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Lacunary maximal operators on convex boundaries
Abstract
We disprove the cap-sufficiency proposal in Wright's Problem 16 for 1<p<(√(13)−1)/2 and for p>3. For each fixed exponent in these ranges, one smooth strictly convex domain has finite original directional cap integral, while the lacunary maximal operator of complete arclength fails restricted weak type. We also give a geometric sufficient condition for strong Lp boundedness, for every finite 1<p<∞, on an arbitrary given bounded convex domain. It requires one logarithmic integral of a curvature lower envelope, without a differentiability assumption on the boundary or a restriction on the number of curvature zeros. Both results concern unnormalised true arclength, one fixed interior origin and all integer dyadic dilations.
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BibTeX
@misc{cen2026lacunary,
title = {{Lacunary maximal operators on convex boundaries}},
author = {Xi Cen and Xinfeng Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32873},
eprint = {2610.32873},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32873},
}
RIS
TY - UNPB
TI - Lacunary maximal operators on convex boundaries
AU - Xi Cen
AU - Xinfeng Wu
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32873
UR - https://arxiv.org/abs/2610.32873
AB - We disprove the cap-sufficiency proposal in Wright's Problem 16 for 1<p<(√(13)−1)/2 and for p>3. For each fixed exponent in these ranges, one smooth strictly convex domain has finite original directional cap integral, while the lacunary maximal operator of complete arclength fails restricted weak type. We also give a geometric sufficient condition for strong L^p boundedness, for every finite 1<p<∞, on an arbitrary given bounded convex domain. It requires one logarithmic integral of a curvature lower envelope, without a differentiability assumption on the boundary or a restriction on the number of curvature zeros. Both results concern unnormalised true arclength, one fixed interior origin and all integer dyadic dilations.
ER -