Abstract
Let \(\Omega \) be a bounded non-smooth domain in \(\mathbb {R}^n\) that satisfies the measure density condition. In this paper, the authors study the interrelations of three basic types of Besov spaces \(B_{p,q}^s(\Omega )\), \(\mathring{B}_{p,q}^s(\Omega )\) and \(\widetilde{B}_{p,q}^s(\Omega )\) on \(\Omega \), which are defined, respectively, via the restriction, completion and supporting conditions with \(p,q\in [1,\infty )\) and \(s\in (0,1)\). The authors prove that \(B_{p,q}^s(\Omega )=\mathring{B}_{p,q}^s(\Omega )=\widetilde{B}_{p,q}^s(\Omega )\), if \(\Omega \) supports a fractional Besov–Hardy inequality, where the latter is proved under certain conditions on fractional Besov capacity or Aikawa’s dimension of the boundary of \(\Omega \).
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The authors are very grateful to the anonymous referee for her/his very careful reading and so many valuable comments which essentially improve the quality of the article.
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This project is supported by the National Natural Science Foundation of China (Grant Nos. 12071431 and 11771395) and the Zhejiang Provincial Natural Science Foundation of China (Grant No. LR22A010006).
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Cao, J., Jin, Y., Yu, Z. et al. Fractional Besov spaces and Hardy inequalities on bounded non-smooth domains. Annali di Matematica (2024). https://doi.org/10.1007/s10231-024-01430-6
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DOI: https://doi.org/10.1007/s10231-024-01430-6