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
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.4$: A glimpse of distribution theory. Multiplication by $C^\infty$ functions