localization for distributions
Definition Suppose is an open set in and is a distribution . Then we say that vanishes on an open set , if the restriction of to is the zero distribution on . In other words, vanishes on , if for all . (Here is the set of smooth function with compact support in .) Similarly, we say that two distributions are equal, or coincide on , if vanishes on . We then write: on .
Here, is an arbitrary index set. If are distributions on , such that on each , then (on U).
Proof. Suppose . Our aim is to show that . First, we have for some compact . It follows (http://planetmath.org/YIsCompactIfAndOnlyIfEveryOpenCoverOfYHasAFiniteSubcover) that there exist a finite collection of :s from the open cover, say , such that . By a smooth partition of unity, there are smooth functions such that
for all .
for all and all ,
for all .
and the theorem follows.
- 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|
|Date of creation||2013-03-22 13:46:17|
|Last modified on||2013-03-22 13:46:17|
|Last modified by||drini (3)|