论文标题
来自Sobolev类型空间的Poincaré不平等和紧凑的嵌入到度量空间上的加权$ l^Q $空间
Poincaré inequalities and compact embeddings from Sobolev type spaces into weighted $L^q$ spaces on metric spaces
论文作者
论文摘要
我们研究了从公制空间上的Sobolev类型空间的嵌入的紧凑性和界限，相对于另一种措施，我们研究了$ l^q $空间。所考虑的Sobolev空间可以是分数顺序的，并且某些陈述也允许采取任何措施。我们的结果是使用覆盖家庭和当地庞加莱型不平等的序列以一般形式提出的。我们展示了如何构建这种合适的掩护和庞加莱的不平等现象。对于本地加倍措施，我们证明了两加权庞加莱不平等的自我改善特性，这也适用于较低维度的措施。 我们同时处理各种Sobolev空间，例如牛顿，分数Hajłasz和Poincaré型空间，用于相当一般的测量和集合，包括分形边界的分形和域。通过考虑在此类域的边界上进行较低维度的度量，我们获得了上述空间的痕量嵌入。在牛顿空间的情况下，我们将嵌入到$ l^q $空间相对于另一种措施中的$ l^q $空间时，是紧凑的。我们的工具由具体示例说明。对于满足适当维度条件的措施，我们在$ {\ bf r}^n $中恢复了在域和分形集中的几个经典嵌入定理。
We study compactness and boundedness of embeddings from Sobolev type spaces on metric spaces into $L^q$ spaces with respect to another measure. The considered Sobolev spaces can be of fractional order and some statements allow also nondoubling measures. Our results are formulated in a general form, using sequences of covering families and local Poincaré type inequalities. We show how to construct such suitable coverings and Poincaré inequalities. For locally doubling measures, we prove a self-improvement property for two-weighted Poincaré inequalities, which applies also to lower-dimensional measures. We simultaneously treat various Sobolev spaces, such as the Newtonian, fractional Hajłasz and Poincaré type spaces, for rather general measures and sets, including fractals and domains with fractal boundaries. By considering lower-dimensional measures on the boundaries of such domains, we obtain trace embeddings for the above spaces. In the case of Newtonian spaces we exactly characterize when embeddings into $L^q$ spaces with respect to another measure are compact. Our tools are illustrated by concrete examples. For measures satisfying suitable dimension conditions, we recover several classical embedding theorems on domains and fractal sets in ${\bf R}^n$.
