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$1/q=1/p-\gamma/n$\mathrm{.}If$u,v^{q/p}\in A_1$orif$uv^{-q/{p’}}\in A_1$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-$\delta$, $0<\delta \le 1$.