac{1}{p(·)}}\ge {\displaystyle \frac{1}{q(·)}}+{\displaystyle \frac{\alpha q_{+}}{q(·)}}$ a.e., thus ${\displaystyle \frac{\alpha q_{+}p\left(\xi \right)}{q\left(\xi \right)}}+{\displaystyle \frac{p\left(\xi \right)}{q\left(\xi \right)}}-1\le 0$ a.e. and we get

$$\begin{array}{c}\hfill |I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|\le C_5^{*}(C_0+1){\left({\displaystyle \frac{M\mathcal{F}\left(\xi \right)}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)}^{{\displaystyle \frac{p\left(\xi \right)}{q\left(\xi \right)}}}{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}.\end{array}$$

(6)

If $\xi \in {⅁}_2,$ then ${\displaystyle \frac{1}{p(·)}}<{\displaystyle \frac{1}{q(·)}}+{\displaystyle \frac{\alpha q_{+}}{q(·)}}$ a.e.; thus, ${\displaystyle \frac{\alpha q_{+}p\left(\xi \right)}{q\left(\xi \right)}}+{\displaystyle \frac{p\left(\xi \right)}{q\left(\xi \right)}}-1>0$ a.e., and we get

$$\begin{array}{c}\hfill |I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|\le C_5^{*}(C_0+1){\left({\displaystyle \frac{\mathcal{M}\mathcal{F}\left(\xi \right)}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)}^{{\displaystyle \frac{p\left(\xi \right)}{q\left(\xi \right)}}}{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}.\end{array}$$

(7)

By following (6) and (7), we get for any $\xi \in R_{+},$

$${\left({\displaystyle \frac{|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|}{C_6{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)}^{q\left(\xi \right)}\le {\left({\displaystyle \frac{\mathcal{M}\mathcal{F}\left(\xi \right)}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)}^{p\left(\xi \right)}.$$

From the definition of modular, we get

$${\Phi }_{q(·)}\left({\displaystyle \frac{I_{\alpha ,\gamma }\mathcal{F}}{C_6{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)\le {\Phi }_{p(·)}\left({\displaystyle \frac{\mathcal{M}\mathcal{F}}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right).$$

Thus, by following the boundedness of the Hardy–Littlewood maximal operator and the definition of the norm in variable Lebesgue spaces, we get

$${\Phi }_{q(·)}\left({\displaystyle \frac{I_{\alpha ,\gamma }\mathcal{F}}{C_6{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)\le {\Phi }_{p(·)}\left({\displaystyle \frac{\mathcal{M}\mathcal{F}}{C_0{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}}}\right)\le 1.$$

Thus,

$$C_6{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\in \{\lambda >0:{\Phi }_{q(·)}\left(\left(I_{\alpha ,\gamma }\mathcal{F}\right)/\lambda \right)\le 1\}.$$

Hence,

$$\parallel I_{\alpha ,\gamma }{\mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}\le C_6{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}.$$

□

Corollary 4.

Let $1<p<\infty ,1<q<\infty$ and ${\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{q}}=\alpha$ and the pair of exponents $(p,q)$ satisfies the $({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{q}}=\alpha )$-boundedness property then the Bessel–Riesz operator $I_{\alpha ,\gamma }:L^p\left({\mathbb{R}}_{+}\right)\to L^q\left({\mathbb{R}}_{+}\right)$ is bounded.

Proof. 

The proof follows from the proof of Theorem 7 using the exponent functions constant such that $p(·)=p$ and $q(·)=q.$ □

Remark 2.

Since Theorem 6 describes the boundedness of the Bessel–Riesz operators under the condition

$${\displaystyle \frac{1}{p(·)}}={\displaystyle \frac{1}{q(·)}}+{\displaystyle \frac{\alpha q_{+}}{q(·)}}\mathrm{a}.\mathrm{e}.,$$

whenever the exponent functions do not satisfy this condition, Theorem 7 provides an alternative result concerning the boundedness of the Bessel–Riesz operators. Moreover, Theorem 7 does not cover Theorem 6, as Theorem 7 requires an additional condition involving the Hardy–Littlewood maximal function, which is not needed in Theorem 6.

Theorem 8.

Let $p(·),q(·)\in \mathfrak{P}\left({\mathbb{R}}_{+}\right),$ with $p\left(\xi \right)=1$ for a.e. $\xi \in {\mathbb{R}}_{+}$ and ${\displaystyle \frac{1}{q_{+}}}=1-\alpha$; then, the Bessel–Riesz operator $I_{\alpha ,\gamma }:L^{p(·)}\left({\mathbb{R}}_{+}\right)\to L^{q(·)}\left({\mathbb{R}}_{+}\right)$ is $(1,q(·\left)\right)$ weak type; further, for any $\mathcal{F}\in L^{p(·)}\left({\mathbb{R}}_{+}\right)$ there exists a constant $C>0,$ such that

$$\parallel t{\chi }_{\{\xi \in {\mathbb{R}}_{+}:|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|>t\}}{\parallel }_{L^{q(·)}\left({\mathbb{R}}_{+}\right)}\le C{\parallel \mathcal{F}\parallel }_{L^{p(·)}\left({\mathbb{R}}_{+}\right)}.$$

Proof. 

