Boundedness of Bessel–Riesz Operator in Variable Lebesgue Measure Spaces

Abstract

In this manuscript, we establish the boundedness of the Bessel–Riesz operator Iα,γf in variable Lebesgue spaces Lp(·). We prove that Iα,γf is bounded from Lp(·) to Lp(·) and from Lp(·) to Lq(·). We explore various scenarios for the boundedness of Iα,γf under general conditions, including constraints on the Hardy–Littlewood maximal operator M. To prove these results, we employ the boundedness of M, along with Hölder’s inequality and classical dyadic decomposition techniques. Our findings unify and generalize previous results in classical Lebesgue spaces. In some cases, the results are new even for constant exponents in Lebesgue spaces.

Keywords

• variable Lebesgue spaces
• Bessel–Riesz kernel
• Bessel–Riesz operator
• boundedness
• Hölder’s inequality
• dyadic decomposition