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 q ℓ 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 T _L¹w(Rᵈ)→ L1,∞w(Rᵈ) [w]₁(1+log [w]FW). ‖ T ‖ L w 1 ( R d ) → L w 1 , ∞ ( R d ) ≲ [ w ] 1 ( 1 + log [ w ] FW ) .