Is the Laplacian of any test function in $C_0^\infty$ still a test function?

Question

As we know, a test function on $\mathbb{R}^n$ is a $C^\infty(\mathbb{R}^n)$ function with compact support, and sometimes we denote the family of such functions by $C_0^\infty(\mathbb{R}^n)$, which is a vector space under pointwise addition and multilication by scalars. Now, if $f$ is a test function on $\mathbb{R}^n$, it is intriguing to ask whether $\Delta f$, the Laplacian of $f$, is still in $C_0^\infty(\mathbb{R}^n)$. For sure, $\Delta f$ must be still infinitely differentiable. How about its support $\mathrm{supp}(\Delta f)$? Is it still bounded and thus compact? Thank you.

Answer 1

Thank you, @Jochen. It occurs to me that $$\mathrm{supp}(\Delta f)\subseteq\mathrm{supp}(f).$$ Now, since $f$ is supported compactly, one finds that $\Delta f\in C_0^\infty(\mathbb{R}^n)$.