On Variational Inequalities for Averages over Hypersurfaces
Abstract
This paper establishes sharp variational estimates for families of averaging operators over smooth hypersurfaces in ℝn+1. For a hypersurface with k non-zero principal curvatures at a given point, we prove that the r-variation operator Vr(At f : t>0) is bounded on Lp for all exponents satisfying (k+1)/k < p < r(k+1), with operator norm depending explicitly on a translation parameter D. In the special case k=n (i.e., non-vanishing Gaussian curvature), we show that the range p<r(n+1) is optimal, thereby recovering and extending classical results for the sphere. Furthermore, for convex hypersurfaces of finite type (in the sense of Iosevich-Sawyer), we obtain variational bounds in the range p0 < p < 2r, where p0 is the contact index of the associated hypersurface. The proofs combine long-short variational decompositions, stationary phase analysis, vector-valued singular integrals, and local smoothing estimates for Fourier integral operators. These results unify and generalize earlier work on maximal functions and variational inequalities for averages over hypersurfaces.
Related
- Noncommutative Maximal Averages over Submanifolds and Variable Hypersurfaces
- Two-parameter variational estimates for averages over tori
- Lp-Estimates for maximal averages along mixed homogeneous hypersurfaces in ℝ3
- Noncommutative maximal inequalities for polynomial ergodic averages
- Odd order cases of Chowla's averages over arbitrary rectangles
BibTeX
@misc{nie2026variational,
title = {{On Variational Inequalities for Averages over Hypersurfaces}},
author = {Xudong Nie and Panwang Wang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01816},
eprint = {2611.01816},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01816},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Variational Inequalities for Averages over Hypersurfaces
AU - Xudong Nie
AU - Panwang Wang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01816
UR - https://arxiv.org/abs/2611.01816
AB - This paper establishes sharp variational estimates for families of averaging operators over smooth hypersurfaces in ℝ^n+1. For a hypersurface with k non-zero principal curvatures at a given point, we prove that the r-variation operator V_r(A_t f : t>0) is bounded on L^p for all exponents satisfying (k+1)/k < p < r(k+1), with operator norm depending explicitly on a translation parameter D. In the special case k=n (i.e., non-vanishing Gaussian curvature), we show that the range p<r(n+1) is optimal, thereby recovering and extending classical results for the sphere. Furthermore, for convex hypersurfaces of finite type (in the sense of Iosevich-Sawyer), we obtain variational bounds in the range p_0 < p < 2r, where p_0 is the contact index of the associated hypersurface. The proofs combine long-short variational decompositions, stationary phase analysis, vector-valued singular integrals, and local smoothing estimates for Fourier integral operators. These results unify and generalize earlier work on maximal functions and variational inequalities for averages over hypersurfaces.
ER -