Variable-order Bessel–Riesz operators and their boundedness on variable Lebesgue spaces

In this article, we introduce a variable-order Bessel–Riesz integral operator and investigate its boundedness properties on variable Lebesgue spaces. We first establish the boundedness in the diagonal case under suitable regularity assumptions on the exponent function. For the nondiagonal setting, we derive a sharp pointwise estimate, which enables us to obtain boundedness results from $L^{p(\cdot )}({\mathbb{R}}_{+})$ to $L^{q(\cdot )}({\mathbb{R}}_{+})$ under appropriate relations between the exponent functions. Moreover, we establish the boundedness through an analog of Young's inequality. Because the classical Young's inequality generally fails in variable Lebesgue spaces due to the lack of translation invariance, we develop an alternative framework adapted to this setting. In particular, we identify sufficient conditions under which the associated kernel belongs to an appropriate variable Lebesgue space and use these conditions to derive the corresponding boundedness results. To the best of our knowledge, this class of operators has not been studied previously in the literature, and the study provides a systematic way to develop the boundedness of variable-order fractional integral operators in variable Lebesgue spaces.