Order of a distribution - Imported Knowlpedia Content

Definition

Let $T$ be a distribution on an open set $\Omega\subseteq\mathbb R^n$ . It has order at most $m$ if for every compact $K\subset\Omega$ there is $C_K\geq0$ such that

|T(\varphi)|\leq C_K \max_{|\alpha|\leq m}\sup_{x\in K}|\partial^\alpha\varphi(x)|

for every $\varphi\in\mathcal D(\Omega)$ with support in $K$ , where $\alpha=(\alpha_1,\ldots,\alpha_n)$ has nonnegative integer entries and $|\alpha|=\alpha_1+\cdots+\alpha_n$ . Its order is the least such nonnegative integer $m$ ; if no single $m$ works for all compact $K$ , its order is infinite. Thus order records how many derivatives of a test function are needed to control the distribution.

Examples and differentiation

A distribution represented by a locally finite measure has order $0$ . The Dirac distribution $\delta_a$ has order $0$ , while $\partial^\alpha\delta_a$ has order $|\alpha|$ . More generally, distributional differentiation raises the order by at most the number of derivatives:

\operatorname{ord}(\partial^\alpha T) \leq \operatorname{ord}(T)+|\alpha|.

These estimates follow directly from the definition of distributional derivative Hörmander, §2.3.

Local and global scope

Continuity of $T$ on each fixed-support test-function space implies a finite-order estimate on every compact $K$ , but the required integer may grow with $K$ . Consequently every distribution has finite order locally, while not every distribution has one finite global order on a noncompact open set. A compactly supported distribution does have finite order Folland, Chapter 9.

Conventions

Some texts say “order $m$ ” when they only mean “order at most $m$ .” Here “order” means the least admissible $m$ . The zero distribution is bounded by every order estimate; assigning it order $0$ or $-\infty$ is a convention that should be stated when formulas depend on the value.

References

Lars Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., Springer, 2003. Springer DOI record. Relevant: §2.3, distributions of finite order and compact support.
Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., Wiley, 1999. Wiley publisher record. Relevant: Chapter 9, distributions and their order.