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

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