Order of a distribution

The order of a distribution on a compact set is the least nonnegative integer $m$ for which its action is bounded by a constant times the sup norms of test-function derivatives through order $m$.

A distribution can have finite order of a distribution on every compact set without having one order bound valid on the whole domain. For example, the locally finite sum ${\sum }_{j\ge 1}{\partial }^j{\delta }_j$ on the real line has orders growing without bound near the integers.

An order-zero distribution is locally bounded by the sup norm of its test function. Such distributions are precisely the distributions represented locally by signed or complex Radon measures.