Multiplication by a smooth compactly supported function in homogeneous Sobolev spaces

Question

Let $f\in C_c^\infty$ and $g\in \dot{H}^s$, $s>1/2$. Is it true that $\|fg\|_{\dot{H}^s}\le c\|g\|_{\dot{H}^s}$? I think this is true for inhomogeneous Sobolev spaces but I wonder whether the same holds for homogeneous Sobolev spaces. I did not find a reference to this result.

Answer 1

This is also true for homogeneous $H^s$ space. For simplicity look at the case $s=1$, you have, denoting $Q$ the support of function $f$ :

$$||fg||^2_{H^1(\mathbb{R}^d)} = \int_Q |\nabla f|^2 g^2 + |\nabla g|^2 f^2 + f^2 g^2 \ \mathrm{d}x$$ then you have

\begin{align*} ||fg||^2_{H^1} & \leq (C_2 +C_1) \int_{Q} g^2 \ \mathrm{d}x + C_2 \int_Q |\nabla g|^2 \ \mathrm{d}x\\ & \leq (C_2+C_1) (\int_{\mathbb{R^d}} g^2 \ \mathrm{d}x + \int_{\mathbb{R^d}} |\nabla g|^2 \ \mathrm{d}x)\\ & \leq C ||g||_{H^1(\mathbb{R}^d)} \end{align*} where I denoted $C_1$ the max of $|\nabla f|^2$ on $Q$ and $C_2$ the max of $f^2$ on $Q$.