Let $\mathcal{S}(\mathbb{R}^n)$ be a Schwartz space and $T : \mathcal{S}(\mathbb{R}^n) \to \mathbb{C}$ be a continuous linear functional under the Frechet topology of $\mathcal{S}(\mathbb{R}^n)$. That is, $T$ is a tempered distribution.
Now, I wonder if there is any sufficient (or eqiuvalent, hopefully) criterion under which $T$ is identified with a polynomially bounded function.
That is
Is there any sufficient (or equivalent) condition $T$ should satisfy in order to ensure (unique) existence of a measurable function $g : \mathbb{R}^n \to \mathbb{C}$ such that $\lvert g(x) \rvert \leq C(1+\lVert x \rVert^2)^{N}$ for some constant $C>0$, $N \in \mathbb{N}$ and a.e. $x \in \mathbb{R}^n$ and satisfies $T(f) = \int_{\mathbb{R}^n} fg$ for all $f \in \mathcal{S}(\mathbb{R}^n)$?