Product Rule for Distributional Derivatives of Distributions multiplied by Smooth Functions

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Theorem

Let $\alpha \in \map {C^\infty} \R$ be a smooth real function.

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


Then in the distributional sense it holds that:

$\paren {\alpha T}' = \alpha' T + \alpha T'$


Proof

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

By the Product Rule for Derivatives:

$\paren {\alpha \phi}' = \alpha' \phi + \alpha \phi'$

Hence:

\(\ds \map {\paren {\alpha T}'} \phi\) \(=\) \(\ds -\map {\paren {\alpha T} } {\phi'}\) Definition of Distributional Derivative
\(\ds \) \(=\) \(\ds -\map T {\alpha \phi'}\) Definition of Multiplication of Distribution by Smooth Function
\(\ds \) \(=\) \(\ds -\map T {\paren {\alpha \phi}' - \alpha' \phi}\) Product Rule for Derivatives
\(\ds \) \(=\) \(\ds -\map T {\paren {\alpha \phi}'} + \map T {\alpha' \phi}\) Definition of Distribution: Linearity
\(\ds \) \(=\) \(\ds \map {T'} {\alpha \phi} + \map T {\alpha' \phi}\) Definition of Distributional Derivative
\(\ds \) \(=\) \(\ds \alpha \map {T'} \phi + \alpha' \map T \phi\) Definition of Multiplication of Distribution by Smooth Function
\(\ds \) \(=\) \(\ds \map {\paren {\alpha T' + \alpha' T} } \phi\) Addition of Distributions

$\blacksquare$


Sources

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