The maximal function on spaces of homogeneous type, or adjacent dyadic cubes do good
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913160743616512 |
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| author | Shalukhina, Alina |
| author_facet | Shalukhina, Alina |
| contents | We prove that the Hardy--Littlewood maximal operator $M$ is bounded on the variable Lebesgue space $L^{p(\cdot)}(X,d,μ)$, with $1<p_-\le p_+<\infty$, over an unbounded space of homogeneous type $(X,d,μ)$ with a Borel-semiregular measure $μ$, if and only if the averaging operators $T_\mathcal{Q}$ are bounded on $L^{p(\cdot)}(X,d,μ)$ uniformly over all families $\mathcal{Q}$ of pairwise disjoint ``cubes'' from a Hytönen--Kairema dyadic system on $X$. This extends Diening's well-known characterization of the boundedness of $M$ on $L^{p(\cdot)}(\mathbb{R}^n)$ to the setting of spaces of homogeneous type, while also providing a slight refinement of the original result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02508 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The maximal function on spaces of homogeneous type, or adjacent dyadic cubes do good Shalukhina, Alina Classical Analysis and ODEs 42B25, 46E30, 30L99 We prove that the Hardy--Littlewood maximal operator $M$ is bounded on the variable Lebesgue space $L^{p(\cdot)}(X,d,μ)$, with $1<p_-\le p_+<\infty$, over an unbounded space of homogeneous type $(X,d,μ)$ with a Borel-semiregular measure $μ$, if and only if the averaging operators $T_\mathcal{Q}$ are bounded on $L^{p(\cdot)}(X,d,μ)$ uniformly over all families $\mathcal{Q}$ of pairwise disjoint ``cubes'' from a Hytönen--Kairema dyadic system on $X$. This extends Diening's well-known characterization of the boundedness of $M$ on $L^{p(\cdot)}(\mathbb{R}^n)$ to the setting of spaces of homogeneous type, while also providing a slight refinement of the original result. |
| title | The maximal function on spaces of homogeneous type, or adjacent dyadic cubes do good |
| topic | Classical Analysis and ODEs 42B25, 46E30, 30L99 |
| url | https://arxiv.org/abs/2509.02508 |