nLab order of a distribution

Definition

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 $k$:

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

Definition

(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))

Examples

Example

Every compactly supported distribution has finite order.

As a special case of example :

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 $k$.

(Hörmander 90, theorem 2.3.4)

Related concepts

• support of a distribution

• scaling degree of a distribution

References

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