Pointwise Riesz potential bounds for rough singular integrals
Abstract
We give a complete characterization of the kernels Ω∈ Lup1(Sp) for which the rough singular integral TΩ and the associated maximal operator MΩ satisfy pointwise bounds by Int1(|∇ ·|) and M1(|∇ ·|), respectively, where Int1 is the Riesz potential and M1 is the fractional maximal operator. These bounds hold if and only if Ω belongs to the homogeneous dual Sobolev space Ẇ−1,∞(Sp). Equivalently, Ω=divSp V for some bounded vector field V on the sphere, or |∫EΩ|≤ C ℋd−2(∂ E) for every open set E⊆Sp with smooth boundary. On the Lebesgue scale, we show that for d≥3 the condition Ω∈Lupd−1,∞(Sp), together with mean zero, suffices, and that the exponent d−1 is sharp. Finally, in dimension d=2, we use complex-analytic methods to prove a new pointwise bound in terms of the complex derivative ∂̅ f that is valid for every mean-zero Ω∈Lup1(𝕊1). As a consequence, we obtain new Sobolev bounds for rough singular integrals in the plane.
Related
- SPARSE POINTWISE BOUNDS FOR MAXIMAL TRUNCATIONS OF ROUGH SINGULAR INTEGRALS AND SOBOLEV-TYPE INEQUALITIES
- On pointwise convergence of singular integrals of Christ-Journé type
- Weak type (1,1) bounds for rough oscillatory singular integrals
- Sharp weighted weak-type bounds for commutators of singular integrals
- Sparse bounds for maximal rough singular integrals
BibTeX
@misc{moen2026pointwise,
title = {{Pointwise Riesz potential bounds for rough singular integrals}},
author = {Kabe Moen and Carlos Pérez},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31888},
eprint = {2610.31888},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31888},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Pointwise Riesz potential bounds for rough singular integrals
AU - Kabe Moen
AU - Carlos Pérez
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31888
UR - https://arxiv.org/abs/2610.31888
AB - We give a complete characterization of the kernels Ω∈ Lup^1(Sp) for which the rough singular integral T_Ω and the associated maximal operator M_Ω satisfy pointwise bounds by Int_1(|∇ ·|) and M_1(|∇ ·|), respectively, where Int_1 is the Riesz potential and M_1 is the fractional maximal operator. These bounds hold if and only if Ω belongs to the homogeneous dual Sobolev space Ẇ^−1,∞(Sp). Equivalently, Ω=div_Sp V for some bounded vector field V on the sphere, or |∫_EΩ|≤ C ℋ^d−2(∂ E) for every open set E⊆Sp with smooth boundary. On the Lebesgue scale, we show that for d≥3 the condition Ω∈Lup^d−1,∞(Sp), together with mean zero, suffices, and that the exponent d−1 is sharp. Finally, in dimension d=2, we use complex-analytic methods to prove a new pointwise bound in terms of the complex derivative ∂̅ f that is valid for every mean-zero Ω∈Lup^1(𝕊^1). As a consequence, we obtain new Sobolev bounds for rough singular integrals in the plane.
ER -