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}})).

Key words： Singular Radon transform; truncated singular integral; rough kernel; convergence

References

1 Al-Qassem H, Pan Y. On certain estimates for Marcinkiewicz integrals and extrapolation. Collect Math, 2009, 60: 123–145

2 Al-Salman A, Pan Y. Singular integrals with rough kernels in Llog⁡L(Sn−1). J Lond Math Soc, 2002, 66: 153–174

3 Al-Salman A, Pan Y. Singular integrals with rough kernels. Canad Math Bull, 2004, 47: 3–11

4 Chen Y, Ding Y, Liu H. Rough singular integrals supported on submanifolds. J Math Anal Appl, 2010, 368: 677–691

5 Colzani L. Hardy Spaces on Spheres. Ph D Thesis. Washington Univ, St Louis, 1982

6 Fan D, Pan Y. Singular integral operators with rough kernels supported by subvarieties. Amer J Math, 1997, 119: 799–839

7 Frazier M, Jawerth B, Weiss G. Littlewood-Paley Theory and the Study of Function Spaces. CBMS Reg Conf Ser Math, No 79. Providence: Amer Math Soc, 1991

8 Grafakos L, Stefanov A. L^p bounds for singular integrals and maximal singular integrals with rough kernels. Indiana Univ Math J, 1998, 47: 455–469

9 Liu F, Wu H. Rough singular integrals associated to compound mappings on Triebel-Lizorkin spaces and Besov spaces. Taiwanese J Math, 2014, 18: 127–146

10 Liu F, Wu H, Zhang D. Boundedness of certain singular integrals along surfaces on Triebel-Lizorkin spaces. Forum Math, 2015, 27: 3439–3460

11 Liu F, Xue Q, Yabuta K. Rough maximal singular integral and maximal operators supported by subvarieties on Triebel-Lizorkin spaces. Nonlinear Anal, 2018, 171: 41–72

12 Sato S. Estimates for singular integrals and extrapolation. Studia Math, 2009, 192: 219–233

13 Stein E M. Note on the class Llog⁡L. Studia Math, 1969, 32: 305–310

14 Stein E M. Problems in harmonic analysis related to curvature and oscillatory integrals. In: Proceedings of the International Congress of Mathematicians, 1986, Vol 1. Providence: Amer Math Soc, 1987, 196–221

15 Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton: Princeton Univ Press, 1993

16 Triebel H. Theory of Function Spaces. Monogr Math, Vol 78. Basel: Birkhäser, 1983

17 Yabuta K. Triebel-Lizorkin space boundedness of rough singular integrals associated to surfaces. J Inequal Appl, 2015, 107: 1–26