Boundedness of Integral Operator of Generalized Bessel–Riesz Kernel in Variable Exponent Function Spaces

Abstract

In this manuscript, we introduce a class of measurable functions $\mathcal{A}\left({\mathbb{R}}_{+}\right)$, which is utilized to construct a generalized Bessel–Riesz kernel and the corresponding generalized Bessel–Riesz operator. We establish sufficient conditions ensuring that the generalized kernel belongs to variable Lebesgue spaces and derive a pointwise estimate for the associated operator in terms of the variable Lebesgue norm and the Hardy–Littlewood maximal operator. The main results provide boundedness criteria for the generalized Bessel–Riesz operator under appropriate assumptions on the exponent functions, as well as in more general settings where these conditions are relaxed. Furthermore, we demonstrate that boundedness results available in the literature can be recovered as special cases of our framework. In addition, we present an example that lies beyond the scope of existing results, thereby illustrating the wider applicability of our approach. In particular, when the exponent functions are constant, our results reduce to the classical Lebesgue space setting. Overall, this work extends and unifies a range of known results and provides a flexible framework for further developments in operator theory on variable Lebesgue spaces.

1. Introduction

In harmonic analysis, integral operators play a significant role. Among them, the Hardy–Littlewood maximal operator is of fundamental importance due to its ability to control singular integrals, thereby serving as a powerful tool for establishing a wide range of results in analysis. The Hardy–Littlewood maximal operator was first introduced by Hardy and Littlewood [1]. It gained further prominence through the seminal work of Calderón and Zygmund [2] on singular integrals, where the dyadic decomposition technique was also developed. The boundedness of the maximal operator on variable Lebesgue spaces has been extensively studied by various authors. Diening [3] was the first to establish a sufficient condition for its boundedness in the case when the essential supremum of the exponent function is finite. Subsequently, this problem has been extensively studied by many researchers under various settings and frameworks. For further details, the reader is referred to [4,5,6,7].

Moreover, another class of integral operators, known as the Bessel–Riesz operator, has attracted considerable attention from researchers. In this direction, Kurata et al. [8] investigated the boundedness of the product of two operators in generalized Morrey spaces, where one of the operators is the Bessel–Riesz operator. Further results concerning the boundedness of the Bessel operator in Lebesgue and Morrey spaces defined over metric measure spaces were established in [9]. Moreover, following a similar line of research, Nasir et al. [10,11] studied the boundedness of the Bessel–Riesz operator on variable Lebesgue spaces and obtained sufficient conditions ensuring its boundedness in this setting.

Furthermore, a generalized version of the Bessel–Riesz operator has also been widely studied. Saba et al. [12] analyzed the boundedness of the generalized Bessel–Riesz operator in Morrey spaces associated with different measures by employing the doubling condition. In the present manuscript, we investigate the boundedness of the generalized Bessel–Riesz operator in variable Lebesgue spaces. In contrast to [12], our analysis does not rely on the doubling condition. Instead, we introduce a class of measurable functions and, by using elements of this class, define a generalized Bessel–Riesz operator. We then establish its boundedness in variable Lebesgue spaces.

Moreover, we study a class of generalized Bessel–Riesz operators, which extend the classical Bessel–Riesz operators. Consequently, the corresponding results for the classical case are recovered as special instances of our generalized framework. Furthermore, our analysis is carried out in the setting of variable exponent Lebesgue spaces, which naturally extend the classical Lebesgue spaces and provide a more flexible analytical framework. This setting allows us to establish boundedness results within a more general and unified structure.

The organization of the manuscript is as follows. Section 2 is devoted to preliminary results and essential facts concerning variable Lebesgue spaces, including the Hardy–Littlewood maximal operator and its boundedness in this setting. Section 3 contains the main results of the paper. In this section, we first define the class $\mathcal{A}\left({\mathbb{R}}_{+}\right)$ of measurable functions, and then introduce the generalized Bessel–Riesz kernel and its associated operator. By employing the dyadic decomposition technique, Hölder’s inequality, the boundedness of the Hardy–Littlewood maximal operator, and properties of functions belonging to the class $\mathcal{A}\left({\mathbb{R}}_{+}\right)$, we establish the boundedness of the generalized Bessel–Riesz operator in variable Lebesgue spaces. We also show that certain known results (see [10]) can be obtained as special cases of our findings, and we provide an example that is not covered by the results in [10]. Finally, concluding remarks are presented in Section 4.

2. Preliminaries

Let $\mathsf{\Lambda }\subseteq {\mathbb{R}}^n$ and denote by $\mathcal{P}\left(\mathsf{\Lambda }\right)$ the class of all Lebesgue measurable exponent functions $p(·):\mathsf{\Lambda }\to [1,+\infty ]$. For any $p(·)\in \mathcal{P}\left(\mathsf{\Lambda }\right)$ and measurable set $E\subseteq \mathsf{\Lambda }$, we define

$$p_{-}\left(E\right):={essinf}_{\vartheta \in E}p\left(\vartheta \right),p_{+}\left(E\right):={esssup}_{\vartheta \in E}p\left(\vartheta \right).$$

Whenever the underlying set is clear, we simply write $p_{-}$ and $p_{+}$. For a given exponent function $p(·)$, its conjugate exponent $p^{\prime }(·)$ is defined pointwise by

$${\displaystyle \frac{1}{p\left(\vartheta \right)}}+{\displaystyle \frac{1}{p^{\prime }\left(\vartheta \right)}}=1,\vartheta \in \mathsf{\Lambda }.$$

Moreover, it follows that ${\left(p^{\prime }(·)\right)}_{+}={\left(p_{-}\right)}^{\prime }$ and ${\left(p^{\prime }(·)\right)}_{-}={\left(p_{+}\right)}^{\prime }$.

The following results on variable Lebesgue spaces $L^{p(·)}\left(\mathsf{\Lambda }\right)$ are adapted from [4,5]. Moreover, the following definition and proposition correspond to Definition 2.5 and Proposition 2.7 in [4].

Definition 1.

Let $p(·)\in \mathcal{P}\left(\mathsf{\Lambda }\right)$ with $p_{+}<+\infty$ and let ζ be a measurable function on Λ. The modular associated with $p(·)$ is defined by

$${\Psi }_{p(·)}\left(\zeta \right):={\int }_{\mathsf{\Lambda }}{\left|\zeta \left(\vartheta \right)\right|}^{p\left(\vartheta \right)}d\vartheta .$$

Proposition 1.

Let $\mathsf{\Lambda }\subseteq {\mathbb{R}}^n$ and $p(·)\in \mathcal{P}\left(\mathsf{\Lambda }\right)$. Then the modular ${\Psi }_{p(·)}$ satisfies the following properties:

• ${\Psi }_{p(·)}\left(\zeta \right)\ge 0$ and ${\Psi }_{p(·)}\left(\zeta \right)={\Psi }_{p(·)}\left(\right|\zeta \left|\right)$.
• ${\Psi }_{p(·)}\left(\zeta \right)=0$ if and only if $\zeta =0$ almost everywhere in Λ.
• If ${\Psi }_{p(·)}\left(\zeta \right)<+\infty$, then $\zeta \left(\vartheta \right)<+\infty$ for almost every $\vartheta \in \mathsf{\Lambda }$.
• ${\Psi }_{p(·)}$ is convex, that is, for $\alpha ,\beta >0$ with $\alpha +\beta =1$,

  $${\Psi }_{p(·)}(\alpha g+\beta \zeta )\le \alpha {\Psi }_{p(·)}\left(g\right)+\beta {\Psi }_{p(·)}\left(\zeta \right).$$
• ${\Psi }_{p(·)}$ is monotone in the sense that if $\left|g\right(\vartheta \left)\right|\ge \left|\zeta \right(\vartheta \left)\right|$ almost everywhere, then ${\Psi }_{p(·)}\left(g\right)\ge {\Psi }_{p(·)}\left(\zeta \right)$.
• For any $\lambda >0$ such that ${\Psi }_{p(·)}(\zeta /\lambda )<+\infty$, the mapping $\nu \mapsto {\Psi }_{p(·)}(\zeta /\nu )$ is continuous and decreasing on $[\lambda ,+\infty )$, and moreover ${\Psi }_{p(·)}(\zeta /\nu )\to 0$ as $\nu \to +\infty$.

The following two definitions appear as Definition 2.8 and Definition 2.12, respectively, in [4].

Definition 2.

For $p(·)\in \mathcal{P}\left(\mathsf{\Lambda }\right)$, the variable Lebesgue space $L^{p(·)}\left(\mathsf{\Lambda }\right)$ is defined as

$$L^{p(·)}\left(\mathsf{\Lambda }\right):=\left\{\zeta \mathit{measurable}:{\Psi }_{p(·)}\left({\displaystyle \frac{\zeta }{\lambda }}\right)<+\infty ,\mathrm{for}\mathrm{some}\lambda >0\right\}.$$

Definition 3.

