$A_1$ theory of weights for rough homogeneous singular integrals and commutators

TL;DR

This paper establishes quantitative weighted norm inequalities for rough homogeneous singular integrals and their commutators with BMO functions, advancing the understanding of their behavior in weighted Lebesgue spaces.

Contribution

It provides new $A_1-A_$ estimates for rough singular integrals and their commutators, with explicit bounds involving $A_1$ and $A_$ weights.

Findings

Derived bounds for $T_$ in weighted $L^p$ spaces.

Established estimates for commutators with BMO functions.

Quantified dependence on $A_1$ and $A_$ weight constants.

Abstract

Quantitative $A_1-A_\infty$ estimates for rough homogeneous singular integrals $T_{\Omega}$ and commutators of $BMO$ symbols and $T_{\Omega}$ are obtained. In particular the following estimates are proved: % \[ \|T_\Omega \|_{L^p(w)}\le c_{n,p}\|\Omega\|_{L^\infty} [w]_{A_1}^{\frac{1}{p}}\,[w]_{A_{\infty}}^{1+\frac{1}{p'}}\|f\|_{L^p(w)} \] % and % \[ \| [b,T_{\Omega}]f\|_{L^{p}(w)}\leq c_{n,p}\|b\|_{BMO}\|\Omega\|_{L^{\infty}} [w]_{A_1}^{\frac{1}{p}}[w]_{A_{\infty}}^{2+\frac{1}{p'}}\|f\|_{L^{p}\left(w\right)}, \] % for $1<p<\infty$ and $1/p+1/p'=1$.