Boundedness of Bessel–Riesz Operator in Variable Lebesgue Measure Spaces

Muhammad Nasir (), Ali Raza (), Luminiţa-Ioana Cotîrlă and Daniel Breaz ()
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Muhammad Nasir: Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New Muslim Town, Lahore 54600, Pakistan
Ali Raza: Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New Muslim Town, Lahore 54600, Pakistan
Luminiţa-Ioana Cotîrlă: Department of Mathematics, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
Daniel Breaz: Department of Mathematics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania

Mathematics, 2025, vol. 13, issue 3, 1-17

Abstract: In this manuscript, we establish the boundedness of the Bessel–Riesz operator I α , γ f in variable Lebesgue spaces L p ( · ) . We prove that I α , γ f is bounded from L p ( · ) to L p ( · ) and from L p ( · ) to L q ( · ) . We explore various scenarios for the boundedness of I α , γ f 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; Hardy–Littlewood maximal operator (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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