Commutator of x and Distributional Derivative acting on Distribution

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Theorem

Let $T \in \map {\DD'} \R$ be a distribution.

Let:

$\sqbrk {x, \dfrac \d {\d x} } T := x \dfrac {\d T}{\d x} - \dfrac {\d \paren {x T}}{\d x}$

where derivatives are to be understood in the distributional sense.


Then:

$\sqbrk {x, \dfrac \d {\d x} } T = - T$


Proof

Let $\phi \in \map \DD \R$ be a test function.

\(\ds \map {\paren{x \dfrac {\d T}{\d x} - \dfrac {\d \paren {x T} }{\d x} } } \phi\) \(=\) \(\ds \map {\paren {x \dfrac {\d T}{\d x} } } \phi - \map {\paren {\dfrac {\d \paren {x T} }{\d x} } } \phi\) Linearity of distribution
\(\ds \) \(=\) \(\ds \map {\paren {\dfrac {\d T}{\d x} } } {x \phi} - \map {\paren {\dfrac {\d \paren {x T} }{\d x} } } \phi\) Definition of Multiplication of Distribution by Smooth Function
\(\ds \) \(=\) \(\ds - \map T {\paren {x \phi}'} + \map {\paren {x T} } {\phi'}\) Definition of Distributional Derivative
\(\ds \) \(=\) \(\ds - \map T {\paren {x \phi}'} + \map T {x \phi'}\) Definition of Multiplication of Distribution by Smooth Function
\(\ds \) \(=\) \(\ds \map T {-\phi - x \phi' + x \phi'}\) Linearity of distribution
\(\ds \) \(=\) \(\ds \map T {-\phi}\)
\(\ds \) \(=\) \(\ds -\map T \phi\) Linearity of distribution

$\blacksquare$


Sources

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