Hardy--Littlewood Maximal Operator and Two-Layer Muckenhoupt Weights on Infinite Rooted k-Ary Trees

Abstract

Let k≥ 2 be an integer, T an infinite rooted k-ary tree, and M the Hardy--Littlewood maximal operator on T. For any p∈(0,∞), we characterize the weight w such that M is bounded on L^p(w). To this end, we introduce a two-layer Muckenhoupt weight class 𝒜_p and prove that, for any p∈(1/2,∞), the boundedness of M on L^p(w), w∈𝒜_p, and the exponential decay boundedness of spherical averaging operators on L^p(w) are mutually equivalent, and that, when p∈(0,1/2], there exists no weight w such that M is bounded on L^p(w). Moreover, for any p∈(1/2,∞), we establish the quantitative estimate, with the optimal exponent 1/p of the weight constant, for the boundedness of M on L^p(w). For any p∈(1,∞), we also obtain two further equivalent characterizations of the boundedness of M on L^p(w), respectively, in terms of a global Sawyer-type testing condition and an estimate for the weighted product measure of distance incidence sets. As applications, for any p∈(1/2,∞), under the assumption that M is bounded on L^p(w), we establish the boundedness of exponentially decaying kernel operators on L^p(w) and weighted Fefferman--Stein vector-valued inequalities.

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BibTeX RIS

BibTeX

@misc{yang2026hardylittlewood,
  title = {{Hardy--Littlewood Maximal Operator and Two-Layer Muckenhoupt Weights on Infinite Rooted $k$-Ary Trees}},
  author = {Dachun Yang and Wen Yuan and Mingdong Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12844},
  eprint = {2609.12844},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12844},
}

RIS

TY  - UNPB
TI  - Hardy--Littlewood Maximal Operator and Two-Layer Muckenhoupt Weights on Infinite Rooted k-Ary Trees
AU  - Dachun Yang
AU  - Wen Yuan
AU  - Mingdong Zhang
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12844
UR  - https://arxiv.org/abs/2609.12844
AB  - Let k≥ 2 be an integer, T an infinite rooted k-ary tree, and M the Hardy--Littlewood maximal operator on T. For any p∈(0,∞), we characterize the weight w such that M is bounded on L^p(w). To this end, we introduce a two-layer Muckenhoupt weight class 𝒜_p and prove that, for any p∈(1/2,∞), the boundedness of M on L^p(w), w∈𝒜_p, and the exponential decay boundedness of spherical averaging operators on L^p(w) are mutually equivalent, and that, when p∈(0,1/2], there exists no weight w such that M is bounded on L^p(w). Moreover, for any p∈(1/2,∞), we establish the quantitative estimate, with the optimal exponent 1/p of the weight constant, for the boundedness of M on L^p(w). For any p∈(1,∞), we also obtain two further equivalent characterizations of the boundedness of M on L^p(w), respectively, in terms of a global Sawyer-type testing condition and an estimate for the weighted product measure of distance incidence sets. As applications, for any p∈(1/2,∞), under the assumption that M is bounded on L^p(w), we establish the boundedness of exponentially decaying kernel operators on L^p(w) and weighted Fefferman--Stein vector-valued inequalities.
ER  -

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