Distributions and Sobolev Spaces - Core Definitions

Distribution theory, developed by Laurent Schwartz, provides a rigorous framework for generalized functions, enabling differentiation of discontinuous functions and solutions to PDEs in weak form.

DefinitionTest Functions

Let $\Omega \subset \mathbb{R}^n$ be an open set. The space of test functions is $\mathcal{D}(\Omega) = C_c^\infty(\Omega) = \{\phi \in C^\infty(\Omega) : \text{supp}(\phi) \text{ is compact in } \Omega\}$

A sequence $(\phi_n)$ converges to $\phi$ in $\mathcal{D}(\Omega)$ if:

All $\phi_n$ are supported in a common compact set $K \subset \Omega$
$\partial^\alpha \phi_n \to \partial^\alpha \phi$ uniformly on $K$ for all multi-indices $\alpha$

DefinitionDistribution

A distribution on $\Omega$ is a linear functional $T : \mathcal{D}(\Omega) \to \mathbb{C}$ that is continuous with respect to the test function topology.

Equivalently, $T$ is a distribution if for every compact $K \subset \Omega$ , there exist $C > 0$ and $m \in \mathbb{N}$ such that $|T(\phi)| \leq C \sum_{|\alpha| \leq m} \sup_{x \in K} |\partial^\alpha \phi(x)|$ for all $\phi \in \mathcal{D}(\Omega)$ supported in $K$ .

The space of distributions is denoted $\mathcal{D}'(\Omega)$ .

ExampleExamples of Distributions

Regular Distributions: Any locally integrable function $f \in L^1_{loc}(\Omega)$ defines a distribution by $T_f(\phi) = \int_\Omega f(x) \phi(x) \, dx$
Dirac Delta: $\delta_a(\phi) = \phi(a)$ is a distribution but not a function
Principal Value: $\text{P.V.}\frac{1}{x}$ defined by $T(\phi) = \lim_{\varepsilon \to 0^+} \int_{|x| > \varepsilon} \frac{\phi(x)}{x} \, dx$
Derivatives: $T'(\phi) = -T(\phi')$ defines the distributional derivative
Heaviside Function: $H(x) = \begin{cases} 1 & x > 0 \\ 0 & x < 0 \end{cases}$ has derivative $H' = \delta_0$

DefinitionOperations on Distributions

Let $T \in \mathcal{D}'(\Omega)$ . Define:

Derivative: $(\partial^\alpha T)(\phi) = (-1)^{|\alpha|} T(\partial^\alpha \phi)$
Multiplication by smooth function: If $a \in C^\infty(\Omega)$ , then $(aT)(\phi) = T(a\phi)$
Support: $\text{supp}(T)$ is the smallest closed set outside which $T$ vanishes
Convolution: If $T \in \mathcal{D}'(\mathbb{R}^n)$ and $\phi \in \mathcal{D}(\mathbb{R}^n)$ , then $T * \phi \in C^\infty(\mathbb{R}^n)$

TheoremStructure Theorem

Every distribution is locally a finite-order derivative of a continuous function. That is, for every compact $K \subset \Omega$ , there exist continuous functions $f_\alpha$ such that $T = \sum_{|\alpha| \leq m} \partial^\alpha f_\alpha$ in a neighborhood of $K$ .

Remark

Distribution theory allows us to differentiate any locally integrable function arbitrarily many times. This is essential for weak solutions to PDEs, where classical derivatives may not exist but distributional derivatives do.

Distributions provide the natural setting for Fourier analysis, PDE theory, and quantum field theory.