Mixed weak estimates of Sawyer type for fractional integrals and some related operators

We prove mixed weak estimates of Sawyer type for fractional operators. More precisely, let $\mathcal{T}$ be either the maximal fractional function $M_\gamma$ or the fractional integral operator $I_\gamma$, $0<\gamma<n$, $1\leq p<n/\gamma$ and $1/q=1/p-\gamma/n$. If $u,v^{q/p}\in A_1$ or if $uv^{-q/{p'}}\in A_1$ and $v^q\in A_\infty(uv^{-q/{p'}})$ then we obtain that the estimate \begin{equation*} uv^{q/p}\left(\left\{x\in \R^n: \frac{|\mathcal{T}(fv)(x)|}{v(x)}>t\right\}\right)^{1/q}\leq \frac{C}{t}\left(\int_{\R^n}|f(x)|^pu(x)^{p/q}v(x)\,dx\right)^{1/p}, \end{equation*} holds for every positive $t$ and every bounded function with compact support. As an important application of the results above we further more exhibe mixed weak estimates for commutators of Calder\'on-Zygmund singular integral and fractional integral operators when the symbol $b$ is in the class Lipschitz-$\delta$, $0<\delta\leq 1$.