derivative of a distribution

The concept of *derivative of a distribution* is the generalization of the concept of derivative of a smooth function with distributions thought of as generalized functions. The concept is uniquely fixed by enforcing the formula for integration by parts to extend from integrals against compactly supported densities to distributions.

Let $X \subset \mathbb{R}^n$ be an open subset of Euclidean space of dimension $n$. For $\alpha \in \mathbb{N}^n$ a multi-index, write

$\partial_\alpha \coloneqq \frac{\partial^{\alpha_1}}{\partial^{\alpha_1} x^{1}}
\cdots
\frac{\partial^{\alpha_n}}{\partial^{\alpha_n} x^{n}}$

for the corresponding operation of differentiation and write

${\vert \alpha \vert}
\;\coloneqq\;
\underoverset{i = 1}{n}{\sum} \alpha_i$

for the total degree.

For $u \in \mathcal{D}'(X)$ then its derivative $\partial_\alpha u \in \mathcal{D}'(X)$ of order $\alpha$ is defined by

$\langle
\partial_\alpha u, b
\rangle
\;=\;
(-1)^{\vert \alpha \vert} \langle u, \partial_\alpha b\rangle$

for any bump function $b$, where $\partial_\alpha b$ is the ordinary derivative of smooth functions.

(e.g. Hörmander 90, def. 3.1.1 with (3.1.1)’)

**(derivative of distributions retains or shrinks wave front set)**

Taking derivatives of distributions retains or shrinks the wave front set:

For $u \in \mathcal{D}'(\mathbb{R}^n)$ a distribution and $\alpha \in \mathbb{N}^n$ a multi-index with $D^\alpha$ denoting the corresponding partial derivative, then

$WF(D^\alpha u) \subset WF(u)
\,.$

(Hörmander 90, (8.1.10), p. 256)

The distributional derivative of the Heaviside distribution $\Theta \in \mathcal{D}'(\mathbb{R})$ is the delta distribution $\delta \in \mathcal{D}'(\mathbb{R})$:

$\partial \Theta = \delta
\,.$

For $b \in C^\infty_c(\mathbb{R})$ any bump function we compute:

$\begin{aligned}
\int \partial\Theta(x) b(x) \, d x
& =
- \int \Theta(x) \partial b(x)\, dx
\\
& =
- \int_0^\infty \partial b(x) d x
\\
& =
- \left( b(x)\vert_{x \to \infty} - b(0) \right)
\\
& =
b(0)
\\
&
=
\int \delta(x) b(x) \, dx
\,.
\end{aligned}$

Every point-supported distribution $u$, hence with $supp(u) = \{p\}$ for some point $p$, is a finite sum of multiplies of derivatives of the delta distribution at that point:

$u =
\underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha\vert} \leq k } }{\sum} c^\alpha \partial_\alpha \delta(p)$

where $\{c^\alpha \in \mathbb{R}\}_\alpha$, and for $k \in \mathbb{N}$ the order of $u$.

(e.g. Hörmander 90, theorem 2.3.4)

- Lars Hörmander,
*The analysis of linear partial differential operators*, vol. I, Springer 1983, 1990

Last revised on November 7, 2017 at 11:37:49. See the history of this page for a list of all contributions to it.