Order of a distribution on a compact set

For $u\in {\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega })$ and compact $K\subseteq \mathrm{\Omega }$, the order of $u$ on $K$ is the least integer $m\ge 0$ for which there exists $C\ge 0$ satisfying $\mathrm{\mid }u(\phi )\mathrm{\mid }\le C{p}_m(\phi )$ for all $\phi \in {\mathcal{D}}_K$. The set of such integers is nonempty by Local finite order characterization of distributions, so its least element exists without choice. If $u$ vanishes on ${\mathcal{D}}_K$, including the empty or empty-interior compact cases, its order on $K$ is assigned to be zero.

The distribution has global finite order if there is one integer $m\ge 0$ such that for every compact $K\subseteq \mathrm{\Omega }$ there is a finite $C_K\ge 0$ with this bound of order $m$. The exponent is uniform; the constants need not be. Its global order is the least such exponent when one exists, and is infinity otherwise. The zero distribution has global order zero. Compactwise finite order by itself does not assert a uniform exponent over all compacts. An order-zero bound controls test values; it does not by definition identify the distribution with a function or a measure.