Weighted Endpoint Variational and Jump Inequalites for Rough Singular Integrals

Abstract

In this present paper, we study the weighted weak type (1, 1) bounds for rough maximal singular integral operators as well as the the corresponding variation and jump operators. Our first result is the weighted weak type (1, 1) bound for the maximal singular integral T_Ω^*f(x)=sup_ε>0|T_ε,Ωf(x)|=sup_ε>0|∫_|x−y|>ε (Ω(x−y))/(|x−y|^d)f(y)dy|, where Ω∈ L^∞(𝕊^d−1), homogeneous of degree zero, and satisfies the cancellation condition. We show that |T_Ω^*|_{L¹(w)→ L^1,∞(w)}≲[w]_A₁[w]_{A_∞}log([w]_{A_∞}+1), where w belongs to the Muckenhoupt class A₁(ℝ^d). This represents an essential improvement of a result (Honz\'ık, Inter.Math. Res. Not. 2020) and a result (Bhojak and Mohanty, J. Funct. Anal.2023). Our second one is the weighted weak type (1, 1) bounds for variation and jump operators corresponding to {T_ε,Ω}_ε∈2^ℤ and {T_{ε, Ω}^ϕ}_ε∈ℝ⁺ under the condition Ω∈ L^∞(𝕊^d−1), where T_ε,Ω^ϕ represents a smooth truncation of rough singular integral operator.These results of this part are the first weighted weak type (1, 1) variation inequalities and jump inequalities for rough singular integrals.

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BibTeX RIS

BibTeX

@misc{chen2026weighted,
  title = {{Weighted Endpoint Variational and Jump Inequalites for Rough Singular Integrals}},
  author = {Ting Chen and Feng Liu and Zhou Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12361},
  eprint = {2609.12361},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12361},
}

RIS

TY  - UNPB
TI  - Weighted Endpoint Variational and Jump Inequalites for Rough Singular Integrals
AU  - Ting Chen
AU  - Feng Liu
AU  - Zhou Wang
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12361
UR  - https://arxiv.org/abs/2609.12361
AB  - In this present paper, we study the weighted weak type (1, 1) bounds for rough maximal singular integral operators as well as the the corresponding variation and jump operators. Our first result is the weighted weak type (1, 1) bound for the maximal singular integral T_Ω^*f(x)=sup_ε>0|T_ε,Ωf(x)|=sup_ε>0|∫_|x−y|>ε (Ω(x−y))/(|x−y|^d)f(y)dy|, where Ω∈ L^∞(𝕊^d−1), homogeneous of degree zero, and satisfies the cancellation condition. We show that |T_Ω^*|_L^1(w)→ L^1,∞(w)≲[w]_A_1[w]_A_∞log([w]_A_∞+1), where w belongs to the Muckenhoupt class A_1(ℝ^d). This represents an essential improvement of a result (Honz\'ık, Inter.Math. Res. Not. 2020) and a result (Bhojak and Mohanty, J. Funct. Anal.2023). Our second one is the weighted weak type (1, 1) bounds for variation and jump operators corresponding to {T_ε,Ω}_ε∈2^ℤ and {T_ε, Ω^ϕ}_ε∈ℝ^+ under the condition Ω∈ L^∞(𝕊^d−1), where T_ε,Ω^ϕ represents a smooth truncation of rough singular integral operator.These results of this part are the first weighted weak type (1, 1) variation inequalities and jump inequalities for rough singular integrals.
ER  -

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