Rough Marcinkiewicz integrals along certain smooth curves

Abstract

This paper is devoted to the study of the multiple-parameter rough Marcinkiewicz integral operators associated with certain smooth curves. It is shown that the Grafakos-Stefanov type size condition F_α(S^m−1 × Sⁿ⁻¹) of the kernel implies the L^p-boundedness of these Marcinkiewicz integral operators for some α > 1/2 and $\frac{1}{{2\alpha }} < p < 1 + 2\alpha$, which is an essential improvement of certain previous results.

Keywords

Marcinkiewicz integral / rough kernel / smooth curve / Littlewood-Paley theory / Fourier transform estimate

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