Weighted estimates for rough singular integrals with applications to angular integrability

Feng Liu and Dashan Fan

Vol. 301 (2019), No. 1, 267–295

DOI: 10.2140/pjm.2019.301.267

Abstract

We study certain singular integral operators, as well as their corresponding truncated maximal operators, along polynomial curves. Assuming that the kernels of operators are rough not only on the unit sphere but also on the radial direction, we establish certain weighted estimates for these operators. As applications, we obtain that these operators are bounded on the mixed radial-angular spaces $L_{\left|x\right|}^p{L}_{\theta }^{\tilde{p}}\left({ℝ}^n\right)$ and on the vector-valued mixed radial-angular spaces $L_{\left|x\right|}^p{L}_{\theta }^{\tilde{p}}\left({ℝ}^n,{\ell }^{\tilde{p}}\right)$. The bounds are independent of the coefficients of the polynomials in the definition of the operators. Our results we obtained improve theorems of Antonio Córdoba (2016) and Piero D’Ancona and Renato Lucà (2016).

Keywords

singular integral, maximal singular integral, rough kernel, mixed radial-angular space, vector-valued mixed radial-angular space, vector-valued norm inequality

Mathematical Subject Classification 2010

Primary: 42B20

Secondary: 42B25

Milestones

Received: 19 September 2017

Revised: 12 June 2018

Accepted: 26 June 2018

Published: 16 September 2019