Extension and trace results for doubling metric measure spaces and their hyperbolic fillings
Extension and trace results for doubling metric measure spaces and their hyperbolic fillings
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DOI：
10.1016/j.matpur.2021.12.003
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发表时间：
2020-08
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通讯作者：
Anders Bjorn;Jana Bjorn;N. Shanmugalingam
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作者：
Anders Bjorn;Jana Bjorn;N. Shanmugalingam
本文研究了紧化度量空间Z上函数的Besov空间与均匀域X ε上函数的Newton-Sobolev空间之间的联系。这个一致区域是由z的（Gromov）双曲填充的均匀化得到的。为此，我们以Bonk-Kleiner[9]和Bourdon-Pajot[13]的形式构造了双曲填充族。然后，对于每一个参数β> 0，我们构造了加倍测度ν在Z到X ε上的一个升程μ β，证明了μ β是加倍的，并且支持1- poincarcarcarr不等式。然后，我们证明了对于每一个0< θ< 1且p≥1的θ，存在β= p(1−θ) ε的选择，使得Besov空间B p， p θ (Z)是牛顿-索博列夫空间N 1， p （X ε, μ β）的迹空间。最后，我们利用X ε上的势理论的工具，得到了函数在B p， p θ (Z)上的准连续性和L - q- lebesgue点（q= s ν p/(s ν−p θ)）的准处处存在性，其中s ν是与Z上的测度ν相关的二次维数。将此应用于欧几里德空间的紧子集，改进了n上Netrusov[43]的结果。
In this paper we study connections between Besov spaces of functions on a compact metric space Z, equipped with a doubling measure, and the Newton–Sobolev space of functions on a uniform domain X ε. This uniform domain is obtained as a uniformization of a (Gromov) hyperbolic filling of Z. To do so, we construct a family of hyperbolic fillings in the style of Bonk–Kleiner [9] and Bourdon–Pajot [13]. Then for each parameter β> 0 we construct a lift μ β of the doubling measure ν on Z to X ε, and show that μ β is doubling and supports a 1-Poincaré inequality. We then show that for each θ with 0< θ< 1 and p≥ 1 there is a choice of β= p (1− θ) ε such that the Besov space B p, p θ (Z) is the trace space of the Newton–Sobolev space N 1, p (X ε, μ β). Finally, we exploit the tools of potential theory on X ε to obtain fine properties of functions in B p, p θ (Z), such as their quasicontinuity and quasieverywhere existence of L q-Lebesgue points with q= s ν p/(s ν− p θ), where s ν is a doubling dimension associated with the measure ν on Z. Applying this to compact subsets of Euclidean spaces improves upon a result of Netrusov [43] in R n.