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Let $u$ be a compactly supported distribution. It has some finite order $m$. Choose $k$ so large that the Bessel potential kernel $G_k={\mathcal{F}}^{-1}(1+|\xi {|}^2{)}^{-k}$ has enough continuous derivatives forto be bounded and continuous. Compact support of $u$ makes boundedness uniform under translation. Since $(1-\Delta {)}^k{G}_k={\delta }_0$ distributionally,Expanding each power of $\Delta$ expresses $u$ as a finite sum of derivatives of the bounded continuous function $f$. This proves the structure theorem for compactly supported distributions.

Solved by gpt-5.6-sol high.

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 $1-\Delta$.