Let $\Gamma$ be a regular boundary of a $C^{k,1}$ domain $\Omega$ and $H^s(\Gamma)$, $s\in(0,1)$, denote the fractional Sobolev space on $\Gamma$. Suppose I define a multiplication operator $M_\phi:H^s(\Gamma)\to H^s(\Gamma)$ where $M_\phi v=\phi v$. What should be the minimal regularity of $v$ for the map to be continuous?

I am thinking that $v$ being Lipschitz (i.e., $v \in C^{0,1}(\Gamma)$) is enough for $M_\phi$ to be continuous. Can someone please give me a reference or study that deals with this kind of problem. Thanks!

Edit I am only interested on the case when the map $H^\frac12(\Gamma) \to H^\frac12(\Gamma)$ is continuous. More precisely, I want to know whether the following statement is true.

Let $\Omega \subset \mathbb{R}^2$ be a bounded Lipschitz domain and $\Gamma$ be a non-empty subset of $\Omega$. Then, the map $v \mapsto \phi v$ is continuous in $H^\frac12(\Gamma)$ for any $v \in H^\frac12(\Gamma)$ and $\phi \in C^{0,1}(\Gamma)$.

I tried to argue the validity of the above statement as follows.

By McShane-Whitney extension theorem, we know that we can find a $\tilde{\phi} \in C^{0,1}(\bar{\Omega})$ such that $\tilde{\phi}|_{\Gamma} = \phi$. Then, using [Grisvard, Elliptic Problems in Nonsmooth Domains, Theorem, p. 21] and [McLean, Strongly Elliptic Systems and Boundary Integral Equations, Theorem 3.37, p. 102], we have that $v \mapsto \phi v$ a continuous linear map in $H^\frac12(\Gamma)$.

Can someone confirm if my argument is correct?


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