L^p Estimates of certain rough parametric Marcinkiewicz functions

Abstract

We prove L^p estimates of a class of parametric Marcinkiewicz integral operators when their kernels satisfy only the L^{\mathrm{1}}(S^{n-\mathrm{1}}) integrability condition. The obtained L^p estimates resolve a problem left open in previous work. Our argument is based on duality technique and direct estimation of operators. As a consequence of our result, we deduce the L^p boundedness of a class of fractional Marcinkiewicz integral operators.

Key words： Marcinkiewicz integrals; parametric Marcinkiewicz functions; rough kernels; Marcinkiewicz interpolation theorem

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