It has been known since 1996 that a lower bound for the measure, $\mu(B(x,r))\geq br^s$, implies Sobolev embedding theorems for Sobolev spaces $M^{1,p}$ defined on metric-measure spaces. We prove that, in fact Sobolev embeddings for $M^{1,p}$ spaces are equivalent to the lower bound of the measure.

中文翻译：

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对于 M1 的 Sobolev 嵌入，空格相当于度量的下界

自 1996 年以来，人们就知道该度量的下限 $\mu(B(x,r))\geq br^s$，意味着 Sobolev 空间的 Sobolev 嵌入定理 $M^{1,p}$ 定义在度量度量空间。我们证明，实际上 $M^{1,p}$ 空间的 Sobolev 嵌入等效于度量的下界。