Endpoint weak-type bounds beyond Calder\'on-Zygmund theory

We prove weighted weak-type $(r,r)$ estimates for operators satisfying $(r,s)$ limited-range sparse domination of $\ell^q$-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 Calder\'on-Zygmund operators, we provide a new and simple proof of the sharp bound $$\|T\|_{L^1_w(\mathbf{R}^d)\rightarrow L^{1,\infty}_w(\mathbf{R}^d)} \lesssim [w]_1(1+\log [w]_{\text{FW}}).$$