The A_2 theorem with the Dini condition

TL;DR

This paper proves that operators with Calderón--Zygmund kernels satisfying the Dini condition adhere to the A_2 theorem, establishing weighted norm inequalities via domination by dyadic operators.

Contribution

It demonstrates that the Dini condition on the kernel's modulus of continuity ensures the A_2 theorem holds for such operators, extending previous results.

Findings

Operators with Dini-continuous kernels satisfy the A_2 theorem.

Weighted norm inequalities are established via dyadic domination.

The result broadens the class of operators for which the A_2 theorem applies.

Abstract

Let $T$ be an $L^2$-bounded operator having an $\omega$-Calder\'on--Zygmund kernel $K$ with a modulus of continuity $\omega$. If $\omega$ satisfied the Dini condition $\int_0^1\omega(t)\ud t/t<\infty$, then $T$ satisfies the $A_2$ theorem $$ \Norm{Tf}{L^2(w)}\lesssim [w]_{A_2}\Norm{f}{L^2(w)} $$ and many related estimates, as a consequence of a domination by dyadic operators.