标题： 通过双曲填充在度量测度空间中对分数 Hardy 不等式的自我改进 显示英文标题

标题： Self-improvement of fractional Hardy inequalities in metric measure spaces via hyperbolic fillings

摘要： 在本文中，我们证明了关于$(\theta,p)$-分数 Hardy 不等式的自改进结果，在有界域的指数$1<p<\infty$和正则性参数$0<\theta<1$方面，适用于加倍度量测度空间。关键的概念工具是一种类似于 Caffarelli-Silvestre 的论证，它通过迹结果将$Z$上的分数 Sobolev 空间与双曲填充$\overline{X}_{\varepsilon}$中的 Newton-Sobolev 空间联系起来，该双曲填充是$Z$的。 利用这一见解，证明了在$Z$的开子集中的分数 Hardy 不等式与填充区域$\overline{X}_{\varepsilon}$中的经典 Hardy 不等式等价。 然后通过应用关于$p$-Hardy 不等式的新的加权自改进结果得到了主要结果。 指数$p$可以通过经典的 Koskela-Zhong 方法进行自改进，但为了在正则性参数$\theta$中获得自改进，发展了一种新的可调节权重理论。 这推广了 Lehrbäck 和 Koskela 关于$d_\Omega^\beta$-加权$p$-Hardy 不等式的自改进结果，允许使用更广泛的权重类。 利用分数 Hardy 不等式与填充区域中 Hardy 不等式之间的等价性，我们还给出了满足分数 Hardy 不等式的域的新例子。 显示英文摘要

摘要： In this paper, we prove a self-improvement result for $(\theta,p)$-fractional Hardy inequalities, in both the exponent $1<p<\infty$ and the regularity parameter $0<\theta<1$, 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 $\theta$. This generalizes a result of Lehrb\"ack and Koskela on self-improvement of $d_\Omega^\beta$-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.