Let $p(·)\in \mathcal{P}\left(\mathsf{\Lambda }\right)$. The Luxemburg norm on $L^{p(·)}\left(\mathsf{\Lambda }\right)$ is given by

$${\parallel \zeta \parallel }_{L^{p(·)}}:=\inf \left\{\lambda >0:{\Psi }_{p(·)}\left({\displaystyle \frac{\zeta }{\lambda }}\right)\le 1\right\}.$$

With this norm, $L^{p(·)}\left(\mathsf{\Lambda }\right)$ becomes a normed space.

The following two propositions are presented as Proposition 2.10 and Proposition 2.15, respectively, in [4].

Proposition 2.

Let $\mathsf{\Lambda }\subseteq {\mathbb{R}}^n$ and $p(·)\in \mathcal{P}\left(\mathsf{\Lambda }\right)$ with $p_{+}<+\infty$. Then for every $\kappa \ge 1$,

$${\kappa }^{p_{-}}{\Psi }_{p(·)}\left(\zeta \right)\le {\Psi }_{p(·)}\left(\kappa \zeta \right)\le {\kappa }^{p_{+}}{\Psi }_{p(·)}\left(\zeta \right),$$

while for $0<\kappa <1$, the inequalities are reversed.

Proposition 3.

Let $\mathsf{\Lambda }\subseteq {\mathbb{R}}^n$ and $p(·)\in \mathcal{P}\left(\mathsf{\Lambda }\right)$. If $\zeta \in L^{p(·)}\left(\mathsf{\Lambda }\right)$ with ${\parallel \zeta \parallel }_{L^{p(·)}}>0$, then

$${\Psi }_{p(·)}\left({\displaystyle \frac{\zeta }{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)\le 1.$$

If in addition $p_{+}<+\infty$, then equality holds for all non-zero $\zeta \in L^{p(·)}\left(\mathsf{\Lambda }\right)$.

The following corollary appears as Corollary 2.17 in [4].

Corollary 1.

Let $p(·)\in \mathcal{P}\left(\mathsf{\Lambda }\right)$ with $p_{+}<+\infty$. If ${\parallel \zeta \parallel }_{L^{p(·)}}>1$, then

$${\Psi }_{p(·)}{\left(\zeta \right)}^{1/p_{+}}\le {\parallel \zeta \parallel }_{L^{p(·)}}\le {\Psi }_{p(·)}{\left(\zeta \right)}^{1/p_{-}},$$

whereas for ${0<\parallel \zeta \parallel }_{L^{p(·)}}\le 1$,

$${\Psi }_{p(·)}{\left(\zeta \right)}^{1/p_{-}}\le {\parallel \zeta \parallel }_{L^{p(·)}}\le {\Psi }_{p(·)}{\left(\zeta \right)}^{1/p_{+}}.$$

If $p(·)$ is constant, i.e., $p_{-}=p_{+}=p$, then the norm reduces to

$${\parallel \zeta \parallel }_{L^p}={\left({\int }_{\mathsf{\Lambda }}{\left|\zeta \left(\vartheta \right)\right|}^{p}d\vartheta \right)}^{1/p},$$

which is the usual norm on classical Lebesgue spaces.

The following Hölder inequality appears as Theorem 2.33 in [4].

Theorem 1

(Hölder’s Inequality). Let $\mathsf{\Lambda }\subseteq {\mathbb{R}}^n$ and $p(·)\in \mathcal{P}\left(\mathsf{\Lambda }\right)$. If ${\zeta }_1\in L^{p(·)}\left(\mathsf{\Lambda }\right)$ and ${\zeta }_2\in L^{p^{\prime }(·)}\left(\mathsf{\Lambda }\right)$, then ${\zeta }_1{\zeta }_2\in L^1\left(\mathsf{\Lambda }\right)$ and

$${\int }_{\mathsf{\Lambda }}|{\zeta }_1\left(\vartheta \right){\zeta }_2\left(\vartheta \right)|d\vartheta \le C\parallel {\zeta }_1{\parallel }_{L^{p(·)}}{\parallel {\zeta }_2\parallel }_{L^{p^{\prime }(·)}},$$

where C is a constant depending only on $p(·)$.

The following definition appears as Definition 2.2 in [4].

Definition 4.

An exponent function $p(·):\mathsf{\Lambda }\to \mathbb{R}$ is said to satisfy the local log-Hölder continuity condition, denoted by $p(·)\in L{H}_0\left(\mathsf{\Lambda }\right)$, if there exists a constant $C_0>0$ such that for all ${\vartheta }_1,{\vartheta }_2\in \mathsf{\Lambda }$ with $|{\vartheta }_1-{\vartheta }_2|<{\displaystyle \frac{1}{2}}$,

$$|p\left({\vartheta }_1\right)-p\left({\vartheta }_2\right)|\le {\displaystyle \frac{C_0}{-\log |{\vartheta }_1-{\vartheta }_2|}}.$$

It is said to satisfy the log-Hölder condition at infinity, denoted by $p(·)\in L{H}_{\infty }\left(\mathsf{\Lambda }\right)$, if there exist constants $C_{\infty }>0$ and $p_{\infty }$ such that

$$|p\left(\vartheta \right)-p_{\infty }|\le {\displaystyle \frac{C_{\infty }}{\log (e+|\vartheta \left|\right)}},\vartheta \in \mathsf{\Lambda }.$$

If both conditions hold, we write $p(·)\in LH\left(\mathsf{\Lambda }\right)$.

The following results concerning maximal operators are taken from Chapter 3 of [5].

Definition 5.

Let $\zeta \in L_{\mathrm{loc}}^1\left({\mathbb{R}}^n\right)$. The Hardy–Littlewood maximal operator $\mathcal{M}$ is defined by

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

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

Theorem 2.

Let $p(·)\in \mathcal{P}\left({\mathbb{R}}^n\right)$ such that $1/p(·)\in LH\left({\mathbb{R}}^n\right)$. Then there exists a constant $C>0$ such that

$$\parallel {\chi }_{\{\vartheta :\mathcal{M}\zeta (\vartheta )>t\}}{\parallel }_{L^{p(·)}}\le C{\parallel \zeta \parallel }_{L^{p(·)}}.$$

Moreover, if $p_{-}>1$, then

$${\parallel \mathcal{M}\zeta \parallel }_{L^{p(·)}}\le C{\parallel \zeta \parallel }_{L^{p(·)}}.$$

The constant C depends on n, the log-Hölder continuity of $1/p(·)$, and the parameters $p_{-}$ and $p_{\infty }$.

Remark 1.

If $p_{+}<+\infty$, then the condition $1/p(·)\in LH\left(\mathsf{\Lambda }\right)$ is equivalent to $p(·)\in LH\left(\mathsf{\Lambda }\right)$.

3. Main Results

Throughout this section, we assume that $1<p_{-}\le p_{+}<\infty$, $1\le q_{-}\le q_{+}<\infty$, and $1<s_{-}\le s_{+}<\infty$. Moreover, we denote ${\mathbb{R}}_{+}=[0,\infty )$. It is worth noting that constants denoted by the same symbol in different results may represent different values.

Definition 6.

Let $\mathcal{A}\left({\mathbb{R}}_{+}\right)$ be the collection of measurable functions $\rho :[0,+\infty )\to [0,+\infty )$ satisfying:

1. 

$\rho \left(1\right)=1$;

2. 

ρ is non-decreasing on $(0,+\infty )$;

3. 

${\displaystyle \frac{\rho \left(t\right)}{t}}$ is non-increasing on $(0,+\infty )$.

Then for any $\rho (·)\in \mathcal{A}\left({\mathbb{R}}_{+}\right),$ we define the generalized Bessel–Riesz kernel for $0<\gamma <+\infty$ as follows:

$$K_{\rho ,\gamma }\left(v\right)=\left\{\begin{array}{cc}{\displaystyle \frac{\rho \left(\vartheta \right)}{\vartheta {(1+\vartheta )}^{\gamma }}},\hfill & \vartheta \in {\mathbb{R}}_{+}\setminus \left\{0\right\},\hfill \\ 0,\hfill & \vartheta =0.\hfill \end{array}\right.$$

Further, for any $\zeta \in L^{p(·)}\left({\mathbb{R}}_{+}\right)$, we define the generalized Bessel–Riesz operator for $\vartheta \in {\mathbb{R}}_{+}$ as

$$\begin{array}{cc}\hfill I_{\rho ,\gamma }\zeta \left(\vartheta \right)=(K_{\rho ,\gamma }\ast \zeta )\left(\vartheta \right)& ={\int }_{{\mathbb{R}}_{+}}{K}_{\rho ,\gamma }\left(|\vartheta -v|\right)\zeta \left(v\right)dv\hfill \\ \hfill & ={\int }_{{\mathbb{R}}_{+}}{\displaystyle \frac{\rho \left(\right|\vartheta -v\left|\right)}{{|\vartheta -v|[1+|\vartheta -v\left|\right]}^{\gamma }}}\zeta \left(v\right)dv.\hfill \end{array}$$

Lemma 1.

