every locally integrable function is a distribution

Suppose U is an open set in n and f is a locally integrable function on U, i.e., fLloc1(U). Then the mapping

u Uf(x)u(x)𝑑x

is a zeroth order distribution. (See parent entry for notation 𝒟(U).)

(proof) (http://planetmath.org/T_fIsADistributionOfZerothOrder)

If f and g are both locally integrable functions on an open set U, and Tf=Tg, then it follows (see this page (http://planetmath.org/TheoremForLocallyIntegrableFunctions)), that f=g almost everywhere. Thus, the mapping fTf is a linear injectionMathworldPlanetmath when Lloc1 is equipped with the usual equivalence relationMathworldPlanetmath for an Lp-space. For this reason, one usually writes f for the distribution Tf.

Title every locally integrable function is a distribution
Canonical name EveryLocallyIntegrableFunctionIsADistribution
Date of creation 2013-03-22 13:44:25
Last modified on 2013-03-22 13:44:25
Owner matte (1858)
Last modified by matte (1858)
Numerical id 8
Author matte (1858)
Entry type Theorem
Classification msc 46-00
Classification msc 46F05