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If $M$ is a non-compact smooth manifold, then an analogue of Rellich's lemma states that the operator of multiplication by a compactly supported function $f:M\rightarrow\mathbb{C}$ is a compact operator between Sobolev spaces $H^l(M)\rightarrow H^k(M)$ for $l>k$.

Suppose we have such an $f$ together with another non-compact manifold $N$. Define $$\tilde{f}:M\times N\rightarrow\mathbb{C},\qquad\tilde{f}(m,n):=f(m).$$ Is multiplication by $\tilde{f}$ a compact operator $H^l(M\times N)\rightarrow H^k(M\times N)$ for $l>k$?

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    $\begingroup$ Certainly not always. Just take $M=N=\mathbb R$, fix a smooth compactly supported function $g$ on $M\times N=\mathbb R^2$ with $g\tilde f\ne 0$ and consider the bounded in $H^\ell$ family $g_t(x,y)=g(x,y+t)$. Apparently you meant something more interesting but it is not quite clear what. $\endgroup$
    – fedja
    Commented Jul 22, 2017 at 1:36

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