For any $\rho (·)\in \mathcal{A}\left({\mathbb{R}}_{+}\right)$ and $s(·)\in \mathcal{P}\left({\mathbb{R}}_{+}\right),$ if

$${\int }_0^1{\left({\displaystyle \frac{\rho \left(\vartheta \right)}{\vartheta }}\right)}^{s_{+}}d\vartheta +{\int }_1^{+\infty }{\left({\displaystyle \frac{\rho \left(\vartheta \right)}{{\vartheta }^{1+\gamma }}}\right)}^{s_{-}}d\vartheta <+\infty ,$$

(1)

then the generalized Bessel–Riesz kernel belongs to the variable Lebesgue space $L^{s(·)}\left({\mathbb{R}}_{+}\right).$

Proof. 

By definition of the modular and generalized Bessel–Riesz kernel, we have

$$\begin{array}{cc}\hfill {\Psi }_{s(·)}\left(K_{\rho ,\gamma }\right)& ={\int }_{{\mathbb{R}}_{+}}{K}_{\rho ,\gamma }{\left(\vartheta \right)}^{s\left(\vartheta \right)}d\vartheta ={\int }_{{\mathbb{R}}_{+}}{\left({\displaystyle \frac{\rho \left(\vartheta \right)}{\vartheta {(1+\vartheta )}^{\gamma }}}\right)}^{s\left(\vartheta \right)}d\vartheta \hfill \\ \hfill & ={\int }_0^1{\left({\displaystyle \frac{\rho \left(\vartheta \right)}{\vartheta {(1+\vartheta )}^{\gamma }}}\right)}^{s\left(\vartheta \right)}d\vartheta +{\int }_1^{+\infty }{\left({\displaystyle \frac{\rho \left(\vartheta \right)}{\vartheta {(1+\vartheta )}^{\gamma }}}\right)}^{s\left(\vartheta \right)}d\vartheta \hfill \\ \hfill & \le {\int }_0^1{\left({\displaystyle \frac{\rho \left(\vartheta \right)}{\vartheta }}\right)}^{s\left(\vartheta \right)}d\vartheta +{\int }_1^{+\infty }{\left({\displaystyle \frac{\rho \left(\vartheta \right)}{{\vartheta }^{1+\gamma }}}\right)}^{s\left(\vartheta \right)}d\vartheta \hfill \\ \hfill & \le {\int }_0^1{\left({\displaystyle \frac{\rho \left(\vartheta \right)}{\vartheta }}\right)}^{s_{+}}d\vartheta +{\int }_1^{+\infty }{\left({\displaystyle \frac{\rho \left(\vartheta \right)}{{\vartheta }^{1+\gamma }}}\right)}^{s_{-}}d\vartheta ,\hfill \end{array}$$

because $\rho \left(1\right)=1$ and $\rho \left(\vartheta \right)/\vartheta$ is non-increasing. Further, by following the given condition, we get ${\Psi }_{s(·)}\left(K_{\rho ,\gamma }\right)<+\infty$; thus $K_{\rho ,\gamma }\in L^{s(·)}\left({\mathbb{R}}_{+}\right)$.    □

Lemma 2.

Let the relation (1) hold. Then there exist a positive integer $N_r$, depending on r, and a constant $C>0$ such that the following assertions hold.

• If $0<\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}<1$, then

$$\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}^{{\displaystyle \frac{1}{s_{+}}}}\ge C{\displaystyle \frac{\rho {\left(2^{k}r\right)}^{\frac{s_{-}}{s_{+}}}{\left(2^{k}r\right)}^{\frac{1-s_{-}}{s_{+}}}}{{(1+2^{k}r)}^{\gamma }}},k<N_r,$$

and

$$\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}^{{\displaystyle \frac{1}{s_{+}}}}\ge C{\displaystyle \frac{\rho \left(2^{k}r\right){\left(2^{k}r\right)}^{\frac{1-s_{+}}{s_{+}}}}{{(1+2^{k}r)}^{\gamma }}},k\ge N_r.$$

Moreover, if $\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}\ge 1$, then

$$\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}\ge C{\displaystyle \frac{\rho {\left(2^{k}r\right)}^{\frac{s_{-}}{s_{+}}}{\left(2^{k}r\right)}^{\frac{1-s_{-}}{s_{+}}}}{{(1+2^{k}r)}^{\gamma }}},k<N_r,$$

and

$$\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}\ge C{\displaystyle \frac{\rho \left(2^{k}r\right){\left(2^{k}r\right)}^{\frac{1-s_{+}}{s_{+}}}}{{(1+2^{k}r)}^{\gamma }}},k\ge N_r.$$

Proof. 

By definition of the modular and properties of $\rho (·)\in \mathcal{A}\left({\mathbb{R}}_{+}\right)$, we have

$$\begin{array}{cc}\hfill {\Psi }_{s(·)}\left(K_{\rho ,\gamma }\right)& ={\int }_{{\mathbb{R}}_{+}}{\left({\displaystyle \frac{\rho \left(\vartheta \right)}{\vartheta {(1+\vartheta )}^{\gamma }}}\right)}^{s\left(\vartheta \right)}d\vartheta \ge {\int }_{{\mathbb{R}}_{+}}{\displaystyle \frac{\rho {\left(\vartheta \right)}^{s\left(\vartheta \right)}}{{\vartheta }^{s\left(\vartheta \right)}{(1+\vartheta )}^{\gamma s_{+}}}}d\vartheta \hfill \\ \hfill & =\sum _{k\in \mathbb{Z}}{\int }_{2^{k}r\le \vartheta <2^{k+1}r}{\displaystyle \frac{\rho {\left(\vartheta \right)}^{s\left(\vartheta \right)}}{{\vartheta }^{s\left(\vartheta \right)}{(1+\vartheta )}^{\gamma s_{+}}}}d\vartheta \hfill \\ \hfill & \ge \sum _{k\in \mathbb{Z}}{\displaystyle \frac{1}{{(1+2^{k+1}r)}^{\gamma s_{+}}}}{\int }_{2^{k}r\le \vartheta <2^{k+1}r}{\left({\displaystyle \frac{\rho \left(\vartheta \right)}{\vartheta }}\right)}^{s\left(\vartheta \right)}d\vartheta ,\hfill \end{array}$$

and there exists an integer $N_r,$ such that $2^{N_r}r\ge 1,$ and $2^{N_r-1}r<1.$ Thus we have

$$\begin{array}{cc}\hfill {\Psi }_{s(·)}\left(K_{\rho ,\gamma }\right)& \ge {\displaystyle \sum _{k=-\infty }^{N_r-1}}{\displaystyle \frac{1}{{\left(1+2^{k+1}r\right)}^{\gamma s_{+}}}}{\int }_{2^{k}r\le \vartheta <2^{k+1}r}{\left({\displaystyle \frac{\rho \left(\vartheta \right)}{\vartheta }}\right)}^{s_{-}}d\vartheta \hfill \\ \hfill & +{\displaystyle \sum _{k=N_r}^{+\infty }}{\displaystyle \frac{1}{{\left(1+2^{k+1}r\right)}^{\gamma s_{+}}}}{\int }_{2^{k}r\le \vartheta <2^{k+1}r}{\left({\displaystyle \frac{\rho \left(\vartheta \right)}{\vartheta }}\right)}^{s_{+}}d\vartheta \hfill \\ \hfill & \ge {\displaystyle \sum _{k=-\infty }^{N_r-1}}{\displaystyle \frac{\rho {\left(2^{k+1}r\right)}^{s_{-}}}{{\left(2^{k+1}r\right)}^{s_{-}}{\left(1+2^{k+1}r\right)}^{\gamma s_{+}}}}{\int }_{2^{k}r\le \vartheta <2^{k+1}r}d\vartheta \hfill \\ \hfill & +{\displaystyle \sum _{k=N_r}^{+\infty }}{\displaystyle \frac{\rho {\left(2^{k+1}r\right)}^{s_{+}}}{{\left(2^{k+1}r\right)}^{s_{+}}{\left(1+2^{k+1}r\right)}^{\gamma s_{+}}}}{\int }_{2^{k}r\le \vartheta <2^{k+1}r}d\vartheta \hfill \\ \hfill & \ge C_0\left({\displaystyle \sum _{k=-\infty }^{N_r-1}}{\displaystyle \frac{\rho {\left(2^{k}r\right)}^{s_{-}}\left(2^{k}r\right)}{{\left(2^{k}r\right)}^{s_{-}}{\left(1+2^{k}r\right)}^{\gamma s_{+}}}}+{\displaystyle \sum _{k=N_r}^{+\infty }}{\displaystyle \frac{\rho {\left(2^{k}r\right)}^{s_{+}}\left(2^{k}r\right)}{{\left(2^{k}r\right)}^{s_{+}}{\left(1+2^{k}r\right)}^{\gamma s_{+}}}}\right).\hfill \end{array}$$

