Scaling degree and local extension space

Fix an integer $n\ge 1$, put $U={\mathbb{R}}^n\setminus \{0\}$, and use the continuous complex-linear test-function pairing of Distribution. For $t\in {\mathcal{D}}^{\mathrm{\prime }}(U)$ and $\rho >0$, define its dilation at the origin by

$⟨t_{\rho },\phi ⟩:={\rho }^{-n}⟨t,\phi (\cdot \mathrm{/}\rho )⟩,\phi \in \mathcal{D}(U)\mathrm{.}$

Positive dilations preserve $U$, so the displayed pairing is defined for every test function in $\mathcal{D}(U)$. If $t$ is the regular distribution induced by a locally integrable function $g$, change of variables gives $⟨t_{\rho },\phi ⟩={\int }_{U}g(\rho x)\phi (x)dx$.

Convergence $s_{\rho }\to 0$ in ${\mathcal{D}}^{\mathrm{\prime }}(U)$ as $\rho \downarrow 0$ means $⟨s_{\rho },\phi ⟩\to 0$ for every fixed $\phi \in \mathcal{D}(U)$. The scaling degree at the origin is

${\mathrm{sd}}_0(t):=\inf \{\omega \in \mathbb{R}:{\rho }^{\omega }{t}_{\rho }\to 0\text{ in }{\mathcal{D}}^{\mathrm{\prime }}(U)\}\mathrm{.}$

The infimum of the empty set is $+\mathrm{\infty }$; if every real $\omega$ belongs to the set, the infimum is $-\mathrm{\infty }$. A finite scaling degree means a real value $\delta \in \mathbb{R}$. For a distribution $T\in {\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^n)$ use the same dilation formula on tests in $\mathcal{D}({\mathbb{R}}^n)$. An extension of $t$ is a distribution $T$ on ${\mathbb{R}}^n$ whose restriction to $U$ equals $t$. The degree of divergence associated with a finite scaling degree $\delta$ in dimension $n$ is $\delta -n$. Exact homogeneity of scaling degree $\delta$ means ${\rho }^{\delta }{T}_{\rho }=T$ for every $\rho >0$.

For this fixed $t$, define its local extension family by

$\mathcal{E}(t):=\{T\in {\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^n):T{\mathrm{\mid }}_U=t\}\mathrm{.}$

Here $T{\mathrm{\mid }}_U=t$ means $⟨T,\tilde{\phi }⟩=⟨t,\phi ⟩$ for every $\phi \in \mathcal{D}(U)$, where $\tilde{\phi }\in \mathcal{D}({\mathbb{R}}^n)$ is its smooth extension by zero. When $\delta ={\mathrm{sd}}_0(t)\in \mathbb{R}$, the scaling-degree-preserving local extension family is

${\mathcal{E}}_{\delta }(t):=\{T\in \mathcal{E}(t):{\mathrm{sd}}_0(T)=\delta \}\mathrm{.}$

Existence of members and the affine structure of these families require separate results; the set definitions alone assert neither.

The definitions use Euclidean dilations about the specified point only. They do not assert that a given distribution arises from a product of fields or that an extension is physically selected.