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The Hardy-Littlewood-type equivalence problem and its applications
Abstract
The main purpose of this paper is to resolve the Hardy-Littlewood-type equivalence problem. In particular, we show that a Hardy-Littlewood-type theorem holds for arbitrary majorants in the case p∈(1,∞), and that the equivalence holds in the case p=1. For the proof, we employ new functional analytic approaches based on Fourier series, the Hilbert transform, and the uniform boundedness principle. Those proof methods are completely different from those of Dyakonov [Acta Math., 178 (1997), 143-167] and Pavlović [Acta Math., 183 (1999), 141-143]. In addition, we present several representative applications that improve a series of known results.
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BibTeX
@misc{chen2026hardylittlewoodtype,
title = {{The Hardy-Littlewood-type equivalence problem and its applications}},
author = {Jiaolong Chen and Shaolin Chen and Hamada Hamada and Qianyun Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31132},
eprint = {2610.31132},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31132},
}
RIS
TY - UNPB
TI - The Hardy-Littlewood-type equivalence problem and its applications
AU - Jiaolong Chen
AU - Shaolin Chen
AU - Hamada Hamada
AU - Qianyun Li
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31132
UR - https://arxiv.org/abs/2610.31132
AB - The main purpose of this paper is to resolve the Hardy-Littlewood-type equivalence problem. In particular, we show that a Hardy-Littlewood-type theorem holds for arbitrary majorants in the case p∈(1,∞), and that the equivalence holds in the case p=1. For the proof, we employ new functional analytic approaches based on Fourier series, the Hilbert transform, and the uniform boundedness principle. Those proof methods are completely different from those of Dyakonov [Acta Math., 178 (1997), 143-167] and Pavlović [Acta Math., 183 (1999), 141-143]. In addition, we present several representative applications that improve a series of known results.
ER -