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The Sobolev embedding theorem says that, supposing $k>0$ and $0<p<n$, if $\frac{1}{p}-\frac{1}{n}=\frac{1}{q}$, then $W^{1,p}(\mathbb R^n)\subset L^q(\mathbb R^n)$ continuously for all $q>p$.

I am interested in the critical case where $n=p$.

I want to prove that in the critical case $W^{1,p}(\mathbb R^n)\subset L^q(\mathbb R^n)$ for all $q\geq p$.

Idea of proof: Suppose $p\geq 2$ and fix $\widetilde q>p$ and define $\varepsilon:=\frac{p^2}{p+\widetilde q}$. Then $0<\varepsilon<1$ and hence if $f\in W^{1,p}(\mathbb R^n)$ then $f\in W^{1,p-\varepsilon}(\mathbb R^n)$. So applying the Sobolev embedding theorem for $p-\varepsilon<n$, we have $f\in L^{q}$ if $\frac{1}{p-\varepsilon}-\frac{1}{n}=\frac{1}{q}$. But rearranging this gives $q=\widetilde q$. So we are done.

Can we deduce the statement if $1\leq p<2$?

Arctic Char
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schrodingerscat
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1 Answers1

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Yes, the critical Sobolev embedding $W^{1,n}(\mathbb{R}^n) \hookrightarrow L^q(\mathbb{R}^n)$ for every $n \leq q < \infty$ is true. You can find it, for example, as case (3) of Theorem 2.31 in the following reference.

Demengel, Françoise; Demengel, Gilbert, Functional spaces for the theory of elliptic partial differential equations. Transl. from the French by Reinie Erné, Universitext. Berlin: Springer (ISBN 978-1-4471-2806-9/pbk; 978-1-4471-2807-6/ebook). xviii, 465 p. (2012). ZBL1239.46001.

Also, note that for the particular case $n = 1$, you actually have $W^{1,1}(\mathbb{R}) \hookrightarrow L^\infty(\mathbb{R})$. For $n > 1$, this embedding fails, but you can recover it partially working with BMO spaces.

cs89
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