Structure theorem for compactly supported distributions

Every compactly supported distribution is a finite sum of distributional derivatives of bounded continuous functions. One proof convolves it with a sufficiently high-order Bessel potential and then applies a power of $1-\Delta$.

The Bessel potential of order $s$ is the Fourier multiplier $(1+|\xi {|}^2{)}^{-s/2}$. Sufficiently high order turns a compactly supported finite-order distribution into a bounded continuous function.