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Weak type (1,1) bounds for rough oscillatory singular integrals
Abstract
In their seminal work, F. Ricci and E. M. Stein [Proc. Nat. Acad. Sci. USA 83, 1--3 (1986)] introduced the oscillatory singular integral TPf(x)=p.v.∫ℝn eiP(x,y)(Ω(x−y))/(|x−y|n)f(y) dy, and S. Chanillo and M. Christ [Duke Math. J. 55, 141--155 (1987)] established its weak-type (1,1) inequality for smooth Ω. For the rough variant, however, the weak-type (1,1) inequality remained a long-standing open question. In the present paper, we settle this problem: specifically, we prove that the rough oscillatory singular integral is of weak type (1,1) with Ω∈ Llog+L(𝕊n−1). Moreover, we establish weak-type (1,1) bounds for the rough maximal oscillatory operator associated with TP, a result of independent interest.
Related
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- Weak type (1,1) estimates for noncommutative rough singular integrals
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- SPARSE POINTWISE BOUNDS FOR MAXIMAL TRUNCATIONS OF ROUGH SINGULAR INTEGRALS AND SOBOLEV-TYPE INEQUALITIES
- Pointwise Riesz potential bounds for rough singular integrals
BibTeX
@misc{chen2026weak,
title = {{Weak type $(1,1)$ bounds for rough oscillatory singular integrals}},
author = {Yanping Chen and Zhengyang Ji},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02789},
eprint = {2610.02789},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02789},
}
RIS
TY - UNPB
TI - Weak type (1,1) bounds for rough oscillatory singular integrals
AU - Yanping Chen
AU - Zhengyang Ji
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.02789
UR - https://arxiv.org/abs/2610.02789
AB - In their seminal work, F. Ricci and E. M. Stein [Proc. Nat. Acad. Sci. USA 83, 1--3 (1986)] introduced the oscillatory singular integral T_Pf(x)=p.v.∫_ℝ^n e^iP(x,y)(Ω(x−y))/(|x−y|^n)f(y) dy, and S. Chanillo and M. Christ [Duke Math. J. 55, 141--155 (1987)] established its weak-type (1,1) inequality for smooth Ω. For the rough variant, however, the weak-type (1,1) inequality remained a long-standing open question. In the present paper, we settle this problem: specifically, we prove that the rough oscillatory singular integral is of weak type (1,1) with Ω∈ Llog^+L(𝕊^n−1). Moreover, we establish weak-type (1,1) bounds for the rough maximal oscillatory operator associated with T_P, a result of independent interest.
ER -