Endpoint weak-type bounds beyond Calderón-Zygmund theory

We prove weighted weak-type $(r,r)$ estimates for operators satisfying $(r,s)$ limited-range sparse domination of ${\mathrm{\ell }}^q$-type. Our results contain improvements for operators satisfying limited-range and square function sparse domination. In the case of operators $T$ satisfying standard sparse form domination such as Calderón-Zygmund operators, we provide a new and simple proof of the sharp bound $$\mathrm{\parallel }T{\mathrm{\parallel }}_{L_w^1({\mathbf{R}}^d)\to L_w^{1,\mathrm{\infty }}({\mathbf{R}}^d)}\lesssim [w{]}_1(1+\log [w{]}_{\text{FW}})\mathrm{.}$$