Thus for any $k<N_r,$ we have

$${\Psi }_{s(·)}\left(K_{\rho ,\gamma }\right)\ge C_0{\displaystyle \sum _{k=-\infty }^{N_r-1}}{\displaystyle \frac{\rho {\left(2^{k}r\right)}^{s_{-}}\left(2^{k}r\right)}{{\left(2^{k}r\right)}^{s_{-}}{\left(1+2^{k}r\right)}^{\gamma s_{+}}}}\ge C_0{\displaystyle \frac{\rho {\left(2^{k}r\right)}^{s_{-}}\left(2^{k}r\right)}{{\left(2^{k}r\right)}^{s_{-}}{\left(1+2^{k}r\right)}^{\gamma s_{+}}}}.$$

(2)

Similarly, for any $k\ge N_r,$ we have

$$\begin{array}{cc}\hfill {\Psi }_{s(·)}\left(K_{\rho ,\gamma }\right)& \ge C_0{\displaystyle \sum _{k=N_r}^{+\infty }}{\displaystyle \frac{\rho {\left(2^{k}r\right)}^{s_{+}}\left(2^{k}r\right)}{{\left(2^{k}r\right)}^{s_{+}}{\left(1+2^{k}r\right)}^{\gamma s_{+}}}}\ge C_0{\displaystyle \frac{\rho {\left(2^{k}r\right)}^{s_{+}}\left(2^{k}r\right)}{{\left(2^{k}r\right)}^{s_{+}}{\left(1+2^{k}r\right)}^{\gamma s_{+}}}}.\hfill \end{array}$$

(3)

Since $s_{+}<+\infty ,$ for $0<\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(.)}}<1,$ by following Propositions 2 and 3, we get

$$1={\Psi }_{s(·)}\left(K_{\rho ,\gamma }/{\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}\right)\ge {\displaystyle \frac{1}{\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}^{s_{-}}}}{\Psi }_{s(·)}\left(K_{\rho ,\gamma }\right),$$

or

$$\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}\ge \parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}^{s_{-}}\ge {\Psi }_{s(·)}\left(K_{\rho ,\gamma }\right),$$

and thus by following (2), for any $k<N_r,$

$$\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}^{{\displaystyle \frac{1}{s_{+}}}}\ge C{\displaystyle \frac{\rho {\left(2^{k}r\right)}^{\frac{s_{-}}{s_{+}}}{\left(2^{k}r\right)}^{\frac{1-s_{-}}{s_{+}}}}{{\left(1+2^{k}r\right)}^{\gamma }}}$$

and by following (3), for any $k\ge N_r,$

$$\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}^{{\displaystyle \frac{1}{s_{+}}}}\ge C{\displaystyle \frac{\rho \left(2^{k}r\right){\left(2^{k}r\right)}^{\frac{1-s_{+}}{s_{+}}}}{{\left(1+2^{k}r\right)}^{\gamma }}}.$$

Further, for $\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(.)}}\ge 1,$ by following Propositions 2 and 3, we get

$$\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}^{s_{+}}\ge {\Psi }_{s(·)}\left(K_{\rho ,\gamma }\right),$$

and thus by following (2) and (3), we get the required estimates.    □

Lemma 3.

For any $\vartheta \in {\mathbb{R}}^{+}$ and $\zeta \in L^{p(·)}\left({\mathbb{R}}_{+}\right),$ the following pointwise estimation holds for the generalized Bessel–Riesz operator:

$$|I_{\rho ,\gamma }\zeta \left(\vartheta \right)|\le C\left(\mathcal{M}\zeta \left(\vartheta \right){\displaystyle \sum _{k=-\infty }^{N_r-1}}{\displaystyle \frac{\rho \left(2^{k}r\right)}{{(1+2^{k}r)}^{\gamma }}}+{\parallel \zeta \parallel }_{L^{p(.)}}{\displaystyle \sum _{k=N_r}^{+\infty }}{\displaystyle \frac{\rho \left(2^{k}r\right){\left(2^{k}r\right)}^{-\frac{1}{p_{+}}}}{{(1+2^{k}r)}^{\gamma }}}\right).$$

Proof. 

Since for any $\vartheta \in {\mathbb{R}}^{+}$ and $\zeta \in L^{p(·)}\left({\mathbb{R}}^{+}\right),$ by definition of the generalized Bessel–Riesz operator, we have

$$I_{\rho ,\gamma }\zeta \left(\vartheta \right)={\int }_{{\mathbb{R}}^{+}}{K}_{\rho ,\gamma }\left(\right|\vartheta -v\left|\right)\zeta \left(v\right)dv=\sum _{k\in \mathbb{Z}}{\int }_{2^{k}r\le |\vartheta -v|<2^{k+1}r}{K}_{\rho ,\gamma }\left(\right|\vartheta -v\left|\right)\zeta \left(v\right)dv,$$

there exists an integer $N_r,$ such that $2^{N_r}r\ge 1,$ and $2^{N_r-1}r<1$; thus we have

$$\begin{array}{cc}\hfill I_{\rho ,\gamma }\zeta \left(\vartheta \right)& ={\displaystyle \sum _{k=-\infty }^{N_r-1}}{\int }_{2^{k}r\le |\vartheta -v|<2^{k+1}r}{K}_{\rho ,\gamma }\left(\right|\vartheta -v\left|\right)\zeta \left(v\right)dv\hfill \\ \hfill & +{\displaystyle \sum _{k=N_r}^{+\infty }}{\int }_{2^{k}r\le |\vartheta -v|<2^{k+1}r}{K}_{\rho ,\gamma }\left(\right|\vartheta -v\left|\right)\zeta \left(v\right)dv.\hfill \end{array}$$

(4)

Now, by the properties of $\rho (·)\in \mathcal{A}\left({\mathbb{R}}_{+}\right)$ and definition of the maximal operator, we get

$$\begin{array}{cc}\hfill |I_1\left(\vartheta \right)|& \le {\displaystyle \sum _{k=-\infty }^{N_r-1}}{\int }_{2^{k}r\le |\vartheta -v|<2^{k+1}r}{K}_{\rho ,\gamma }\left(\right|\vartheta -v\left|\right)\left|\zeta \left(v\right)\right|dv\hfill \\ \hfill & ={\displaystyle \sum _{k=-\infty }^{N_r-1}}{\int }_{2^{k}r\le |\vartheta -v|<2^{k+1}r}{\displaystyle \frac{\rho \left(\right|\vartheta -v\left|\right)\left|\zeta \right(v\left)\right|}{{|\vartheta -v|[1+|\vartheta -v\left|\right]}^{\gamma }}}dv\hfill \\ \hfill & \le {\displaystyle \sum _{k=-\infty }^{N_r-1}}{\displaystyle \frac{\rho \left(2^{k}r\right)}{{(1+2^{k}r)}^{\gamma }}}{\displaystyle \frac{1}{2^{k}r}}{\int }_{2^{k}r\le |\vartheta -v|<2^{k+1}r}\left|\zeta \left(v\right)\right|dv\hfill \\ \hfill & \le \mathcal{M}\zeta \left(\vartheta \right){\displaystyle \sum _{k=-\infty }^{N_r-1}}{\displaystyle \frac{\rho \left(2^{k}r\right)}{{(1+2^{k}r)}^{\gamma }}}.\hfill \end{array}$$

(5)

Further,

$$\begin{array}{cc}\hfill |I_2\left(\vartheta \right)|& \le {\displaystyle \sum _{k=N_r}^{+\infty }}{\int }_{2^{k}r\le |\vartheta -v|<2^{k+1}r}{K}_{\rho ,\gamma }\left(\right|\vartheta -v\left|\right)\left|\zeta \left(v\right)\right|dv\hfill \\ \hfill & ={\displaystyle \sum _{k=N_r}^{+\infty }}{\int }_{2^{k}r\le |\vartheta -v|<2^{k+1}r}{\displaystyle \frac{\rho \left(\right|\vartheta -v\left|\right)\left|\zeta \right(v\left)\right|}{{|\vartheta -v|[1+|\vartheta -v\left|\right]}^{\gamma }}}dv\hfill \\ \hfill & \le {\displaystyle \sum _{k=N_r}^{+\infty }}{\displaystyle \frac{\rho \left(2^{k}r\right)}{2^{k}r{(1+2^{k}r)}^{\gamma }}}{\int }_{2^{k}r\le |\vartheta -v|<2^{k+1}r}\left|\zeta \left(v\right)\right|dv.\hfill \end{array}$$

