Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be a measurable $L^\infty$ function. Assume further that $f$ is not zero a.e. and $f(-x)=f(x)$ for a.e. $x \in \mathbb{R}^n$.
Then, I wonder if we can conclude that there exists some $c \in (0,\infty)$ such that \begin{equation} \int_{\mathbb{R}^n} f(x)e^{-c\lVert x \rVert^2}d^nx \neq 0 \end{equation} as an element of $\mathbb{R}^m$.
I asked a similar question before : If $f(x) \neq 0$ on $[0,\infty)$, does there exists some $\alpha \in [0,\infty)$ such that $\int_0^\infty f(x)e^{-\alpha x^2}dx \neq 0$?
However, I cannot extend to multivariate cases based on the condition $f(-x)=f(x)$..
Could anyone please help me?