A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces

Abstract

In this paper, we consider the maximal operator related to the Laplace-Bessel differential operator (BB-maximal operator) on Lp(⋅),γ(Rk,+n){L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces. We will give a necessary condition for the boundedness of the BB-maximal operator on variable exponent Lebesgue spaces. Moreover, we will obtain that the BB-maximal operator is not bounded on Lp(⋅),γ(Rk,+n){L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces in the case of p−=1{p}_{-}=1. We will also prove the boundedness of the fractional maximal function associated with the Laplace-Bessel differential operator (fractional BB-maximal function) on Lp(⋅),γ(Rk,+n){L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces.

Keywords

• b-maximal operator
• generalized translation operator
• singular integrals
• variable exponent lebesgue spaces
• 42a50
• 42b25
• 42b35