localization for distributions

Definition Suppose U is an open set in ℝn and T is a distribution Tβˆˆπ’Ÿβ€²β’(U). Then we say that T vanishes on an open set VβŠ‚U, if the restrictionPlanetmathPlanetmath of T to V is the zero distribution on V. In other words, T vanishes on V, if T⁒(v)=0 for all v∈C0∞⁒(V). (Here C0∞⁒(V) is the set of smooth function with compact support in V.) Similarly, we say that two distributions S,Tβˆˆπ’Ÿβ€²β’(U) are equal, or coincide on V, if S-T vanishes on V. We then write: S=T on V.

TheoremMathworldPlanetmath[1, 3] Suppose U is an open set in ℝn and {Ui}i∈I is an open cover of U, i.e.,


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

Proof. Suppose uβˆˆπ’Ÿβ’(U). Our aim is to show that S⁒(u)=T⁒(u). First, we have supp⁑uβŠ‚K for some compactPlanetmathPlanetmath KβŠ‚U. It follows (http://planetmath.org/YIsCompactIfAndOnlyIfEveryOpenCoverOfYHasAFiniteSubcover) that there exist a finite collectionMathworldPlanetmath of Ui:s from the open cover, say U1,…,UN, such that KβŠ‚βˆͺi=1NUi. By a smooth partition of unity, there are smooth functionsMathworldPlanetmath Ο•1,…,Ο•N:U→ℝ such that

  1. 1.

    supp⁑ϕiβŠ‚Ui for all i.

  2. 2.

    Ο•i⁒(x)∈[0,1] for all x∈U and all i,

  3. 3.

    βˆ‘i=1NΟ•i⁒(x)=1 for all x∈K.

From the first property, and from a property for the supportMathworldPlanetmath of a function (http://planetmath.org/SupportOfFunction), it follows that supp⁑ϕi⁒uβŠ‚supp⁑ϕi∩supp⁑uβŠ‚Ui. Therefore, for each i, S⁒(Ο•i⁒u)=T⁒(Ο•i⁒u) since S and T conicide on Ui. Then


and the theorem follows. β–‘


  • 1 G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.
  • 2 W. Rudin, Functional AnalysisMathworldPlanetmathPlanetmath, McGraw-Hill Book Company, 1973.
  • 3 L. HΓΆrmander, The AnalysisMathworldPlanetmath of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis), 2nd ed, Springer-Verlag, 1990.
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