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

In this manuscript, we introduce a class of measurable functions A(R+), 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.