What question did this study set out to answer?

The aim is to establish weighted weak-type estimates for operators that adhere to limited-range sparse domination.

July 6, 2026Open Access

Endpoint Weak-type Bounds Beyond Calderón-Zygmund Theory

ZNZoe NieraethBasque Center for Applied Mathematics CSCody B. StockdaleClemson University

Key Points

The aim is to establish weighted weak-type estimates for operators that adhere to limited-range sparse domination.
Proved weighted weak-type estimates for operators under (r,s) domination.
Introduced new proofs for sharp bounds on boundedness of operators in specific weighted function spaces.
Considered operators satisfying standard sparse form domination, specifically Calderón-Zygmund operators.
Established that ||T||_{L^1_w(ℝ^d)→L^{1,∞}_w(ℝ^d)} ≤ [w]_1(1+log[w]_{FW}) with detailed conditions.
Provided improvements on bounds under limited-range and square function sparse domination.

Abstract

Abstract 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 ₋℉ₖ (ₑ㵧) ₋^₁, w (Rᵈ) w₁ (1+ w ₅ₖ). ‖ T ‖ L w 1 (R d) → L w 1, ∞ (R d) ≲ w 1 (1 + log w FW).

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