For any $\mathcal{F}\in L^{p(·)}\left({\mathbb{R}}_{+}\right)$, since $p\left(\xi \right)=1$ for almost every $\xi \in {\mathbb{R}}_{+}$, it follows that $\mathcal{F}\in L^1\left({\mathbb{R}}_{+}\right)$. The case $\mathcal{F}=0$ is trivial, so we assume $\mathcal{F}\ne 0$. Now, by applying the dyadic decomposition technique, there exists an integer $N_r$ satisfying $2^{N_r}r\ge 1$ and $2^{N_r-1}r<1$, such that

$$\begin{array}{cc}\hfill I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)& ={\int }_{{\mathbb{R}}_{+}}{\displaystyle \frac{{|\xi -v|}^{\alpha -1}\mathcal{F}\left(v\right)}{{[1+|\xi -v\left|\right]}^{\gamma }}}dv=\sum _{k\in \mathbb{Z}}{\int }_{2^{k}r\le |\xi -v|<2^{k+1}r}{\displaystyle \frac{{|\xi -v|}^{\alpha -1}\mathcal{F}\left(v\right)}{{[1+|\xi -v\left|\right]}^{\gamma }}}dv\hfill \\ \hfill & =\sum _{k=-\infty }^{N_r-1}{\int }_{2^{k}r\le |\xi -v|<2^{k+1}r}{\displaystyle \frac{{|\xi -v|}^{\alpha -1}\mathcal{F}\left(v\right)}{{[1+|\xi -v\left|\right]}^{\gamma }}}dv+\sum _{k=N_r}^{\infty }{\int }_{2^{k}r\le |\xi -v|<2^{k+1}r}{\displaystyle \frac{{|\xi -v|}^{\alpha -1}\mathcal{F}\left(v\right)}{{[1+|\xi -v\left|\right]}^{\gamma }}}dv.\hfill \end{array}$$

Now, let

$$\begin{array}{cc}\hfill |g_1\left(\xi \right)|& \le \sum _{k=-\infty }^{N_r-1}{\int }_{2^{k}r\le |\xi -v|<2^{k+1}r}{\displaystyle \frac{{|\xi -v|}^{\alpha -1}\left|\mathcal{F}\left(v\right)\right|}{{[1+|\xi -v\left|\right]}^{\gamma }}}dv\hfill \\ \hfill & \le \mathcal{M}\mathcal{F}\left(\xi \right)\sum _{k=-\infty }^{N_r-1}{\displaystyle \frac{{\left(2^{k}r\right)}^{\alpha }}{{[1+2^{k}r]}^{\gamma }}}\le C_7{r}^{\alpha }\mathcal{M}\mathcal{F}\left(\xi \right).\hfill \end{array}$$

(8)

Also,

$$\begin{array}{cc}\hfill |g_2\left(\xi \right)|& \le \sum _{k=N_r}^{\infty }{\int }_{2^{k}r\le |\xi -v|<2^{k+1}r}{\displaystyle \frac{{|\xi -v|}^{\alpha -1}\left|\mathcal{F}\left(v\right)\right|}{{[1+|\xi -v\left|\right]}^{\gamma }}}dv\hfill \\ \hfill & \le {\parallel \mathcal{F}\parallel }_{L^1}\sum _{k=N_r}^{\infty }{\displaystyle \frac{{\left(2^{k}r\right)}^{\alpha -1}}{{[1+2^{k}r]}^{\gamma }}}\le C_8{r}^{\alpha -1}{\parallel \mathcal{F}\parallel }_{L^1}.\hfill \end{array}$$

(9)

Since for any $\xi \in {\mathbb{R}}_{+},$ we have

$$\mu \left(\{\xi \in {\mathbb{R}}_{+}:|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|>\beta \}\right)\le \mu \left(\{\xi \in {\mathbb{R}}_{+}:|g_1\left(\xi \right)|>\beta /2\}\right)+\mu \left(\{\xi \in {\mathbb{R}}_{+}:|g_2\left(\xi \right)|>\beta /2\}\right).$$

Also, by following the fact that the maximal operator satisfies the (1,1) weak-type relation and using (8), we get

$${\int }_{\{\xi \in {\mathbb{R}}_{+}:|g_1\left(\xi \right)|>\beta /2\}}d\xi \le {\int }_{\{\xi \in {\mathbb{R}}_{+}:\mathcal{M}\mathcal{F}\left(\xi \right)>\beta /\left(2C_7{r}^{\alpha }\right)\}}d\xi \le {\displaystyle \frac{2C_7{r}^{\alpha }}{\beta }}{\parallel \mathcal{F}\parallel }_{L^1\left({\mathbb{R}}_{+}\right)}.$$

