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The Zygmund conjecture and Rey's exponential integrability conjecture
Abstract
We prove the Zygmund conjecture in all parameters: the maximal operator associated with products of cubes with side lengths (s1,…,sm−1,ϕ(s1,…,sm−1)), where ϕ is any positive coordinatewise nondecreasing function, satisfies the weak L(log L)m−2 estimate for every m≥3. The estimate is new for m>3. Our approach is to prove Rey's recent conjecture on the exponential integrability of the overlap function h𝒢=∑I∈𝒢1I, with exponent 1/(m−1), for sparse incomparable families 𝒢 of m-parameter dyadic rectangles, m≥2. The Zygmund conjecture, even in the continuous form above, follows by strengthening a reduction given by Rey.
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BibTeX
@misc{martikainen2026zygmund,
title = {{The Zygmund conjecture and Rey's exponential integrability conjecture}},
author = {Henri Martikainen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23895},
eprint = {2609.23895},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23895},
}
RIS
TY - UNPB
TI - The Zygmund conjecture and Rey's exponential integrability conjecture
AU - Henri Martikainen
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.23895
UR - https://arxiv.org/abs/2609.23895
AB - We prove the Zygmund conjecture in all parameters: the maximal operator associated with products of cubes with side lengths (s_1,…,s_m−1,ϕ(s_1,…,s_m−1)), where ϕ is any positive coordinatewise nondecreasing function, satisfies the weak L(log L)^m−2 estimate for every m≥3. The estimate is new for m>3. Our approach is to prove Rey's recent conjecture on the exponential integrability of the overlap function h_𝒢=∑_I∈𝒢1_I, with exponent 1/(m−1), for sparse incomparable families 𝒢 of m-parameter dyadic rectangles, m≥2. The Zygmund conjecture, even in the continuous form above, follows by strengthening a reduction given by Rey.
ER -