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

TL;DR

This paper establishes mixed weak estimates of Sawyer type for fractional operators and their commutators, extending the understanding of weighted inequalities in harmonic analysis.

Contribution

It introduces new mixed weak estimates for fractional operators and their commutators with Lipschitz symbols, broadening the scope of weighted inequalities in the field.

Findings

Proves mixed weak estimates for fractional maximal and integral operators.

Extends results to commutators with Lipschitz symbols.

Provides conditions on weights for the estimates to hold.

Abstract

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-$\delta$, $0<\delta\leq 1$.