What is the definition of the order of a distribution?

Question

A linear functional $T$ on $\mathcal{D}(\Omega)$ is a distribution if $\phi_n \to 0$ in $\mathcal{D}(\Omega)$ $\Rightarrow$ $T(\phi_n) \to 0$ in $\mathbb{R}$.

But I cannot find what the order of a distribution means.

Answer 1

In general, $\mathcal{D}(\Omega):=\bigcup_{K \in \mathcal{K}(\Omega)} \mathcal{D}_K(\Omega)$ (where $\mathcal{K}(\Omega)$ denotes the union of all compacts set content in a open subset $\Omega \subset \mathbb{R}^n$), Note that $\Omega \subseteq \bigcup_{j \in \mathbb{N}} K_j$, where $\lbrace K_j \rbrace$ is an increasing sequence of compact in $\Omega$. Now $\varphi_k \rightarrow \varphi$ in $\mathcal{D}(\Omega)$ means that

(1) $\exists K \in \mathcal{K}(\Omega) \forall k \in \mathbb{N}: \mathrm{supp}(\varphi_k), \mathrm{supp}(\varphi) \subset K$.

(2) $\forall \alpha \in \mathbb{N}^n: D^\alpha \varphi_k \rightarrow D^\alpha \varphi$ uniformly on $K$.

and the distributions are linear continuous functional $T:\mathcal{D}(\Omega) \longrightarrow \mathbb{C}$ such that $T(\varphi_k) \rightarrow T(\varphi)$ every time that $\varphi_k \rightarrow \varphi$ in $\mathcal{D}(\Omega)$. It can be shown that this condition of continuity of distribution $T$ it's equivalent the following condition

(3) $\forall K \in \mathcal{K}(\Omega) \exists N_K\in \mathbb{N}$ and $\exists C_K > 0 : |T(\varphi)| \leq C_K \left \| \varphi \right \|_{K,N_K}$

where

$\left \| \varphi \right \|_{K,N} := \sup_{x \in K, |\alpha| \leq N} |D^\alpha \varphi(x)|$ for all $\varphi \in \mathcal{D}_K(\Omega)$.

The order of distribution $T$ is the smallest such integer $N \in \mathbb{N}$ which is independent of $K$, i.e. the smallest $N$ such that

(3') $\forall K \in \mathcal{K}(\Omega) \exists C_K > 0 : |T(\varphi)| \leq C_K \left \| \varphi \right \|_{K,N}$ holds.

If this integer does not exist, the order of the distribution is defined as infinity.

Answer 2

Let us consider a distribution $T$ on $\mathbb{R}^n$. One can prove that for every compact $K \subset \mathbb{R}^n$, there exists $N$, $c(\Omega,N)$ such that $$ |\langle T, \phi \rangle| \leq c \lVert \phi \rVert_{C^N(\Omega)} $$ for all $\phi \in \mathcal{D}$. The order of the distribution $T$ is the least such $N$ that is good for all compact sets $K$. Note that we allow the constant $c$ to vary with $\Omega$.

From a more abstract perspective, it means that the distribution $T$ factors through the space $C_C^N(\mathbb{R}^n)$. In other words, let $i$ be the inclusion of $C_C^\infty(\mathbb{R}^n)$ into $C_C^N(\mathbb{R}^n)$. Then $T$ factors as $T = T' \circ i$, $$ C_C^\infty(\mathbb{R}^n) \overset{i}\to C_C^N(\mathbb{R}^n) \overset{T'}\to \mathbb{R} $$ for some continuous map $T'$.