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It is well known that $\|fg\|_s \lesssim \|f\|_{s_1} \|g\|_{s_2}$ for functions $f: {\mathbb R}^n \rightarrow {\mathbb R}$ under certain conditions on $s$, $s_1$, $s_2$ (i.e. $s_1$, $s_2 \geq s$ and $s \leq s_1+s_2 -n/2$). Here $\|\cdot\|_t$ is the usual norm for $H^t({\mathbb R}^n)$ and all indices are non-negative.

My question is: Does the result also hold if we replace ${\mathbb R}^n$ with a (not necessarily bounded) domain $\Omega \subset {\mathbb R}^n$? (I need a result which is not restricted to integers.)

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    $\begingroup$ If your domain is sufficiently nice, can you not use an extension theorem (e.g. Stein's) and the full-space product estimate? $\endgroup$ Jul 13, 2019 at 18:39
  • $\begingroup$ Thanks Matt, yes, that works. $\endgroup$
    – olih
    Jul 14, 2019 at 19:05
  • $\begingroup$ Do you have a precise reference for the initial result? $\endgroup$ Apr 1, 2022 at 20:34
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    $\begingroup$ @BeniBogosel reference for the initial result: arxiv.org/abs/1512.07379 $\endgroup$
    – nakajuice
    Dec 13, 2023 at 18:41

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Let $n\ge 1$ be an integer and $s>n/2$. Then you have $H^s(\mathbb R^n)\subset L^\infty(\mathbb R^n)$ and for $f,g\in H^s(\mathbb R^n)$, $$ \Vert fg \Vert_{H^s(\mathbb R^n)}\le c_n\bigl(\Vert f \Vert_{L^\infty(\mathbb R^n)} \Vert g \Vert_{H^s(\mathbb R^n)} +\Vert f \Vert_{H^s(\mathbb R^n)}\Vert g \Vert_{L^\infty(\mathbb R^n)}\bigr). $$ Of course the above estimate can be localized. If $\Omega$ is an open subset of $\mathbb R^n$, $s>n/2$ and $f,g\in H^s_{\text{loc}}(\Omega)$, $\chi, \psi\in C^\infty_c(\Omega)$, you get $$ \Vert \chi f\psi g \Vert_{H^s(\mathbb R^n)}\le c_n\bigl(\Vert \chi f \Vert_{L^\infty(\mathbb R^n)} \Vert \psi g \Vert_{H^s(\mathbb R^n)} +\Vert \chi f \Vert_{H^s(\mathbb R^n)}\Vert \psi g \Vert_{L^\infty(\mathbb R^n)}\bigr), $$ and this proves that $fg\in H^s_{\text{loc}}(\Omega)\subset L^\infty_{\text{loc}}(\Omega)$.

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  • $\begingroup$ Bazin: The estimate $\|fg\|_s \lesssim \|f\|_s \|g\|_s$ for $s>n/2$ follows directly since $H^s(\Omega)$ is a Banach algebra in that case. I was specifically interested (i) in situations where the Sobolev indices on the left and right are different, and (ii) not just for $H^s_\mathrm{loc}(\Omega)$; and indeed the key is Stein's extension theorem, as Matt pointed out. $\endgroup$
    – olih
    Jul 14, 2019 at 19:09

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