Abstract
Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\) with \(n\ge2\) and \(s\in(0,1)\). Assume that \(\phi \colon [0, \infty) \to [0, \infty)\) is a Young function obeying the doubling condition with the constant \(K_\phi< 2^{\frac{n}{s}}\). We demonstrate that \(\Omega\) supports a \((\phi_\frac{n}{s}, \phi)\)-Poincaré inequality if it is a John domain. Alternatively, assume further that \(\Omega\) is a bounded domain that is quasiconformally equivalent to a uniform domain (for \(n\geq3\)) or a simply connected domain (for \(n=2\)), then we show that \(\Omega\) is a John domain if a \((\phi_\frac{n}{s}, \phi)\)-Poincaré inequality holds.
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The authors are partially supported by National Natural Science Foundation of China (No. 12201238).
The second author is also supported by GuangDong Basic and Applied Basic Research Foundation (Grant No. 2022A1515111056), Bureau of Science and Technology of Huizhou Municipality (Grant No. 2023EQ050040) and the Professorial and Doctoral Scientific Research Foundation of Huizhou University (Grant No. 2021JB035).
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Feng, S., Liang, T. A \((\phi_\frac{n}{s}, \phi)\)-Poincaré inequality on John domains. Anal Math (2024). https://doi.org/10.1007/s10476-024-00038-5
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DOI: https://doi.org/10.1007/s10476-024-00038-5