Tempered Distributions

Notations#

Let \(k\in\mathbb{N}_+\). For an open set \(U\subset \mathbb{R}^n\), denote by \(C^k(U)\) the space of \(k\)-times continuously differentiable functions, and \(C^\infty(U) = \cap_{k\in\mathbb{N}}C^k(U)\). By \(C_b^k(U)\) denote the space of functions which are bounded together withtheir derivatives up to order \(k\). By \(C^k_c(U)\) the space of compactly supported elements of \(C^k(U)\). All functions will be complex-valued, unless otherwise noted.

The standard multi-index notation: let \(x = (x_1, \ldots, x_n) \in \mathbb{R}^n\), and let \(\alpha = (\alpha_1,\ldots,\alpha_n)\in \mathbb{N}^n_0\). Put

\[x^\alpha := \prod^n_{j=1} {x_j}^{\alpha_j},\quad \partial^\alpha_x := \partial^{\alpha_1}_{x_1}\ldots \partial^{\alpha_n}_{x_n}, \quad D^{\alpha}_x := D^{\alpha_1}_{x_1}\ldots D^{\alpha_n}_{x_n}, \quad D:=\frac{1}{i}\partial.\]

We take \(|\alpha| = \sum_j \alpha_j\) and \(\alpha ! := \prod^n_{j=1}\alpha_j !\).

The Japanese bracket will be used:

\[\langle x\rangle = (1+|x|^2)^{1/2}.\]

Schwartz space#

Definition (Schwartz functions). The space \(\mathcal{S}(\mathbb{R}^n)\) of Schwartz functions consists of all \(\phi\in C^\infty(\mathbb{R}^n)\) s.t. for all \(k\in \mathbb{N}\), \(||\phi||^s_k :=\underset{x\in\mathbb{R}^n, |\alpha|+|\beta| \leq k}{\text{sup}} |x^\alpha D^\beta \phi(x)| < \infty\).

Example. The function \(\exp(-|x|^2)\) is a Schwartz function. There is a continuous inclusion \(C^\infty_c(\mathbb{R}^n) \hookrightarrow \mathcal{S}(\mathbb{R}^n)\) with dense range. 

The Schwartz space, \(\mathcal{S}(\mathbb{R}^n)\), equipped with the countably many seminorms \(||\cdot||^s_k\), is a Frechet space. Induce from the seminorms a norm:

\[||\phi||_k = \underset{|\alpha|+|\beta| \leq k}{\text{max}}\underset{x\in\mathbb{R}^n}{\text{sup}}|x^\alpha D^\beta \phi(x)|,\]

the topology on \(\mathcal{S}(\mathbb{R}^n\) is then given by the metric

\[d(\phi,\psi) = \sum_k 2^{-k}\frac{||\phi -\psi||_k}{1+||\phi -\psi||_k}.\]

We have continuous linear operators

\[\begin{align*}x_j &: \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}(\mathbb{R}^n):\phi\mapsto x_j \phi \\D_j &: \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}(\mathbb{R}^n):\phi\mapsto D_j\phi\end{align*}\]

and furthermore integration \(\int: \mathcal{S}^(\mathbb{R}^n)\to\mathbb{C}\) is a continuous linear map. This follows from

\[\left|\int \phi(x) dx \right| = \left|\int \langle x\rangle^{-n-1} (\langle x\rangle^{n+1}\phi(x)) dx\right| \leq C_n ||\phi||^s_{n+1}.\]

Given \(a\in C^\infty_b(\mathbb{R}^n)\), pointwise multiplication by \(a\) is a continuous linear operator.

By \(\boxtimes\) we denote the exterior product. Namely, \(\phi\boxtimes\psi \in \mathcal{S}(\mathbb{R}^{2n})\) for \(\phi,\psi \in \mathcal{S}(\mathbb{R}^n)\), so that \(\phi\boxtimes\psi (x,y) = \phi(x)\psi(y)\).

Fourier transform#

The Fourier transform of \(\phi\in\mathcal{S}(\mathbb{R}^n)\) is defined by

\[(\mathcal{F}\phi)(\xi) = \hat{\phi}(\xi) := \int e^{-ix\cdot \xi}\phi(x) dx, \quad\xi\in\mathbb{R}^n,\]

with the inverse Fourier transform of \(\psi\in\mathcal{S}(\mathbb{R}^n)\) given by

\[(\mathcal{F}^{-1}\psi)(\xi) := (2\pi)^{-n} \int e^{ix\cdot \xi}\psi(\xi) d\xi,\quad x\in\mathbb{R}^n.\]

