Showing $u \in H^s$ and $\varphi \in C^1$ implies $u\varphi \in H^s$ (product rule)

Question

Let $\Omega$ be bounded and open set in $\mathbb{R}^n$.

As a start, I pose this question:

For $u \in H^s(\Omega)=W^{s,2}(\Omega)$, define the Holder seminorm type quantity

$$F(u) = \int_\Omega\int_\Omega\frac{|u(x) - u(y)|^2}{|x-y|^{n+2s}}\;\mathrm{d}x\mathrm{d}y$$ and this integral exists by definition of being in the space $H^s$.

I want to show that if $\varphi \in C^1(\Omega)$ then $F(\varphi u)$ is finite.

How can I do this? can I approximate $\varphi$ by functiosn in $C_c^\infty$ in some way so that i get bounds on $\varphi$ and its derivatiave?

If anyone has a reference for this basic result I would be happy. Thank you.

Answer 1

In the stated form, it is not true. Take $u = 1$, then you would get $F(\varphi) < \infty$ which may not be true if $\varphi$ has some severe singularities at the boundary of $\Omega$.

It does indeed hold for $\varphi \in W^{1,\infty}(\Omega) = C^{0,1}(\bar\Omega)$, as pointed out by Siminore. In the cases $s = 0$ and $s = 1$ this is trivial. But I guess also $s \in (0,1)$ is straight-forward.