Convergence of truncated rough singular integrals supported by subvarieties on Triebel-Lizorkin spaces

Abstract

Let be a function of homogeneous of degree zero and satisfy the cancellation condition on the unit sphere. Suppose that h is a radial function. Let {{\displaystyle T}}_{h,\mathrm{\Omega },p} be the classical singular Radon transform, and let {{\displaystyle T}}_{h,\mathrm{\Omega },{p}}^{\epsilon } be its truncated operator with rough kernels associated to polynomial mapping p which is defined by {{\displaystyle T}}_{h,\mathrm{\Omega },p}^{\epsilon }f(x)=\left|{{\displaystyle \int }}_{\left|y\right|\rangle \epsilon }f(x-p(y))h(\left|y\right|)\mathrm{\Omega }(y){\left|y\right|}^{-n}\mathrm{d}y\right|. In this paper, we show that for any \alpha \in (-\infty ,\infty ) and (p,q) satisfying certain index condition, the operator {{\displaystyle T}}_{h,\mathrm{\Omega },p}^{\epsilon } enjoys the following convergence properties {{\displaystyle \lim }}_{\epsilon \to \mathrm{0}}{\Vert {{\displaystyle T}}_{h,\mathrm{\Omega },p}^{\epsilon }f-{{\displaystyle T}}_{h,\mathrm{\Omega },p}f\Vert }_{{{\displaystyle \dot{F}}}_{\alpha }^{p,q}({ℝ}^d)}=\mathrm{0} and {{\displaystyle \lim }}_{\epsilon \to \mathrm{0}}{\Vert {{\displaystyle T}}_{h,\mathrm{\Omega },p}^{\epsilon }f-{{\displaystyle T}}_{h,\mathrm{\Omega },p}f\Vert }_{{{\displaystyle \dot{B}}}_{\alpha }^{p,q}({ℝ}^d)}=\mathrm{0} provided that \mathrm{\Omega }\in {L({\log }^{+}L)}^{\beta }(S^{n-\mathrm{1}}) for some \beta \in (\mathrm{0},\mathrm{1}] or \mathrm{\Omega }\in H^{\mathrm{1}}(S^{n-\mathrm{1}}), or \mathrm{\Omega }\in ({{\displaystyle \cup }}_{\mathrm{1}\langle q\langle \infty }{{\displaystyle B}}_q^{(\mathrm{0},\mathrm{0})}(S^{n-\mathrm{1}})).

Keywords

Singular Radon transform / truncated singular integral / rough kernel / convergence

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