Self-improvement of fractional Hardy inequalities in metric measure spaces via hyperbolic fillings
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| Format: | Preprint |
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2024
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| author | Eriksson-Bique, Sylvester Kline, Josh |
| author_facet | Eriksson-Bique, Sylvester Kline, Josh |
| contents | In this paper, we prove a self-improvement result for $(θ,p)$-fractional Hardy inequalities, in both the exponent $1<p<\infty$ and the regularity parameter $0<θ<1$, for bounded domains in doubling metric measure spaces. The key conceptual tool is a Caffarelli-Silvestre-type argument, which relates fractional Sobolev spaces on $Z$ to Newton-Sobolev spaces in the hyperbolic filling $\overline{X}_{\varepsilon}$ of $Z$ via trace results. Using this insight, it is shown that a fractional Hardy inequality in an open subset of $Z$ is equivalent to a classical Hardy inequality in the filling $\overline{X}_{\varepsilon}$. The main result is then obtained by applying a new weighted self-improvement result for $p$-Hardy inequalities. The exponent $p$ can be self-improved by a classical Koskela-Zhong argument, but a new theory of regularizable weights is developed to obtain the self-improvement in the regularity parameter $θ$. This generalizes a result of Lehrbäck and Koskela on self-improvement of $d_Ω^β$-weighted $p$-Hardy inequalities by allowing a much broader class of weights. Using the equivalence of fractional Hardy inequalities with Hardy inequalities in the fillings, we also give new examples of domains satisfying fractional Hardy inequalities. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_02848 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Self-improvement of fractional Hardy inequalities in metric measure spaces via hyperbolic fillings Eriksson-Bique, Sylvester Kline, Josh Analysis of PDEs Classical Analysis and ODEs Metric Geometry 35R11, (26D10, 28A75, 30L15, 31C15, 31E05, 35A23, 46E35) In this paper, we prove a self-improvement result for $(θ,p)$-fractional Hardy inequalities, in both the exponent $1<p<\infty$ and the regularity parameter $0<θ<1$, for bounded domains in doubling metric measure spaces. The key conceptual tool is a Caffarelli-Silvestre-type argument, which relates fractional Sobolev spaces on $Z$ to Newton-Sobolev spaces in the hyperbolic filling $\overline{X}_{\varepsilon}$ of $Z$ via trace results. Using this insight, it is shown that a fractional Hardy inequality in an open subset of $Z$ is equivalent to a classical Hardy inequality in the filling $\overline{X}_{\varepsilon}$. The main result is then obtained by applying a new weighted self-improvement result for $p$-Hardy inequalities. The exponent $p$ can be self-improved by a classical Koskela-Zhong argument, but a new theory of regularizable weights is developed to obtain the self-improvement in the regularity parameter $θ$. This generalizes a result of Lehrbäck and Koskela on self-improvement of $d_Ω^β$-weighted $p$-Hardy inequalities by allowing a much broader class of weights. Using the equivalence of fractional Hardy inequalities with Hardy inequalities in the fillings, we also give new examples of domains satisfying fractional Hardy inequalities. |
| title | Self-improvement of fractional Hardy inequalities in metric measure spaces via hyperbolic fillings |
| topic | Analysis of PDEs Classical Analysis and ODEs Metric Geometry 35R11, (26D10, 28A75, 30L15, 31C15, 31E05, 35A23, 46E35) |
| url | https://arxiv.org/abs/2412.02848 |