Sharp off-diagonal weighted weak type estimates for sparse operators

TL;DR

This paper establishes sharp weak type weighted estimates for a class of sparse operators, including fractional and singular integral operators, with a focus on optimal bounds involving Muckenhoupt weights.

Contribution

The paper introduces the first sharp weak type weighted bounds for sparse operators, extending known results to fractional and square function operators with optimal weight dependence.

Findings

Proved sharp weak type weighted estimates for sparse operators.

Derived optimal bounds involving $A_{p,q}$ and $A_$ weight classes.

Established conditions for fractional square functions on $L^q(w^q)$ with $p>2$.

Abstract

We prove sharp weak type weighted estimates for a class of sparse operators that includes majorants of standard $\alpha$-fractional singular integrals, fractional integral operators, Marcinkiewicz integral operators, and square functions. These bounds are knows to be sharp in many cases, and our main new result is the optimal bound $$[w]_{A_{p,q}}^{\frac{1}{q}}[w^{q}]_{A_{\infty}}^{\frac{1}{2}-\frac{1}{p}}\lesssim[w]_{A_{p,q}}^{\frac{1}{2}-\frac{\alpha}{d}}$$ for proper conditions which satisfy that three index $p$, $q$ and $\alpha$ ensure weak type norm of fractional square functions on $L^{q}(w^{q})$ with $p>2$.