Distributions and Sobolev Spaces - Examples and Constructions

The Fourier transform extends to tempered distributions, providing powerful tools for solving PDEs and analyzing regularity.

DefinitionTempered Distributions

The space of Schwartz functions (rapidly decreasing) is $\mathcal{S}(\mathbb{R}^n) = \{\phi \in C^\infty(\mathbb{R}^n) : \sup_x |x^\alpha \partial^\beta \phi(x)| < \infty \text{ for all } \alpha, \beta\}$

The space of tempered distributions is $\mathcal{S}'(\mathbb{R}^n)$ , the continuous dual of $\mathcal{S}$ .

Every distribution with compact support is tempered, and every tempered distribution extends to a distribution. The Fourier transform maps $\mathcal{S}$ to itself and extends to an isomorphism $\mathcal{S}' \to \mathcal{S}'$ .

DefinitionFourier Transform of Distributions

For $T \in \mathcal{S}'(\mathbb{R}^n)$ , define $\hat{T} \in \mathcal{S}'$ by $\hat{T}(\phi) = T(\hat{\phi})$ where $\hat{\phi}(\xi) = \int_{\mathbb{R}^n} e^{-2\pi i x \cdot \xi} \phi(x) \, dx$ .

ExampleFourier Transforms of Distributions

Dirac Delta: $\hat{\delta}_0(\phi) = \delta_0(\hat{\phi}) = \hat{\phi}(0) = \int \phi = 1(\phi)$ , so $\hat{\delta}_0 = 1$
Constant Function: Similarly, $\hat{1} = \delta_0$
Exponential: $\widehat{e^{2\pi i a \cdot x}} = \delta_a$
Derivative: $\widehat{\partial_j T} = 2\pi i \xi_j \hat{T}$ (multiplication by $\xi_j$ )
Multiplication: $\widehat{x_j T} = \frac{1}{2\pi i} \partial_{\xi_j} \hat{T}$

TheoremCharacterization of Sobolev Spaces via Fourier Transform

For $s \in \mathbb{R}$ and $1 < p < \infty$ , the Sobolev space $H^s(\mathbb{R}^n)$ can be characterized as $H^s(\mathbb{R}^n) = \{u \in \mathcal{S}' : (1 + |\xi|^2)^{s/2} \hat{u}(\xi) \in L^2(\mathbb{R}^n)\}$

with norm $\|u\|_{H^s}^2 = \int_{\mathbb{R}^n} (1 + |\xi|^2)^s |\hat{u}(\xi)|^2 \, d\xi$

This Fourier characterization extends Sobolev spaces to non-integer and negative orders, crucial for elliptic regularity theory.

ExampleSolving PDEs via Fourier Transform

Consider $-\Delta u + u = f$ on $\mathbb{R}^n$ . Taking Fourier transforms: $(4\pi^2 |\xi|^2 + 1) \hat{u} = \hat{f}$ $\hat{u}(\xi) = \frac{\hat{f}(\xi)}{4\pi^2 |\xi|^2 + 1}$

For $f \in L^2$ , this gives $u \in H^2$ with $\|u\|_{H^2} \lesssim \|f\|_{L^2}$

The Fourier transform converts differentiation into multiplication, transforming PDEs into algebraic equations.

Remark

Tempered distributions and Fourier analysis provide the technical machinery for microlocal analysis, where the structure of singularities is analyzed in phase space $(x, \xi)$ . This leads to pseudodifferential operators and wavefront sets, fundamental tools in modern PDE theory.

The combination of distribution theory, Sobolev spaces, and Fourier analysis forms the backbone of linear PDE theory.