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

Esra Kaya

doi:10.1515/math-2021-0041

Abstract

Abstract In this paper, we consider the maximal operator related to the Laplace-Bessel differential operator ( B B -maximal operator) on L p ( ⋅ ) , γ ( R k , + 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 B B -maximal operator on variable exponent Lebesgue spaces. Moreover, we will obtain that the B B -maximal operator is not bounded on L p ( ⋅ ) , γ ( R k , + 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 B B -maximal function) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces.

Highlights

This paper is associated with the maximal function generated by the Laplace-Bessel differential operator ∑ ∑ ΔB ≔ k i=1 Bi + n i=k+1 ∂2 ∂xi2 = γi ∂, xi ∂xi1 ≤ k ≤ n, which is known as an important differential operator in harmonic analysis
The boundedness of the B-maximal operator plays an important role in obtaining the boundedness of the convolutiontype singular integral operators related to the Laplace-Bessel differential operator
We will obtain that a necessary condition for the boundedness of the B-maximal operator on Lp(⋅),γ( nk,+) variable exponent Lebesgue spaces, using + translation according to n-variables

Summary

Introduction

1 ≤ k ≤ n, which is known as an important differential operator in harmonic analysis. On variable exponent Lebesgue spaces, the boundedness of some operators in harmonic analysis, such as maximal operator, and singular integral operator, has an important role. The boundedness of the B-maximal operator plays an important role in obtaining the boundedness of the convolutiontype singular integral operators related to the Laplace-Bessel differential operator. We will obtain that a necessary condition for the boundedness of the B-maximal operator on Lp(⋅),γ( nk,+) variable exponent Lebesgue spaces, using + translation according to n-variables.

Notations and preliminaries

The fractional B-maximal function on variable exponent Lebesgue spaces