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

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: % \[ \|T_Ω\|_{L^p(w)}\le c_{n,p}\|Ω\|_{L^\infty} [w]_{A_1}^{\frac{1}{p}}\,[w]_{A_{\infty}}^{1+\frac{1}{p'}}\|f\|_{L^p(w)} \] % and % \[ \| [b,T_Ω]f\|_{L^{p}(w)}\leq c_{n,p}\|b\|_{BMO}\|Ω\|_{L^{\infty}} [w]_{A_1}^{\frac{1}{p}}[w]_{A_{\infty}}^{2+\frac{1}{p'}}\|f\|_{L^{p}\left(w\right)}, \] % for $1<p<\infty$ and $1/p+1/p^{\prime }=1$.

19 pages

References in corpus (1)

• A Bellman function counterexample to the $A_1$ conjecture: the blow-up of the weak norm estimates of weighted singular operators