Estimates for imaginary powers of Laplace operator in variable Lebesgue spaces and applications

In this paper we study some estimates of norms in variable exponent Lebesgue spaces for singular integral operators that are imaginary powers of the Laplace operator in Rn. Using the Mellin transform argument, from these estimates we obtain the boundedness for a family of maximal operators in variable exponent Lebesgue spaces, which are closely related to the (weak) solution of the wave equation.