Quantitative Weighted Bounds for a Class of Singular Integral Operators

Abstract

In this article, the authors consider the weighted bounds for the singular integral operator defined by

$${T_A}f(x) = {\rm{p}}.{\rm{v}}.\int_{\mathbb{R}^{n}} {{{{\rm{\Omega }}(x - y)} \over {{\rm{|}}x - y{{\rm{|}}^{n + 1}}}}\left( {A(x) - A(y) - \nabla A(y)} \right)f(y){\rm{d}}y} ,$$

where Ω is homogeneous of degree zero and has vanishing moment of order one, and A is a function on ℝⁿ such that ▽A ∈ BMO(ℝⁿ). By sparse domination, the authors obtain some quantitative weighted bounds for T_a when Ω satisfies regularity condition of L^r-Dini type for some r ∈ (1, ∞).