By using Hölder’s inequality and the fact that ${\left(p^{\prime }(·)\right)}_{-}={\left(p_{+}\right)}^{\prime }$, we get

$$\begin{array}{cc}\hfill |I_2\left(\vartheta \right)|& \le C_0{\parallel \zeta \parallel }_{L^{p(·)}}{\displaystyle \sum _{k=N_r}^{+\infty }}{\displaystyle \frac{\rho \left(2^{k}r\right){\left(2^{k}r\right)}^{\frac{1}{{\left(p^{\prime }\right)}_{-}}}}{2^{k}r{(1+2^{k}r)}^{\gamma }}}=C_0{\parallel \zeta \parallel }_{L^{p(·)}}{\displaystyle \sum _{k=N_r}^{+\infty }}{\displaystyle \frac{\rho \left(2^{k}r\right){\left(2^{k}r\right)}^{-\frac{1}{p_{+}}}}{{(1+2^{k}r)}^{\gamma }}}.\hfill \end{array}$$

(6)

Thus by combining (4)–(6), we get

$$\begin{array}{c}\hfill |I_{\rho ,\gamma }\zeta \left(\vartheta \right)|\le C\left(\mathcal{M}\zeta \left(\vartheta \right){\displaystyle \sum _{k=-\infty }^{N_r-1}}{\displaystyle \frac{\rho \left(2^{k}r\right)}{{(1+2^{k}r)}^{\gamma }}}+{\parallel \zeta \parallel }_{L^{p(·)}}{\displaystyle \sum _{k=N_r}^{+\infty }}{\displaystyle \frac{\rho \left(2^{k}r\right){\left(2^{k}r\right)}^{-\frac{1}{p_{+}}}}{{(1+2^{k}r)}^{\gamma }}}\right).\end{array}$$

 □

Theorem 3.

Let $p(·),q(·),s(·)\in \mathcal{P}\left({\mathbb{R}}^{+}\right)$ and (1) hold; if ${\displaystyle \frac{1}{q_{+}}}\le c\left({\displaystyle \frac{1}{p_{+}}}+{\displaystyle \frac{1}{s_{+}}}-1\right)$ and ${\displaystyle \frac{q(·)}{p(·)}}=1+{\displaystyle \frac{c{q}_{+}(s_{-}-1)}{s_{+}}}$ a.e. for some constant $c>0,$ and further, $p(.)\in LH\left({\mathbb{R}}_{+}\right)$, then the generalized Bessel–Riesz operator $I_{\rho ,\gamma }:L^{p(·)}\left({\mathbb{R}}_{+}\right)\to L^{q(·)}\left({\mathbb{R}}_{+}\right)$ is bounded. Moreover, for any $\zeta \in L^{p(.)}\left({\mathbb{R}}^{+}\right)$ there exists a positive constant $C>0$, such that for $0<\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(.)}}<1,$ we have

$$\parallel I_{\rho ,\gamma }{\zeta \parallel }_{L^{q(·)}}\le C\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}^{1/s_{+}}{\parallel \zeta \parallel }_{L^{p(·)}}$$

and for $\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}\ge 1,$ we have

$$\parallel I_{\rho ,\gamma }{\zeta \parallel }_{L^{q(·)}}\le C\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}{\parallel \zeta \parallel }_{L^{p(·)}}.$$

Proof. 

First we find two estimations depending upon the norm of the generalized Bessel–Riesz kernel. If $0<\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(.)}}<1,$ then by following Lemma 2, we have

$$\begin{array}{cc}\hfill \mathcal{M}\zeta \left(\vartheta \right){\displaystyle \sum _{k=-\infty }^{N_r-1}}{\displaystyle \frac{\rho \left(2^{k}r\right)}{{(1+2^{k}r)}^{\gamma }}}& \le C_0\mathcal{M}\zeta \left(\vartheta \right){\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}^{1/s_{+}}{\displaystyle \sum _{k=-\infty }^{N_r-1}}{\displaystyle \frac{\rho {\left(2^{k}r\right)}^{1-\frac{s_{-}}{s_{+}}}}{{\left(2^{k}r\right)}^{\frac{1-s_{-}}{s_{+}}}}},\hfill \end{array}$$

and since $\rho (·)$ in non-decreasing and $\rho \left(1\right)=1,$

$$\mathcal{M}\zeta \left(\vartheta \right){\displaystyle \sum _{k=-\infty }^{N_r-1}}{\displaystyle \frac{\rho \left(2^{k}r\right)}{{(1+2^{k}r)}^{\gamma }}}\le C_0{r}^{{\displaystyle \frac{s_{-}-1}{s_{+}}}}\mathcal{M}\zeta \left(\vartheta \right){\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}^{1/s_{+}}{\displaystyle \sum _{k=-\infty }^{N_r-1}}{2}^{{\displaystyle \frac{k}{s_{+}}}(s_{-}-1)},$$

and since $s_{-}>1,$

$$\mathcal{M}\zeta \left(\vartheta \right){\displaystyle \sum _{k=-\infty }^{N_r-1}}{\displaystyle \frac{\rho \left(2^{k}r\right)}{{(1+2^{k}r)}^{\gamma }}}\le C_1{r}^{{\displaystyle \frac{s_{-}-1}{s_{+}}}}\mathcal{M}\zeta \left(\vartheta \right){\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}^{1/s_{+}}.$$

(7)

Further, by following Lemma 2,

$${\parallel \zeta \parallel }_{L^{p(·)}}{\displaystyle \sum _{k=N_r}^{+\infty }}{\displaystyle \frac{\rho \left(2^{k}r\right){\left(2^{k}r\right)}^{-\frac{1}{p_{+}}}}{{(1+2^{k}r)}^{\gamma }}}\le C_2{\parallel \zeta \parallel }_{L^{p(·)}}{\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}^{1/s_{+}}{\displaystyle \sum _{k=N_r}^{+\infty }}{\displaystyle \frac{{\left(2^{k}r\right)}^{-\frac{1}{p_{+}}}}{{\left(2^{k}r\right)}^{\frac{1-s_{+}}{s_{+}}}}},$$

and by using the fact that $1-{\displaystyle \frac{1}{p_{+}}}-{\displaystyle \frac{1}{s_{+}}}\le -{\displaystyle \frac{1}{c{q}_{+}}}$, we get

$$\begin{array}{cc}\hfill {\parallel \zeta \parallel }_{L^{p(·)}}{\displaystyle \sum _{k=N_r}^{+\infty }}{\displaystyle \frac{\rho \left(2^{k}r\right){\left(2^{k}r\right)}^{-\frac{1}{p_{+}}}}{{(1+2^{k}r)}^{\gamma }}}& \le C_2{\parallel \zeta \parallel }_{L^{p(·)}}{\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}^{1/s_{+}}{\displaystyle \sum _{k=N_r}^{+\infty }}{\left(2^{k}r\right)}^{1-{\displaystyle \frac{1}{p_{+}}}-{\displaystyle \frac{1}{s_{+}}}}\hfill \\ \hfill & \le C_2{\parallel \zeta \parallel }_{L^{p(·)}}{\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}^{1/s_{+}}{\displaystyle \sum _{k=N_r}^{+\infty }}{\left(2^{k}r\right)}^{-{\displaystyle \frac{1}{c{q}_{+}}}}\hfill \\ \hfill & =C_2{r}^{-{\displaystyle \frac{1}{c{q}_{+}}}}{\parallel \zeta \parallel }_{L^{p(·)}}{\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}^{1/s_{+}}{\displaystyle \sum _{k=N_r}^{+\infty }}{2}^{-{\displaystyle \frac{k}{c{q}_{+}}}},\hfill \end{array}$$

and since the series on the right is convergent, we have

$${\parallel \zeta \parallel }_{L^{p(·)}}{\displaystyle \sum _{k=N_r}^{+\infty }}{\displaystyle \frac{\rho \left(2^{k}r\right){\left(2^{k}r\right)}^{-\frac{1}{p_{+}}}}{{(1+2^{k}r)}^{\gamma }}}\le C_3{r}^{-{\displaystyle \frac{1}{c{q}_{+}}}}{\parallel \zeta \parallel }_{L^{p(·)}}{\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}^{1/s_{+}}.$$

(8)

If $\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(.)}}\ge 1,$ by following Lemma 2, and by using a similar procedure, we get

$$\mathcal{M}\zeta \left(\vartheta \right){\displaystyle \sum _{k=-\infty }^{N_r-1}}{\displaystyle \frac{\rho \left(2^{k}r\right)}{{(1+2^{k}r)}^{\gamma }}}\le C_4{r}^{{\displaystyle \frac{s_{-}-1}{s_{+}}}}\mathcal{M}\zeta \left(\vartheta \right){\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}$$

(9)

and

$${\parallel \zeta \parallel }_{L^{p(.)}}{\displaystyle \sum _{k=N_r}^{+\infty }}{\displaystyle \frac{\rho \left(2^{k}r\right){\left(2^{k}r\right)}^{-\frac{1}{p_{+}}}}{{(1+2^{k}r)}^{\gamma }}}\le C_5{r}^{-{\displaystyle \frac{1}{c{q}_{+}}}}{\parallel \zeta \parallel }_{L^{p(·)}}{\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}.$$

(10)

