Marcinkiewicz integrals with rough kernels in H^{\mathrm{1}}（{\mathrm{S}}^{n-\mathrm{1}}）

Abstract

This paper is devoted to studying the Marcinkiewicz integral operators associated to polynomial compound curves. Some new bounds for the above operators on the Lebesgue, Triebel-Lizorkin, and Besov spaces are established by assuming that their rough kernels are given by\mathrm{\Omega }\in H^{\mathrm{1}}（{\mathrm{S}}^{n-\mathrm{1}}） andh\in {\mathrm{\Delta }}_{}（{ℝ}_{+}）for some\text{γ>1}: It should be pointed out that the bounds are independent of h,\mathrm{\Omega },\gamma and the coefficients of the polynomials in the definition of the operators.

Key words： Marcinkiewicz integral; polynomial curve; H^{\mathrm{1}}（{\mathrm{S}}^{n-\mathrm{1}}）; Triebel-Lizorkin space

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