Bessel–Riesz Operator in Variable Lebesgue Spaces Lp(·)(R+)

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.