cutoff function for fundamental solution

Consider the fundamental solution $h(x)=c_{n,s}|x|^{-(n-2)}$ defined for $x\in\mathbb{R}^{n}$, $x\neq 0$ with $c_{n,s}$ a normalizing constant. Is there some way to define this function so that it is zero outside a bounded domain $\Omega\subset\mathbb{R}^{n}$, i.e., $h(x)=0$ for $x\in\mathbb{R}^{n}\backslash\overline{\Omega}$.

My approach: If we consider $f$ a cutoff function such that $0\leq f\leq 1$, $f\in C^{\infty}(\mathbb{R}^{n})$, $f=0$ in $\overline{\mathbb{R}^{n}\backslash\Omega}$, $f=1$ in $\Omega$, then $H(x):=fh(x)$ satisfies $H\in C^{\infty}(\mathbb{R}^{n})$, $H=0$ in $\overline{\mathbb{R}^{n}\backslash\Omega}$ and $H=h$ in $\Omega$.I'm not sure if this is ok, as I have a singularity and there may be issues with the boundary of $\Omega$.

Any idea it will be great, thanks!