The A_2 theorem with the Dini condition

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 ${\int }_0^1\omega (t)\text{ud}t\mathrm{/}t<\mathrm{\infty }$, then $T$ satisfies the $A_2$ theorem $$\text{Norm}Tf{L}^2(w)\lesssim [w{]}_{A_2}\text{Norm}f{L}^2(w)$$ and many related estimates, as a consequence of a domination by dyadic operators.