Weighted Weak-type Inequalities For Fractionally Sparsely Dominated Operators

TL;DR

This paper develops quantitative weak type bounds for operators dominated by fractional sparse operators, enhancing the understanding of their mapping properties in harmonic analysis.

Contribution

It introduces new weak type estimates for bilinear fractional sparse operators, extending previous results and providing precise bounds for their mapping behavior.

Findings

Established bounds for restricted weak type $L^{p,1}\rightarrow L^{q, infty}$

Derived multiplier weak type estimates for fractional sparse operators

Extended existing theory with refined quantitative bounds

Abstract

In this paper, we establish quantitative weak type estimates for operators that are dominated by (fractional) sparse operators in bilinear sense. Specifically, we derive bounds for both the restricted weak type $L^{p,1}\rightarrow L^{q,\infty}$ and the multiplier weak type, the latter of which has been previously considered by Cruz-Uribe and Sweeting. These estimates provide a precise quantification of the mapping properties of the considered operators, extending and refining the existing theory.