Now before proceeding further, first we estimate the following expression. Let

$$A=r^{{\displaystyle \frac{s_{-}-1}{s_{+}}}}\mathcal{M}\zeta \left(\vartheta \right)+r^{-{\displaystyle \frac{1}{c{q}_{+}}}}{\parallel \zeta \parallel }_{L^{p(·)}}.$$

If we choose $r={\left({\displaystyle \frac{\mathcal{M}\zeta \left(\vartheta \right)}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)}^{-{\displaystyle \frac{p\left(\vartheta \right)c{q}_{+}}{q\left(\vartheta \right)}}},$ then

$$\begin{array}{cc}\hfill A& ={\left({\displaystyle \frac{\mathcal{M}\zeta \left(\vartheta \right)}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)}^{-c{q}_{+}{\displaystyle \frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}}\left({\displaystyle \frac{s_{-}-1}{s_{+}}}\right)}\mathcal{M}\zeta \left(\vartheta \right)+{\left({\displaystyle \frac{\mathcal{M}\zeta \left(\vartheta \right)}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)}^{{\displaystyle \frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}}}{\parallel \zeta \parallel }_{L^{p(·)}}\hfill \\ \hfill & ={\displaystyle \frac{{\parallel \zeta \parallel }_{L^{p(·)}}^{c{q}_{+}\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}\left(\frac{s_{-}-1}{s_{+}}\right)}}{{\left(\mathcal{M}\zeta \left(\vartheta \right)\right)}^{c{q}_{+}\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}\left(\frac{s_{-}-1}{s_{+}}\right)-1}}}+{\displaystyle \frac{{\left(\mathcal{M}\zeta \left(\vartheta \right)\right)}^{\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}}}{{\parallel \zeta \parallel }_{L^{p(·)}}^{\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}-1}}}\hfill \\ \hfill & ={\displaystyle \frac{{\parallel \zeta \parallel }_{L^{p(·)}}^{c{q}_{+}\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}\left(\frac{s_{-}-1}{s_{+}}\right)+\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}-1}+{\left(\mathcal{M}\zeta \left(\vartheta \right)\right)}^{c{q}_{+}\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}\left(\frac{s_{-}-1}{s_{+}}\right)+\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}-1}}{{\left(\mathcal{M}\zeta \left(\vartheta \right)\right)}^{c{q}_{+}\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}\left(\frac{s_{-}-1}{s_{+}}\right)-1}{\parallel \zeta \parallel }_{L^{p(·)}}^{\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}-1}}},\hfill \end{array}$$

and by following ${\displaystyle \frac{q\left(\vartheta \right)}{p\left(\vartheta \right)}}=1+{\displaystyle \frac{c{q}_{+}(s_{-}-1)}{s_{+}}}$ a.e., we get

$$\begin{array}{cc}\hfill A& ={\displaystyle \frac{2}{{\left(\mathcal{M}\zeta \left(\vartheta \right)\right)}^{-\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}}{\parallel \zeta \parallel }_{L^{p(·)}}^{\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}-1}}}=2{\displaystyle \frac{{\left(\mathcal{M}\zeta \left(\vartheta \right)\right)}^{\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}}}{{\parallel \zeta \parallel }_{L^{p(·)}}^{\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}-1}}}=2{\left({\displaystyle \frac{\mathcal{M}\zeta \left(\vartheta \right)}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)}^{{\displaystyle \frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}}}{\parallel \zeta \parallel }_{L^{p(·)}},\hfill \end{array}$$

and consequently,

$$r^{{\displaystyle \frac{s_{-}-1}{s_{+}}}}\mathcal{M}\zeta \left(\vartheta \right)+r^{-{\displaystyle \frac{1}{c{q}_{+}}}}{\parallel \zeta \parallel }_{L^{p(·)}}=2{\left({\displaystyle \frac{\mathcal{M}\zeta \left(\vartheta \right)}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)}^{{\displaystyle \frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}}}{\parallel \zeta \parallel }_{L^{p(·)}},a.e.$$

(11)

Thus, by applying Lemma 3 together with the relations (7), (8) and (11), we obtain

$${\displaystyle \frac{|I_{\rho ,\gamma }\zeta \left(\vartheta \right)|/C_6{\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}^{1/s_{+}}}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\le {\left({\displaystyle \frac{\mathcal{M}\zeta \left(\vartheta \right)}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)}^{{\displaystyle \frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}}},$$

or

$${\left({\displaystyle \frac{|I_{\rho ,\gamma }\zeta \left(\vartheta \right)|/C_6{\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}^{1/s_{+}}}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)}^{q\left(\vartheta \right)}\le {\left({\displaystyle \frac{\mathcal{M}\zeta \left(\vartheta \right)}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)}^{p\left(\vartheta \right)},$$

and by following the monotonicity of the modular functional,

$${\Psi }_{q(·)}\left({\displaystyle \frac{I_{\rho ,\gamma }\zeta /C_6{\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}^{1/s_{+}}}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)\le {\Psi }_{p(·)}\left({\displaystyle \frac{\mathcal{M}\zeta }{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right).$$

(12)

Similarly, by applying Lemma 3 together with the relations (9)–(11), we obtain

$${\displaystyle \frac{|I_{\rho ,\gamma }\zeta \left(\vartheta \right)|/C_7{\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\le {\left({\displaystyle \frac{\mathcal{M}\zeta \left(\vartheta \right)}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)}^{{\displaystyle \frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}}},$$

and further by using the monotonicity of the modular functional, we get

$${\Psi }_{q(·)}\left({\displaystyle \frac{I_{\rho ,\gamma }\zeta /C_7{\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)\le {\Psi }_{p(·)}\left({\displaystyle \frac{\mathcal{M}\zeta }{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right).$$

(13)

Since $p_{+}<\infty$ and $p(·)$ is log-Hölder continuous, it follows from the boundedness of the Hardy–Littlewood maximal $\mathcal{M}$ operator and homogeneity of the norm that ${\Psi }_{p(·)}\left(\mathcal{M}\zeta /{\parallel \zeta \parallel }_{L^{p(·)}}\right)\le 1.$ Then (12) and (13) provide the required results.    □

Corollary 2.

If all exponent functions in Theorem 3 are constants, that is, $q(·)=q_{-}=q_{+}=q,p(·)=p_{-}=p_{+}=p,\mathrm{and}s(·)=s_{-}=s_{+}=s,$ with $c=1$, then the inequality ${\displaystyle \frac{1}{q_{+}}}\le c\left({\displaystyle \frac{1}{p_{+}}}+{\displaystyle \frac{1}{s_{+}}}-1\right)$ reduces to ${\displaystyle \frac{1}{q}}\le {\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{s}}-1.$ Moreover, the relation ${\displaystyle \frac{q(·)}{p(·)}}=1+{\displaystyle \frac{c{q}_{+}(s_{-}-1)}{s_{+}}}$ simplifies to ${\displaystyle \frac{1}{q}}={\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{s}}-1.$ Thus in the constant exponent case if the relations ${\displaystyle \frac{1}{q}}={\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{s}}-1$ and

$${\int }_0^1{\left({\displaystyle \frac{\rho \left(t\right)}{t}}\right)}^{s}dt+{\int }_1^{+\infty }{\left({\displaystyle \frac{\rho \left(t\right)}{t^{1+\gamma }}}\right)}^{s}dt<+\infty ,$$

hold then the generalized Bessel–Riesz operator $I_{\rho ,\gamma }:L^p\left({\mathbb{R}}_{+}\right)\to L^q\left({\mathbb{R}}_{+}\right)$ is bounded.

Definition 7.

The pair of exponent functions $\left(q\right(·),p(·\left)\right)$ is said to satisfy the boundedness property if for any $\vartheta \in \mathsf{\Omega }=\{\vartheta \in {\mathbb{R}}_{+}:\mathcal{M}\zeta \left(\vartheta \right)\ge \parallel \zeta {\parallel }_{L^{p(·)}}\}$ the exponents satisfy ${\displaystyle \frac{q\left(\vartheta \right)}{p\left(\vartheta \right)}}\le 1+{\displaystyle \frac{q_{+}(s_{-}-1)}{s_{+}}}$ a.e. and for any $\vartheta \in \mho =\{\vartheta \in {\mathbb{R}}_{+}:\mathcal{M}\zeta \left(\vartheta \right)<\parallel \zeta {\parallel }_{L^{p(·)}}\}$ the exponents satisfy ${\displaystyle \frac{q\left(\vartheta \right)}{p\left(\vartheta \right)}}>1+{\displaystyle \frac{q_{+}(s_{-}-1)}{s_{+}}}$ a.e.

Theorem 4.

