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Following https://en.wikipedia.org/wiki/Sobolev_inequality we have:

Let $k,l$ be nonnegative integers with $k>l$ and $p,q$ real numbers with $1\leq p< q<\infty$ and $1/q=1/p-(k-l)/n$ then $L^p_k \subset L^q_l$ with the inclusion continuous.

In the case of Sobolev spaces over a compact manifold $X$ of dimension $n$ for the inclusion to be continuous we need $k>l$ and $k -n/p \geq l - n/q$.

I would like to apply the Sobolev embedding theorem for $\mathbb{R}^n$ in the case $p=q=2$ but in the statement above I need $p <q$. Are the hypothesis for the compact case enough so that the embedding is true in the case of $\mathbb{R}^n$?

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