Disproof of Wright's Problems 14 and 15 on lacunary maximal functions
Abstract
We disprove the proposed weak endpoint bounds in James Wright's Problems 14 and 15 from the 2017 collection of harmonic-analysis problems. For every d≥1, we exhibit a fixed compactly supported singular probability measure with Fourier decay of every order 0<ε<1/2, for which the dyadic convolution maximal operator has infinite weak L1 and infinite H1 to weak L1 operator norms. For every finite nonempty set of dyadic scales, both optimal weak constants equal the number of scales, with the Hardy constant measured in the specified atomic norm. Increasingly separated renewal digits provide uniform real-frequency Fourier decay while allowing the disjoint output regions needed for the positive and mean-zero tests. We first establish the fixed measure, then prove the two endpoint estimates using separate normalized tests. Finally we prove the two Fourier lemmas used in the construction.
Related
- Fourier decay and endpoint estimates for lacunary maximal operators
- Lacunary maximal operators on convex boundaries
- Maximal volume of convex bodies avoiding random points and lacunary dilates
- Failure of the H1× H1⟶ L1/2,∞ estimate for the lacunary bilinear spherical maximal function and applications
- On variation functions of lacunary series
BibTeX
@misc{cen2026disproof,
title = {{Disproof of Wright's Problems 14 and 15 on lacunary maximal functions}},
author = {Xi Cen and Guixiang Hong and Xinfeng Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23563},
eprint = {2610.23563},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23563},
}
RIS
TY - UNPB
TI - Disproof of Wright's Problems 14 and 15 on lacunary maximal functions
AU - Xi Cen
AU - Guixiang Hong
AU - Xinfeng Wu
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23563
UR - https://arxiv.org/abs/2610.23563
AB - We disprove the proposed weak endpoint bounds in James Wright's Problems 14 and 15 from the 2017 collection of harmonic-analysis problems. For every d≥1, we exhibit a fixed compactly supported singular probability measure with Fourier decay of every order 0<ε<1/2, for which the dyadic convolution maximal operator has infinite weak L^1 and infinite H^1 to weak L^1 operator norms. For every finite nonempty set of dyadic scales, both optimal weak constants equal the number of scales, with the Hardy constant measured in the specified atomic norm. Increasingly separated renewal digits provide uniform real-frequency Fourier decay while allowing the disjoint output regions needed for the positive and mean-zero tests. We first establish the fixed measure, then prove the two endpoint estimates using separate normalized tests. Finally we prove the two Fourier lemmas used in the construction.
ER -