Weighted Weak-type Inequalities For Fractionally Sparsely Dominated Operators

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.