The question is rather vague but one characterisation is the following: for each compact interval $I$ (I am working on the real line for simplicity) and each uniformly bounded sequence $(x_n)$ of smooth functions with support in $I$ which converges in $L^1$ we have $T(x_n)\to 0$. This can easily be reformulated in terms of inequalities if that is more to your taste.

The question is rather vague but one characterisation is the following: for each compact interval $I$ (I am working on the real line for simplicity) and each uniformly bounded sequence $(x_n)$ of smooth functions with support in $I$ which converges in $L^1$ to $0$ we have $T(x_n)\to 0$. This can easily be reformulated in terms of inequalities if that is more to your taste.

Edited as suggested by comment.