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arXiv · 1903.05793

Sobolev embedding for $M^{1,p}$ spaces is equivalent to a lower bound of the measure

Ryan Alvarado·Przemysław Górka·Piotr Hajłasz
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Abstract

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.

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Ryan Alvarado, Przemysław Górka, Piotr Hajłasz. 2019-03-14. Sobolev embedding for $M^{1,p}$ spaces is equivalent to a lower bound of the measure. https://arxiv.org/abs/1903.05793

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