Let the relation (1) hold and the pair $\left(q\right(·),p(·\left)\right)$ satisfy the boundedness property. Further, if ${\displaystyle \frac{1}{q_{+}}}={\displaystyle \frac{1}{p_{+}}}+{\displaystyle \frac{1}{s_{+}}}-1$ and $p(·)\in LH\left({\mathbb{R}}^{+}\right)$, then the generalized Bessel–Riesz operator $I_{\rho ,\gamma }:L^{p(·)}\left({\mathbb{R}}_{+}\right)\to L^{q(·)}\left({\mathbb{R}}_{+}\right)$ is bounded. Moreover, for any $\zeta \in L^{p(·)}\left({\mathbb{R}}_{+}\right)$ there exists a constant $C>0$, such that for $0<\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}<1,$ we have

$$\parallel I_{\rho ,\gamma }{\zeta \parallel }_{L^{q(·)}}\le C\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}^{1/s_{+}}{\parallel \zeta \parallel }_{L^{p(·)}}$$

and for $\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}\ge 1,$

$$\parallel I_{\rho ,\gamma }{\zeta \parallel }_{L^{q(·)}}\le C\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}{\parallel \zeta \parallel }_{L^{p(·)}}.$$

Proof. 

First, we estimate $T=r^{{\displaystyle \frac{s_{-}-1}{s_{+}}}}\mathcal{M}\zeta \left(\vartheta \right)+r^{-{\displaystyle \frac{1}{q_{+}}}}{\parallel \zeta \parallel }_{L^{p(·)}}$ for $r={\left({\displaystyle \frac{\mathcal{M}\zeta \left(\vartheta \right)}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)}^{-q_{+}{\displaystyle \frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}}}$ as follows:

$$\begin{array}{cc}\hfill T& \le {\left({\displaystyle \frac{\mathcal{M}\zeta \left(\vartheta \right)}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)}^{-q_{+}{\displaystyle \frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}}{\displaystyle \frac{s_{-}-1}{s_{+}}}}\mathcal{M}\zeta \left(\vartheta \right)+{\left({\displaystyle \frac{\mathcal{M}\zeta \left(\vartheta \right)}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)}^{{\displaystyle \frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}}}{\parallel \zeta \parallel }_{L^{p(·)}}\hfill \\ \hfill & ={\displaystyle \frac{{\parallel \zeta \parallel }_{L^{p(·)}}^{\frac{p\left(\vartheta \right)q_{+}}{q\left(\vartheta \right)}\frac{s_{-}-1}{s_{+}}}}{{\left(\mathcal{M}\zeta \left(\vartheta \right)\right)}^{\frac{p\left(\vartheta \right)q_{+}}{q\left(\vartheta \right)}\frac{s_{-}-1}{s_{+}}}}}+{\displaystyle \frac{{\left(\mathcal{M}\zeta \left(\vartheta \right)\right)}^{\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}}}{{\parallel \zeta \parallel }_{L^{p(·)}}^{\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}-1}}}\hfill \\ \hfill & ={\displaystyle \frac{{\parallel \zeta \parallel }_{L^{p(·)}}^{\frac{p\left(\vartheta \right)q_{+}}{q\left(\vartheta \right)}\frac{s_{-}-1}{s_{+}}+\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}-1}+{\left(\mathcal{M}\zeta \left(\vartheta \right)\right)}^{\frac{p\left(\vartheta \right)q_{+}}{q\left(\vartheta \right)}\frac{s_{-}-1}{s_{+}}+\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}-1}}{{\left(\mathcal{M}\zeta \left(\vartheta \right)\right)}^{\frac{p\left(\vartheta \right)q_{+}}{q\left(\vartheta \right)}\frac{s_{-}-1}{s_{+}}-1}{\parallel \zeta \parallel }_{L^{p(·)}}^{\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}-1}}}.\hfill \end{array}$$

Since for any $\vartheta \in {\mathbb{R}}_{+},$ either $\vartheta \in \mathsf{\Omega }$ or $\vartheta \in \mho ,$

$$\begin{array}{cc}\hfill T& \le {\displaystyle \frac{{\left(\mathcal{M}\zeta \left(\vartheta \right)\right)}^{\frac{p\left(\vartheta \right)q_{+}}{q\left(\vartheta \right)}\frac{s_{-}-1}{s_{+}}+\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}-1}+{\left(\mathcal{M}\zeta \left(\vartheta \right)\right)}^{\frac{p\left(\vartheta \right)q_{+}}{q\left(\vartheta \right)}\frac{s_{-}-1}{s_{+}}+\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}-1}}{{\left(\mathcal{M}\zeta \left(\vartheta \right)\right)}^{\frac{p\left(\vartheta \right)q_{+}}{q\left(\vartheta \right)}\frac{s_{-}-1}{s_{+}}-1}{\parallel \zeta \parallel }_{L^{p(·)}}^{\frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}-1}}}\hfill \\ \hfill & =2{\left({\displaystyle \frac{\mathcal{M}\zeta \left(\vartheta \right)}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)}^{{\displaystyle \frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}}}{\parallel \zeta \parallel }_{L^{p(·)}}.\hfill \end{array}$$

(14)

If $0<\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(.)}}<1,$ then by Lemma 3, we get

$$|I_{\rho ,\gamma }\zeta \left(\vartheta \right)|\le C*\left(r^{{\displaystyle \frac{s_{-}-1}{s_{+}}}}\mathcal{M}\zeta \left(\vartheta \right)+r^{-{\displaystyle \frac{1}{q_{+}}}}{\parallel \zeta \parallel }_{L^{p(.)}}\right){\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}^{1/s_{+}}.$$

By following (14),

$${\displaystyle \frac{|I_{\rho ,\gamma }\zeta \left(\vartheta \right)|/C\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}^{1/s_{+}}}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\le {\left({\displaystyle \frac{\mathcal{M}\zeta \left(\vartheta \right)}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)}^{{\displaystyle \frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}}},$$

and therefore, by monotonicity of the modular functional

$${\Psi }_{q(·)}\left({\displaystyle \frac{I_{\rho ,\gamma }\zeta /C{\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}^{1/s_{+}}}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)\le {\Psi }_{p(·)}\left({\displaystyle \frac{\mathcal{M}\zeta }{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right).$$

It follows from the boundedness of the maximal operator that ${\Psi }_{p(·)}\left({\displaystyle \frac{\mathcal{M}\zeta }{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)\le 1.$ Therefore,

$$\parallel I_{\rho ,\gamma }{\zeta \parallel }_{L^{q(·)}}\le {C\parallel \zeta \parallel }_{L^{p(·)}}{\parallel K_{\rho ,\gamma }\parallel }_{L^{s(·)}}^{1/s_{+}}.$$

Similarly for $\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(.)}}\ge 1,$ by following a similar procedure we get the required expression.    □

Remark 2.

Since Theorem 3 holds only when the exponent function satisfies

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

its conclusion does not generally remain valid when the relation ${\displaystyle \frac{q(·)}{p(·)}}=1+{\displaystyle \frac{c{q}_{+}(s_{-}-1)}{s_{+}}}a.e.$ fails to hold for $\vartheta \in {\mathbb{R}}_{+}$. Therefore, in Theorem 4, we consider situations in which boundedness can be established under more general assumptions on the exponent functions, including those not covered by Theorem 3. However, Theorem 3 cannot be recovered as a special case of Theorem 4, since the latter requires additional conditions involving the maximal function expressed in terms of the norm ${\parallel f\parallel }_{L^{p(·)}}$, which are not needed in Theorem 3.

Remark 3.

The estimates in Theorems 3 and 4, obtained by choosing $r={\left({\displaystyle \frac{\mathcal{M}\zeta \left(\vartheta \right)}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)}^{-q_{+}{\displaystyle \frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}}},$ can also be derived by taking $r={\left({\displaystyle \frac{\mathcal{M}\zeta \left(\vartheta \right)}{{\parallel \zeta \parallel }_{L^{p(·)}}}}\right)}^{\left({\displaystyle \frac{p\left(\vartheta \right)}{q\left(\vartheta \right)}}-1\right){\displaystyle \frac{s_{+}}{s_{-}-1}}},$ and then proceeding in a similar manner.

Corollary 3.

If the exponent functions appearing in Definition 7 and Theorem 4 are constant, then the corresponding boundedness results for the generalized Bessel–Riesz operator $I_{\rho ,\gamma }:L^p\left({\mathbb{R}}_{+}\right)\to L^q\left({\mathbb{R}}_{+}\right)$ hold in the classical Lebesgue spaces.

Corollary 4.

