Tf is a distribution of zeroth order


To check that Tf is a distribution of zeroth order (http://planetmath.org/Distribution4), we shall use condition (3) on this page (http://planetmath.org/Distribution4). First, it is clear that Tf is a linear mapping. To see that Tf is continuousMathworldPlanetmath, suppose K is a compact set in U and u𝒟K, i.e., u is a smooth functionMathworldPlanetmath with support in K. We then have

|Tf(u)| = |Kf(x)u(x)𝑑x|
K|f(x)||u(x)|𝑑x
K|f(x)|𝑑x||u||.

Since f is locally integrable, it follows that C=K|f(x)|𝑑x is finite, so

|Tf(u)|C||u||.

Thus f is a distribution of zeroth order ([1], pp. 381).

References

  • 1 S. Lang, Analysis II, Addison-Wesley Publishing Company Inc., 1969.
Title Tf is a distribution of zeroth order
Canonical name TfIsADistributionOfZerothOrder
Date of creation 2013-03-22 13:44:28
Last modified on 2013-03-22 13:44:28
Owner Koro (127)
Last modified by Koro (127)
Numerical id 6
Author Koro (127)
Entry type Proof
Classification msc 46F05
Classification msc 46-00