p.\tmspace-.1667emv.(1x) is a distribution of first order

(Following [1, 2].) Let u𝒟(U). Then suppu[-k,k] for some k>0. For any ε>0, u(x)/x is Lebesgue integrableMathworldPlanetmath in |x|[ε,k]. Thus, by a change of variable, we have

p.v.(1x)(u) = limε0+[ε,k]u(x)-u(-x)x𝑑x.

Now it is clear that the integrand is continuousMathworldPlanetmathPlanetmath for all x{0}. What is more, the integrand approaches 2u(0) for x0, so the integrand has a removable discontinuity at x=0. That is, by assigning the value 2u(0) to the integrand at x=0, the integrand becomes continuous in [0,k]. This means that the integrand is Lebesgue measurable on [0,k]. Then, by defining fn(x)=χ[1/n,k](u(x)-u(-x))/x (where χ is the characteristic functionMathworldPlanetmathPlanetmathPlanetmathPlanetmath), and applying the Lebesgue dominated convergence theoremMathworldPlanetmath, we have

p.v.(1x)(u) = [0,k]u(x)-u(-x)x𝑑x.

It follows that p.v.(1x)(u) is finite, i.e., p.v.(1x) takes values in . Since 𝒟(U) is a vector space, if follows easily from the above expression that p.v.(1x) is linear.

To prove that p.v.(1x) is continuous, we shall use condition (3) on this page (http://planetmath.org/Distribution4). For this, suppose K is a compact subset of and u𝒟K. Again, we can assume that K[-k,k] for some k>0. For x>0, we have

|u(x)-u(-x)x| = |1x(-x,x)u(t)𝑑t|

where |||| is the supremum norm. In the first equality we have used the fundamental theorem of calculusMathworldPlanetmathPlanetmath for the Lebesgue integral (valid since u is absolutely continuousMathworldPlanetmath on [-k,k]). Thus


and p.v.(1x) is a distributionPlanetmathPlanetmathPlanetmath of first order (http://planetmath.org/Distribution4) as claimed.


Title p.\tmspace-.1667emv.(1x) is a distribution of first order
Canonical name operatornamepvfrac1xIsADistributionOfFirstOrder
Date of creation 2013-03-22 13:46:07
Last modified on 2013-03-22 13:46:07
Owner Koro (127)
Last modified by Koro (127)
Numerical id 7
Author Koro (127)
Entry type Proof
Classification msc 46F05
Classification msc 46-00