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We study mixed weak type inequalities for the commutator $[b,T]$, where $b$is a BMO function and $T$ is a Calder\'on-Zygmund operator. More precisely, weprove that for every $t>0$ \begin{equation*}%\label{tesis_teo2.2}uv(\{x\in\R^n: |\frac{[b,T](fv)(x)}{v(x)}|>t\})\leqC\int_{\R^n}\phi(\frac{|f(x)|}{t})u(x)v(x)\,dx, \end{equation*} where$\phi(t)=t(1+\log^{+}{t})$, $u\in A_1$ and $v\in A_{\infty}(u)$. Our techniqueinvolves the classical Calder\'on-Zygmund decomposition, which allow us to givea direct proof. We use this result to prove an analogous inequality for higherorder commutators. We also obtain a mixed estimation for a wide class ofmaximal operators associated to certain Young functions of $L\log L$ type whichare in intimate relation with the commutators. This last estimate involves anarbitrary weight $u$ and a radial function $v$ which is not even locallyintegrable.

Mixed weak estimates of Sawyer type for commutators of singular integrals and related operators