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Jul 22, 2017 at 1:36 comment added fedja 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.
Jul 21, 2017 at 17:04 history asked geometricK CC BY-SA 3.0