Symmetric mollifier on cube instead of ball

Question

It is known that there exists a symmetric, positive mollifier $\phi\in C^\infty_0(B_1(0))$, with $B_1(0)\subset\mathbb{R}^n$ the open unit ball. As far as I know, there also exists some positive smooth mollifier $\psi\in C^\infty_0(Q_1(0))$, with $Q_1(0)\subset\mathbb{R}^n$ the open unit cube: $Q_1(0) = (-1,1)^n$. My question is: can this smooth mollifier on $Q_1(0)$ also be assumed to be symmetric? Or is this only possible on the open ball?

Answer 1

Let $\phi_1$ be the symmetric mollifier in dimension $n=1$ for $B(0, 1)=Q(0, 1)$. Define $\psi(x)=\phi_1(x_1)\phi_1(x_2)\cdots\phi_1(x_n)$. This should be symmetric.