Vol. 301, No. 1, 2019

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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|x|pLθp̃(n) and on the vector-valued mixed radial-angular spaces L|x|pLθp̃(n,p̃). 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
Authors
Feng Liu
College of Mathematics and Systems Science
Shandong University of Science and Technology
Qingdao
China
Dashan Fan
Department of Mathematical Sciences
University of Wisconsin-Milwaukee
Milwaukee, WI
United States