Further, without loss of generality, if we choose ${\displaystyle \frac{\beta }{2}}=C_8{r}^{\alpha -1}{\parallel \mathcal{F}\parallel }_{L^1\left({\mathbb{R}}_{+}\right)},$ then by following (9), $|g_2\left(\xi \right)|\le \beta /2,$ thus $\mu \left(\{\xi \in {\mathbb{R}}_{+}:|g_2\left(\xi \right)|>\beta /2\}\right)=0.$ Hence, we get

$$\mu \left(\{\xi \in {\mathbb{R}}_{+}:|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|>\beta \}\right)\le {\displaystyle \frac{2C_7{r}^{\alpha }}{\beta }}{\parallel \mathcal{F}\parallel }_{L^1\left({\mathbb{R}}_{+}\right)}=C_8^{*}r.$$

Now for any $\beta >0,$ we choose $\lambda >0$, such that $0<\lambda <\beta .$ Thus, from the definition of modular, we have

$$\begin{array}{cc}\hfill {\Phi }_{q(·)}\left({\displaystyle \frac{\beta {\chi }_{\{\xi \in {\mathbb{R}}_{+}:|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|>\beta \}}}{\lambda }}\right)& ={\int }_{{\mathbb{R}}_{+}}{\left({\displaystyle \frac{\beta {\chi }_{\{\xi \in {\mathbb{R}}_{+}:|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|>\beta \}}}{\lambda }}\right)}^{q\left(\xi \right)}d\xi \hfill \\ \hfill & ={\int }_{\{\xi \in {\mathbb{R}}_{+}:|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|>\beta \}}{\left({\displaystyle \frac{\beta }{\lambda }}\right)}^{q\left(\xi \right)}d\xi \hfill \\ \hfill & \le {\left({\displaystyle \frac{\beta }{\lambda }}\right)}^{q_{+}}\mu \left(\{\xi \in {\mathbb{R}}_{+}:|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|>\beta \}\right)\le C_8^{*}r{\left({\displaystyle \frac{\beta }{\lambda }}\right)}^{q_{+}},\hfill \end{array}$$

because ${\displaystyle \frac{\beta }{\lambda }}\ge 1$, we have

$${\left({\displaystyle \frac{\beta }{\lambda }}\right)}^{q(·)}\le {\left({\displaystyle \frac{\beta }{\lambda }}\right)}^{q_{+}},$$

and

$${\int }_{\left\{\xi \in {\mathbb{R}}_{+}:\left|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)\right|>\beta \right\}}d\xi =\mu \left(\left\{\xi \in {\mathbb{R}}_{+}:\left|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)\right|>\beta \right\}\right).$$

Thus, using the fact ${\displaystyle \frac{1}{q_{+}}}=1-\alpha ,$ we have ${\Phi }_{q(·)}\left({\displaystyle \frac{\beta {\chi }_{\{\xi \in {\mathbb{R}}_{+}:|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|>\beta \}}}{\lambda }}\right)\le 1,$ if $\lambda \ge C_9{\parallel \mathcal{F}\parallel }_{L^1\left({\mathbb{R}}_{+}\right)}.$ Consequently,

$$\parallel \beta {\chi }_{\{\xi \in {\mathbb{R}}_{+}:|I_{\alpha ,\gamma }\mathcal{F}\left(\xi \right)|>\beta \}}{\parallel }_{L^{q(·)}\left({\mathbb{R}}_{+}\right)}\le C_9{\parallel \mathcal{F}\parallel }_{L^1\left({\mathbb{R}}_{+}\right)}.$$

□

4. Conclusions

In this manuscript, we have investigated fundamental properties of the Bessel–Riesz operator within the framework of variable Lebesgue spaces. Our main objective was to establish boundedness results for the Bessel–Riesz operator $I_{\alpha ,\gamma }$ under general conditions that go beyond the existing results in the literature, particularly those in [7]. In classical Lebesgue spaces, such boundedness results often rely on Young’s inequality, which is applicable due to the translation invariance of the underlying space. However, since variable exponent spaces lack this property, Young’s inequality is no longer valid in this setting. This challenge necessitated the development of alternative strategies that do not rely on the norm of the Bessel–Riesz kernel but instead utilize functional relations between domain and range spaces.

To address this, we constructed appropriate range spaces $L^{q(·)}\left({\mathbb{R}}_{+}\right)$ for given domain spaces $L^{p(·)}\left({\mathbb{R}}_{+}\right)$, and conversely identified suitable domain spaces for prescribed range spaces, ensuring the boundedness of the Bessel–Riesz operators in both directions. These theoretical findings were substantiated through detailed examples. Furthermore, we extended our results to more general contexts by incorporating the boundedness of the Hardy–Littlewood maximal operator and assuming log-Hölder continuity conditions. This extension allowed us to handle a broader class of exponent functions. Since the maximal operator is not known to be bounded when the essential infimum $p_{-}=1$, our results in that case do not apply directly. Therefore, we also established weak-type $(1,q(·\left)\right)$ boundedness results for the Bessel–Riesz operator to cover this limiting case.

In addition, we formulated corresponding results in the classical Lebesgue spaces, many of which are novel even in that well-established setting. The structure of this manuscript provides a comprehensive treatment of the subject, beginning with the necessary preliminaries and leading to general boundedness results along with their illustrative implications. Collectively, our results advance the theory of integral operators in variable exponent spaces and open avenues for future research in the study of convolution-type operators in non-standard functional frameworks.