nLab derivative of a distribution

Idea

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.

Definition

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)’)

Properties

Proposition

(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)

Examples

Proposition

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

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

Proof

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

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

Example

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)

Related concepts

• product of a distribution with a smooth function

• Fourier transform of distributions

References

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