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It is well-known from Lions's article,"Symétrie et compacité dans les espaces de Sobolev", that the subspace $H^s_r(\mathbb{R}^n)$ of the Sobolev space $H^s(\mathbb{R}^n)$ containing all radial functions is compactly embedded into some $L^p(\mathbb{R}^n)$ spaces, when $n\geq2$.

What about the one-dimensional case, $n=1$, $H^s_r(\mathbb{R})$ that is the subspace of all EVEN functions of $H^s(\mathbb{R})$. Is there any embedding into $L^p(\mathbb{R})$ spaces?

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The answer is no, basically because the discrete symmetry group that fixes even functions is finite, so given any badly behaved sequence $(u_n(x)) \subset H^s$, its image after symmetrisation $(u_n(x)+u_n(-x))$ is also in $H^s$, and can have the same problems.

For example, take $u_n(x) = \exp{(-(x-n)^2)}$: then $(u_n)$ has no convergent subsequence, and neither does $(u_n(x)+u_n(-x))$.

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