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

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 ${\mathbb{R}}^n$. 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