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Consider $H^1(\mathbb{R})$ the standard Sobolev space of functions $f\in L^2$ such that the weak derivative of $f$ is also in $L^2$.

Now, my question is the following: if instead of considering the usual Lebesgue measure you endow $H^1(\mathbb{R})$ with, for instance, the following measure $d\mu=e^{-|x|}dx$, would usual Sobolev embeddings still hold? I am particularly interested in, for example, the embedding $H^1\hookrightarrow L^\infty$ (which is a basic result in the one-dimensional case for the Lebesgue measure).

Can someone recommend me some references to learn a little bit about this kind of questions?

Sharik
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1 Answers1

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These are usually called weighted Sobolev spaces, in the sense that you let $d\mu(x)=w(x)\,dx$ for some function $w:\Omega\to [0,+\infty)$ and then call the resulting space 'weighted Sobolev space with respect to the weight $w$'.

For general weights $w$ these spaces are pathological (they may not even include all smooth compactly supported functions...) so you need to add a few requirements to get reasonable spaces. For instance a good class of weights is that of Muckenhoupt weights - in this case you can show that the usual embedding theorems hold, see e.g. this paper.

An even more general setting is defining Sobolev spaces over arbitrary metric measure spaces and if you require a 'regularity' condition on the measure which introduces a concept of dimension, you can again obtain embedding theorems (see here).

Lorenzo Q
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