localization for distributions

Definition Suppose $U$ is an open set in ${ℝ}^n$ and $T$ is a distribution $T\in {𝒟}^{\prime }(U)$. Then we say that $T$ vanishes on an open set $V\subset U$, if the restriction of $T$ to $V$ is the zero distribution on $V$. In other words, $T$ vanishes on $V$, if $T(v)=0$ for all $v\in C_0^{\mathrm{\infty }}(V)$. (Here $C_0^{\mathrm{\infty }}(V)$ is the set of smooth function with compact support in $V$.) Similarly, we say that two distributions $S,T\in {𝒟}^{\prime }(U)$ are equal, or coincide on $V$, if $S-T$ vanishes on $V$. We then write: $S=T$ on $V$.

Theorem[1, 3] Suppose $U$ is an open set in ${ℝ}^n$ and ${\{U_i\}}_{i\in I}$ is an open cover of $U$, i.e.,

$$U=\bigcup _{i\in I}{U}_i.$$

Here, $I$ is an arbitrary index set. If $S,T$ are distributions on $U$, such that $S=T$ on each $U_i$, then $S=T$ (on U).

Proof. Suppose $u\in 𝒟(U)$. Our aim is to show that $S(u)=T(u)$. First, we have $\mathrm{supp}u\subset K$ for some compact $K\subset U$. It follows (http://planetmath.org/YIsCompactIfAndOnlyIfEveryOpenCoverOfYHasAFiniteSubcover) that there exist a finite collection of $U_i$:s from the open cover, say $U_1,\mathrm{\dots },U_N$, such that $K\subset {\cup }_{i=1}^N{U}_i$. By a smooth partition of unity, there are smooth functions ${\varphi }_1,\mathrm{\dots },{\varphi }_N:U\to ℝ$ such that

1. $\mathrm{supp}{\varphi }_i\subset U_i$ for all $i$.

2. ${\varphi }_i(x)\in [0,1]$ for all $x\in U$ and all $i$,

3. ${\sum }_{i=1}^N{\varphi }_i(x)=1$ for all $x\in K$.

From the first property, and from a property for the support of a function (http://planetmath.org/SupportOfFunction), it follows that $\mathrm{supp}{\varphi }_{i}u\subset \mathrm{supp}{\varphi }_i\cap \mathrm{supp}u\subset U_i$. Therefore, for each $i$, $S({\varphi }_{i}u)=T({\varphi }_{i}u)$ since $S$ and $T$ conicide on $U_i$. Then

$S(u)={\displaystyle \sum _{i=1}^N}S({\varphi }_{i}u)={\displaystyle \sum _{i=1}^N}T({\varphi }_{i}u)=T(u),$

and the theorem follows. $\mathrm{\square }$

Title	localization for distributions
Canonical name	LocalizationForDistributions
Date of creation	2013-03-22 13:46:17
Last modified on	2013-03-22 13:46:17
Owner	drini (3)
Last modified by	drini (3)
Numerical id	9
Author	drini (3)
Entry type	Definition
Classification	msc 46-00
Classification	msc 46F05