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July 6, 2026Potential Analysis0 citationsOpen Access

Endpoint Weak-type Bounds Beyond Calderón-Zygmund Theory

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ZNZoe NieraethCSCody B. Stockdale

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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Cite This Study

Nieraeth et al. (2026) studied this question.

synapsesocial.com/papers/6a4b4578997070ff83b5b2cahttps://doi.org/10.1007/s11118-026-10294-9
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