A metric counterpart of the Gu-Yung formula
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916168091041792 |
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| author | Buccheri, Stefano Górny, Wojciech |
| author_facet | Buccheri, Stefano Górny, Wojciech |
| contents | In this note we consider a generalisation to the metric setting of the recent work [Gu-Yung, JFA 281 (2021), 109075]. In particular, we show that under relatively weak conditions on a metric measure space $(X,d,ν)$, it holds true that \[ \bigg[ \frac{u(x)-u(y)}{d(x,y)^{\frac{s}{p}}} \bigg]_{L^p_w(X \times X, ν\otimes ν)} \approx \| u \|_{L^p(X,ν)}, \] where $s$ is a generalised dimension associated to $X$ and $[\cdot]_{L^p_w}$ is the weak Lebesgue norm. We provide some counterexamples which show that our assumptions are optimal. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_13475 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A metric counterpart of the Gu-Yung formula Buccheri, Stefano Górny, Wojciech Functional Analysis Metric Geometry 26D10, 26A33, 28C15, 35A23, 46E30 In this note we consider a generalisation to the metric setting of the recent work [Gu-Yung, JFA 281 (2021), 109075]. In particular, we show that under relatively weak conditions on a metric measure space $(X,d,ν)$, it holds true that \[ \bigg[ \frac{u(x)-u(y)}{d(x,y)^{\frac{s}{p}}} \bigg]_{L^p_w(X \times X, ν\otimes ν)} \approx \| u \|_{L^p(X,ν)}, \] where $s$ is a generalised dimension associated to $X$ and $[\cdot]_{L^p_w}$ is the weak Lebesgue norm. We provide some counterexamples which show that our assumptions are optimal. |
| title | A metric counterpart of the Gu-Yung formula |
| topic | Functional Analysis Metric Geometry 26D10, 26A33, 28C15, 35A23, 46E30 |
| url | https://arxiv.org/abs/2403.13475 |