Some estimates for imaginary powers of the Laplace operator in variable Lebesgue spaces and applications

TL;DR

This paper establishes norm estimates for imaginary powers of the Laplace operator in variable Lebesgue spaces, leading to boundedness results for related maximal operators and applications to wave equations.

Contribution

It introduces new bounds for singular integral operators' imaginary powers in variable Lebesgue spaces using Mellin transform techniques.

Findings

Boundedness of imaginary powers of Laplace operator in variable Lebesgue spaces

Maximal operators related to wave equations are bounded in these spaces

Applications to solutions of wave equations in variable exponent contexts

Abstract

In this paper we study some estimates of norms in variable exponent Lebesgue spaces for a singular integral operators that are imaginary powers of the Laplace operator in $\R^n$. Using Mellin transform argument, from this estimates we obtain boundedness for a family of maximal operators in variable exponent Lebesgue spaces, which are closely related to the (weak) solution of the wave equation.42B25, 42B20, 46E30, 44A10, 42B10, 35L05