PreprintarXiv licencearXiv
Fourier decay and endpoint estimates for lacunary maximal operators
Abstract
We study endpoint estimates for lacunary maximal operators associated with compactly supported measures. For every d≥1, we construct a compactly supported, absolutely continuous probability measure on ℝd with polynomial Fourier decay whose associated lacunary maximal operator fails to map H1(ℝd) boundedly into L1,∞(ℝd). In particular, it fails to be of weak type (1,1). This gives a negative answer to the weak type (1,1) question posed by Seeger and Wright in 2011 and to Open Problems 14 and 15, contributed by Wright at the 2016 Conference in Harmonic Analysis, held at the University of Wisconsin--Madison in honor of Professor Michael Christ.
Related
- Endpoint estimates for spherical maximal operators acting on radial functions in hyperbolic spaces
- Lacunary maximal operators on convex boundaries
- Endpoint maximal estimates for the half-wave propagator
- Strong-type endpoint obstructions for bilinear spherical maximal operators
- Endpoint gradient bounds for centered convex maximal operators and probability kernels
BibTeX
@misc{zhou2026fourier,
title = {{Fourier decay and endpoint estimates for lacunary maximal operators}},
author = {Xiaojing Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21197},
eprint = {2610.21197},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21197},
}
RIS
TY - UNPB
TI - Fourier decay and endpoint estimates for lacunary maximal operators
AU - Xiaojing Zhou
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21197
UR - https://arxiv.org/abs/2610.21197
AB - We study endpoint estimates for lacunary maximal operators associated with compactly supported measures. For every d≥1, we construct a compactly supported, absolutely continuous probability measure on ℝ^d with polynomial Fourier decay whose associated lacunary maximal operator fails to map H^1(ℝ^d) boundedly into L^1,∞(ℝ^d). In particular, it fails to be of weak type (1,1). This gives a negative answer to the weak type (1,1) question posed by Seeger and Wright in 2011 and to Open Problems 14 and 15, contributed by Wright at the 2016 Conference in Harmonic Analysis, held at the University of Wisconsin--Madison in honor of Professor Michael Christ.
ER -