Definition Suppose $U$ is an open set in $\sR^n$ and $T$ is a distribution $T\in \cD'(U)$ . 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 $S,T\in \cD'(U)$ 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 $\sR^n$ 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 $u\in \cD(U)$ . Our aim is to show that $S(u)=T(u)$ . First, we have $\supp u \subset K$ for some compact $K\subset U$ . It follows 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 $\phi_1, \ldots, \phi_N: U\to \sR$ such that -40-JG for all $i$ .

1. $\phi_i(x) \in [0,1]$ for all $x\in U$ and all $i$ ,
2. $\sum_{i=1}^N \phi_i(x) = 1$ for all $x\in K$ .

Bibliography

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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.