nLab
order of a distribution

Contents

Definition

Let X nX \subset \mathbb{R}^n be an open subset of Euclidean space. One way to state the continuity condition on a distribution uu on XX, being a continuous linear functional on the space C c (X)C^\infty_c(X) of bump functions is to require that for all compact subspaces KK there exist constants CC and kk such that the absolute value of u(b)u(b) is bounded by CC times the suprema of the sums of the absolute values of all derivatives of bb of order bounded by kk:

KXcompact(C Kk K(bC c (X)(|u(b)|C K|α|k Ksup( αb)))). \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(X)}{\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 u𝒟(X)u \in \mathcal{D}'(X) has order k\leq k \in \mathbb{N} if kk serves as a global choice (valid for all compact KXK \subset X) in the above formula, hence if:

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

If uu is of order k\leq k for some kk \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 kk if the order of the derivatives is bounded by kk.

(Hörmander 90, theorem 2.3.4)

References

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

Last revised on August 6, 2017 at 22:39:31. See the history of this page for a list of all contributions to it.