Finite dimensionality of Besov spaces and potential-theoretic decomposition of metric spaces
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arXiv
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| Format: | Preprint |
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2024
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| author | Kumagai, Takashi Shanmugalingam, Nageswari Shimizu, Ryosuke |
| author_facet | Kumagai, Takashi Shanmugalingam, Nageswari Shimizu, Ryosuke |
| contents | In the context of a metric measure space $(X,d,μ)$, we explore the potential-theoretic implications of having a finite-dimensional Besov space. We prove that if the dimension of the Besov space $B^θ_{p,p}(X)$ is $k>1$, then $X$ can be decomposed into $k$ number of irreducible components (Theorem 1.1). Note that $θ$ may be bigger than $1$, as our framework includes fractals. We also provide sufficient conditions under which the dimension of the Besov space is $1$. We introduce critical exponents $θ_p(X)$ and $θ_p^{\ast}(X)$ for the Besov spaces. As examples illustrating Theorem 1.1, we compute these critical exponents for spaces $X$ formed by glueing copies of $n$-dimensional cubes, the Sierpiński gaskets, and of the Sierpiński carpet. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01292 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite dimensionality of Besov spaces and potential-theoretic decomposition of metric spaces Kumagai, Takashi Shanmugalingam, Nageswari Shimizu, Ryosuke Functional Analysis Metric Geometry 31E05, 28A80 (Primary) 46E36, 31C25 (Secondary) In the context of a metric measure space $(X,d,μ)$, we explore the potential-theoretic implications of having a finite-dimensional Besov space. We prove that if the dimension of the Besov space $B^θ_{p,p}(X)$ is $k>1$, then $X$ can be decomposed into $k$ number of irreducible components (Theorem 1.1). Note that $θ$ may be bigger than $1$, as our framework includes fractals. We also provide sufficient conditions under which the dimension of the Besov space is $1$. We introduce critical exponents $θ_p(X)$ and $θ_p^{\ast}(X)$ for the Besov spaces. As examples illustrating Theorem 1.1, we compute these critical exponents for spaces $X$ formed by glueing copies of $n$-dimensional cubes, the Sierpiński gaskets, and of the Sierpiński carpet. |
| title | Finite dimensionality of Besov spaces and potential-theoretic decomposition of metric spaces |
| topic | Functional Analysis Metric Geometry 31E05, 28A80 (Primary) 46E36, 31C25 (Secondary) |
| url | https://arxiv.org/abs/2409.01292 |