# localization for distributions

Definition Suppose $U$ is an open set in $\mathbb{R}^{n}$ and $T$ is a distribution $T\in\mathcal{D}^{\prime}(U)$. 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 $v\in C_{0}^{\infty}(V)$. (Here $C_{0}^{\infty}(V)$ is the set of smooth function with compact support in $V$.) Similarly, we say that two distributions $S,T\in\mathcal{D}^{\prime}(U)$ 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 $\mathbb{R}^{n}$ 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 $u\in\mathcal{D}(U)$. Our aim is to show that $S(u)=T(u)$. First, we have $\operatorname{supp}u\subset K$ 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 $U_{1},\ldots,U_{N}$, such that $K\subset\cup_{i=1}^{N}U_{i}$. By a smooth partition of unity, there are smooth functions $\phi_{1},\ldots,\phi_{N}:U\to\mathbb{R}$ such that

1. 1.

$\operatorname{supp}\phi_{i}\subset U_{i}$ for all $i$.

2. 2.

$\phi_{i}(x)\in[0,1]$ for all $x\in U$ and all $i$,

3. 3.

$\sum_{i=1}^{N}\phi_{i}(x)=1$ 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 $\operatorname{supp}\phi_{i}u\subset\operatorname{supp}\phi_{i}\cap% \operatorname{supp}u\subset U_{i}$. Therefore, for each $i$, $S(\phi_{i}u)=T(\phi_{i}u)$ since $S$ and $T$ conicide on $U_{i}$. Then

 $\displaystyle S(u)=\sum_{i=1}^{N}S(\phi_{i}u)=\sum_{i=1}^{N}T(\phi_{i}u)=T(u),$

and the theorem follows. $\Box$

## References

• 1 G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.
• 2 W. Rudin, Functional Analysis, McGraw-Hill Book Company, 1973.
• 3 L. Hörmander, The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis), 2nd ed, Springer-Verlag, 1990.
Title localization for distributions LocalizationForDistributions 2013-03-22 13:46:17 2013-03-22 13:46:17 drini (3) drini (3) 9 drini (3) Definition msc 46-00 msc 46F05