Mixed Weak Estimates of Sawyer Type for Commutators of Generalized Singular Integrals and Related Operators

Abstract

We study mixed weak-type inequalities for the commutator $[b,T]$, where $b$ is a BMO function, and $T$ is a Calderón–Zygmund operator. More precisely, we prove that, for every $t>0$,

$$uv\left(\right\{x\in {\mathbb{R}}^n:\left|\frac{[b,T](fv)(x)}{v(x)}\right|>t\left\}\right)\le C{\int }_{{\mathbb{R}}^n}\Phi \left(\frac{|f(x)|}{t}\right)u(x)v(x)dx,$$ where $\Phi (t)=t(1+{log}^{+}t)$, $u\in A_1$, and $v\in A_{\infty }(u)$. Our technique involves the classical Calderón–Zygmund decomposition, which allows us to give a direct proof without taking into account the associated maximal operator. We use this result to prove an analogous inequality for higher-order commutators.

For a given Young function $\varphi$ we also consider singular integral operators $T$ whose kernels satisfy a $L^{\varphi }$-Hörmander property, and we find sufficient conditions on $\varphi$ such that a mixed weak estimate holds for $T$ and also for its higher order commutators $T_b^m$.

We also obtain a mixed estimation for a wide class of maximal operators associated to certain Young functions of $LlogL$ type which are in intimate relation with the commutators. This last estimate involves an arbitrary weight $u$ and a radial function $v$ which is not even locally integrable.