If we take $\rho \left(\vartheta \right)={\vartheta }^{\alpha }$ with $0<\alpha <1$, then $\rho \left(1\right)=1$, ρ is non-decreasing, and $\rho \left(\vartheta \right)/\vartheta$ is non-increasing on $(0,+\infty )$. Thus, ${\vartheta }^{\alpha }\in \mathcal{A}\left({\mathbb{R}}_{+}\right)$, and the definitions of the generalized Bessel–Riesz kernel and the generalized Bessel–Riesz operator coincide with the corresponding definitions in [10]. Moreover, the condition

$${\int }_0^1{\left({\displaystyle \frac{\rho \left(\vartheta \right)}{\vartheta }}\right)}^{s_{+}}d\vartheta +{\int }_1^{+\infty }{\left({\displaystyle \frac{\rho \left(\vartheta \right)}{{\vartheta }^{1+\gamma }}}\right)}^{s_{-}}d\vartheta ={\int }_0^1{\vartheta }^{(\alpha -1)s_{+}}d\vartheta +{\int }_1^{+\infty }{\vartheta }^{(\alpha -1-\gamma )s_{-}}d\vartheta <+\infty$$

is equivalent to

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

Hence, Lemma 3 in [10] appears as a special case of our Lemma 1. Furthermore, for this particular choice of $\rho \left(\vartheta \right)$, Lemma 7 in [10] also follows as a special case of our Lemma 3. In addition, for this choice of $\rho (·)$, the boundedness results in Theorems 3 and 4 correspond to Theorems 7 and 8 in [10], respectively, but with slightly different conditions on the exponent function, which we refine here to handle more general cases.

Example 1.

If we take $\rho \left(\vartheta \right)={\displaystyle \frac{2\vartheta }{1+\vartheta }}$ for $\vartheta \in [0,+\infty )$, then $\rho \left(1\right)=1$. Moreover,

$${\rho }^{\prime }\left(\vartheta \right)={\displaystyle \frac{2}{{(1+\vartheta )}^2}}\ge 0\mathit{for}\mathit{all}\vartheta \in (0,+\infty ),$$

which shows that $\rho (·)$ is non-decreasing on $(0,+\infty )$. On the other hand,

$${\left({\displaystyle \frac{\rho \left(\vartheta \right)}{\vartheta }}\right)}^{\prime }={\displaystyle \frac{-2}{{(1+\vartheta )}^2}}\le 0\mathit{for}\mathit{all}\vartheta \in (0,+\infty ),$$

so that ${\displaystyle \frac{\rho \left(\vartheta \right)}{\vartheta }}$ is non-increasing on $(0,+\infty )$. Hence, $\rho (·)\in \mathcal{A}\left({\mathbb{R}}_{+}\right)$. For this particular choice of $\rho (·)$, the generalized Bessel–Riesz operator takes the form

$$I_{\rho ,\gamma }\zeta \left(\vartheta \right)=(K_{\rho ,\gamma }\ast \zeta )\left(\vartheta \right)={\int }_{{\mathbb{R}}_{+}}{\displaystyle \frac{2}{{[1+|\vartheta -v\left|\right]}^{1+\gamma }}}\zeta \left(v\right)dv.$$

To ensure that the Bessel–Riesz kernel $K_{\rho ,\gamma }$ belongs to $L^{s(·)}\left({\mathbb{R}}_{+}\right)$, the following condition on $\rho (·)$ must be satisfied:

$${\int }_0^1{\left({\displaystyle \frac{\rho \left(\vartheta \right)}{\vartheta }}\right)}^{s_{+}}d\vartheta +{\int }_1^{+\infty }{\left({\displaystyle \frac{\rho \left(\vartheta \right)}{{\vartheta }^{1+\gamma }}}\right)}^{s_{-}}d\vartheta ={\int }_0^1{\left({\displaystyle \frac{2}{1+\vartheta }}\right)}^{s_{+}}d\vartheta +{\int }_1^{+\infty }{\left({\displaystyle \frac{2}{{\vartheta }^{\gamma }(1+\vartheta )}}\right)}^{s_{-}}d\vartheta <+\infty .$$

This condition is equivalent to

$$1<{\displaystyle \frac{1}{\gamma }}<s_{-}\le s_{+}.$$

If this condition holds, then by Lemma 2, there exists an integer $N_r$ such that the following estimates for the norm of the Bessel–Riesz kernel hold:

• If $0<\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}<1$, then

$$\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}^{{\displaystyle \frac{1}{s_{+}}}}\ge {\displaystyle \frac{C{\left(2^{k}r\right)}^{\frac{1}{s_{+}}}}{{\left(1+2^{k}r\right)}^{\gamma +\frac{s_{-}}{s_{+}}}}},k<N_r,$$

and

$$\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}^{{\displaystyle \frac{1}{s_{+}}}}\ge {\displaystyle \frac{C{\left(2^{k}r\right)}^{\frac{1}{s_{+}}}}{{\left(1+2^{k}r\right)}^{1+\gamma }}},k\ge N_r.$$

Moreover, if $\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}\ge 1$, then

$$\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}\ge {\displaystyle \frac{C{\left(2^{k}r\right)}^{\frac{1}{s_{+}}}}{{\left(1+2^{k}r\right)}^{\gamma +\frac{s_{-}}{s_{+}}}}},k<N_r,$$

and

$$\parallel K_{\rho ,\gamma }{\parallel }_{L^{s(·)}}\ge {\displaystyle \frac{C{\left(2^{k}r\right)}^{\frac{1}{s_{+}}}}{{\left(1+2^{k}r\right)}^{1+\gamma }}},k\ge N_r.$$

Furthermore, Lemma 3 provides the following estimate for the Bessel–Riesz operator corresponding to this choice of $\rho (·)$:

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

If $1<{\displaystyle \frac{1}{\gamma }}<s_{-}\le s_{+},{\displaystyle \frac{1}{q_{+}}}\le c\left({\displaystyle \frac{1}{p_{+}}}+{\displaystyle \frac{1}{s_{+}}}-1\right),$ and ${\displaystyle \frac{q(·)}{p(·)}}=1+{\displaystyle \frac{c{q}_{+}(s_{-}-1)}{s_{+}}}$ a.e. for some constant $c>0$, with $p(·)\in LH\left({\mathbb{R}}_{+}\right)$, then for $\rho \left(\vartheta \right)={\displaystyle \frac{2\vartheta }{1+\vartheta }}$, Theorem 3 guarantees the boundedness of the Bessel–Riesz operator $I_{\rho ,\gamma }:L^{p(·)}\left({\mathbb{R}}_{+}\right)\to L^{q(·)}\left({\mathbb{R}}_{+}\right).$ Moreover, if $1<{\displaystyle \frac{1}{\gamma }}<s_{-}\le s_{+},$ and the pair $\left(q\right(·),p(·\left)\right)$ satisfies the boundedness property, with ${\displaystyle \frac{1}{q_{+}}}={\displaystyle \frac{1}{p_{+}}}+{\displaystyle \frac{1}{s_{+}}}-1$ and $p(·)\in LH\left({\mathbb{R}}_{+}\right)$, then by Theorem 4, for this choice of $\rho (·)$, the Bessel–Riesz operator $I_{\rho ,\gamma }:L^{p(·)}\left({\mathbb{R}}_{+}\right)\to L^{q(·)}\left({\mathbb{R}}_{+}\right)$ is bounded.

4. Conclusions

This manuscript is devoted to the study of the structural properties and boundedness of a generalized Bessel–Riesz operator, defined via convolution with an associated kernel over the Euclidean domain. The proposed generalization is intrinsically connected to the extension of the classical Bessel–Riesz kernel.

To this end, we introduce a class of measurable functions $\mathcal{A}\left({\mathbb{R}}_{+}\right)$, which serves as a foundation for defining a generalized Bessel–Riesz kernel and the corresponding operator. Although related generalizations have appeared in [12], the structural assumptions imposed therein differ significantly from those adopted in the present work through the class $\mathcal{A}\left({\mathbb{R}}_{+}\right)$, thereby leading to a distinct analytical framework.

We establish different auxiliary results that play a crucial role in the development of our main results. In particular, Lemma 1 provides sufficient conditions on the exponent function to ensure that the generalized Bessel–Riesz kernel belongs to the appropriate variable Lebesgue space. Moreover, Lemma 3 yields a key estimate for the generalized Bessel–Riesz operator in terms of the variable Lebesgue norm and the associated maximal operator.

The principal contributions of this manuscript are contained in Theorems 3 and 4. In Theorem 3, we establish the boundedness of the generalized Bessel–Riesz operator under suitable assumptions on the exponent functions. Theorem 4 further extends these results by addressing cases where the assumptions of Theorem 3 are not satisfied, employing alternative techniques to derive corresponding boundedness results.

In addition, we demonstrate that, for an appropriate choice of $\rho (·)$, the classical boundedness results obtained in [10] are recovered as special cases within our framework, albeit under modified assumptions on the exponent functions that naturally arise in the generalized setting. We also present an example that is not encompassed by [10], thereby highlighting the broader applicability and flexibility of our approach. Furthermore, when the exponent functions are constant, our results reduce to the classical theory in Lebesgue spaces.

In conclusion, the results presented herein extend several known results in the literature, provide a unified and flexible framework for the analysis of generalized operators, and suggest new avenues for further research in the theory of operators on variable Lebesgue spaces.

This work mainly establishes the boundedness of the considered operator in variable exponent spaces. However, several important aspects remain open for future investigation. In particular, further in-depth properties such as compactness, interpolation properties, weak-type boundedness, and boundedness on weighted variable exponent spaces are not addressed in this study and are left as potential directions for future work.