Let $X \subset \mathbb{R}^n$ be an open subset of Euclidean space. One way to state the continuity condition on a distribution $u$ on $X$ — being a continuous linear functional on the space $C^\infty_c(X)$ of bump functions — is to require that for all compact subspaces $K$ there exist constants $C$ and $k$ such that for any $b\in C^\infty_c(X)$ with support in $K$ the absolute value of $u(b)$ is bounded by $C$ times the suprema of the sums of the absolute values of all derivatives of $b$ of order bounded by $k$:

$\underset{ {K \subset X} \atop {\text{compact}}}{\forall}
\left(
\underset{ {C_K \in \mathbb{R}} \atop {k_K \in \mathbb{N}} }{\exists}
\left(
\underset{b \in C^\infty_c(K)}{\forall}
\left(
{\vert u(b) \vert}
\leq
C_K
\underset{
\vert \alpha\vert \leq k_K
}
{\sum}
sup(\partial^\alpha b)
\right)
\right)
\right)
\,.$

**(order of a distribution)**

The distribution $u \in \mathcal{D}'(X)$ has *order* $\leq k \in \mathbb{N}$ if $k$ serves as a global choice (valid for all compact $K \subset X$) in the above formula, hence if:

$\underset{ {K \subset X} \atop {\text{compact}}}{\forall}
\left(
\underset{ {C_K \in \mathbb{R}} }{\exists}
\left(
\underset{b \in C^\infty_c(K)}{\forall}
\left(
{\vert u(b) \vert}
\leq
C_K \underset{\vert \alpha\vert \leq k}{\sum}sup(\partial^\alpha b)
\right)
\right)
\right)
\,.$

If $u$ is of order $\leq k$ for some $k \in \mathbb{N}$, it is said to have *finite order*.

(e.g. Hoermander 90, def. 2.1.1, below (2.1.2))

Every compactly supported distribution has finite order.

As a special case of example :

A point-supported distribution is a sum of derivatives of the delta distribution at that point. This has order $k$ if the order of the derivatives is bounded by $k$.

- Lars Hörmander,
*The analysis of linear partial differential operators*, vol. I, Springer (1983, 1990)

Last revised on May 2, 2024 at 14:52:48. See the history of this page for a list of all contributions to it.