A necessary condition for the boundedness of the maximal operator on $L^{p(\cdot)}$ over reverse doubling spaces of homogeneous type
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917616388407296 |
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| author | Karlovych, Oleksiy Shalukhina, Alina |
| author_facet | Karlovych, Oleksiy Shalukhina, Alina |
| contents | Let $(X,d,μ)$ be a space of homogeneous type and $p(\cdot):X\to[1,\infty]$ be a variable exponent. We show that if the measure $μ$ is Borel-semiregular and reverse doubling, then the condition ${\rm ess\,inf}_{x\in X}p(x)>1$ is necessary for the boundedness of the Hardy-Littlewood maximal operator $M$ on the variable Lebesgue space $L^{p(\cdot)}(X,d,μ)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_10915 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A necessary condition for the boundedness of the maximal operator on $L^{p(\cdot)}$ over reverse doubling spaces of homogeneous type Karlovych, Oleksiy Shalukhina, Alina Functional Analysis Classical Analysis and ODEs Let $(X,d,μ)$ be a space of homogeneous type and $p(\cdot):X\to[1,\infty]$ be a variable exponent. We show that if the measure $μ$ is Borel-semiregular and reverse doubling, then the condition ${\rm ess\,inf}_{x\in X}p(x)>1$ is necessary for the boundedness of the Hardy-Littlewood maximal operator $M$ on the variable Lebesgue space $L^{p(\cdot)}(X,d,μ)$. |
| title | A necessary condition for the boundedness of the maximal operator on $L^{p(\cdot)}$ over reverse doubling spaces of homogeneous type |
| topic | Functional Analysis Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2403.10915 |