We have \(||\mathcal{F}\phi||^s_0 \leq C_N ||\phi||_{n+1}^s\), \(||\mathcal{F}^{-1}\phi||^s_0 \leq C_N ||\phi||_{n+1}^s\). This, coupled with

\[\begin{align*}\mathcal{F}(D_{x_j}\phi) & = \xi_j\mathcal{F}\phi\\\mathcal{F}^{-1}(D_{x_j}\phi) & = -\xi_j\mathcal{F}^{-1}\phi\end{align*}\]

and

\[\begin{align*}\mathcal{F}(x_j \phi) & = -D_{\xi_j}\mathcal{F}\phi\\\mathcal{F}^{-1}(\xi_j\phi) & =D_{x_j}\mathcal{F}^{-1}\phi\end{align*}\]

shows that \(||\mathcal{F}\phi||^s_k \leq C_N ||\phi||^s_{k+n+1}, \forall k\in \mathbb{N}\), proving that the (inverse) Fourier transform preserves the Schwartz space:

\[\mathcal{F},\mathcal{F}^{-1} : \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}(\mathbb{R}^n).\]

Tempered distributions#

Definition (tempered distributions). The space \(\mathcal{S}'(\mathbb{R}^n)\) of tempered distributions is the space of all continuous linear functionals \(u:\mathcal{S}(\mathbb{R}^n)\to\mathbb{C}\), equipped with the weak topology: the seminorms are \(|u|_\phi := |u(\phi)|\) for an arbitrary fixed \(\phi\in\mathcal{S}(\mathbb{R}^n)\). We write \(\langle u,\phi\rangle := u(\phi)\).

Fourier transforms can be extended to maps on tempered distributions, also denoted by \(\mathcal{F}\) and \(\mathcal{F}^{-1}\). The two maps, \(\mathcal{F},\mathcal{F}^{-1}:\mathcal{S}'(\mathbb{R}^n) \to \mathcal{S}'(\mathbb{R}^n)\) are given by duality,

\[\langle \mathcal{F}u,\psi\rangle = \langle u,\mathcal{F} \psi\rangle\]

for \(\psi \in \mathcal{S}(\mathbb{R}^n)\) and \(u\in\mathcal{S}'(\mathbb{R}^n)\).

Schwartz representation theorem#

There is an injective map \(T:\mathcal{S}(\mathbb{R}^n)\to\mathcal{S}'(\mathbb{R}^n): \phi\mapsto T_\phi\) that is dense (in the weak topology), where 

\[T_\phi (\psi) := \int \phi(x)\psi(x) dx.\]

Now the maps \(x_i, D_j\) extended by continuity (hence uniquely) to operators \(x_j ,D_j : \mathcal{S}'(\mathbb{R}^n) \to \mathcal{S}'(\mathbb{R}^n)\), defined by duality:

\[\begin{align*}\langle x_i u,\psi\rangle &= \langle u, x_i \psi\rangle,\\\langle D_{x_i} u,\psi\rangle &= -\langle u, D_{x_i} \psi\rangle,\end{align*}\]

and moreover the definition agrees with \(T\), in that \(D_{x_j} T_\phi = T_{D_{x_j}\phi}\) and \(x_j T_\phi = T_{x_j \phi}\). This association \(T\) of a distribution to a function can be extended considerably, for example. if \(g:\mathbb{R}^n \to \mathbb{C}\) is a bounded and continuous function, then \(T_g(\psi) = \int g(x)\psi(x) dx\) still defines a distribution that vanishes iff \(g\) vanishes identically, thus for any \(\alpha,\beta \in \mathbb{N}^n\), \(x^\beta D^\alpha_x g \in \mathcal{S}'(\mathbb{R}^n)\) if \(g:\mathbb{R}^n \to \mathbb{C}\) is bounded and continuous.

Conversely, we have the Schwartz representation theorem:

Theorem. For any \(u\in\mathcal{S}'(\mathbb{R}^n)\) there is a finite collection \(u_{\alpha\beta}:\mathbb{R}^n\to\mathbb{C}\) of bounded continuous functions with \(|alpha| + |beta| \leq k\) s.t.

\[u = \sum_{|\alpha| + |\beta| \leq k} x^\beta D^\alpha_x u_{\alpha\beta}.\]

Which simply says that tempered distributions are just products of polynomials and derivatives of bounded continuous functions.

Schwartz kernel theorem#

Theorem. Let \(O_K : \mathcal{S}(\mathbb{R}^m) \to \mathcal{S}'(\mathbb{R}^n)\) be the linear map \(O_K(\psi)(\phi) = \int K\cdot\phi\boxtimes\psi dx dy\), where \(K\in \mathcal{S}'(\mathbb{R}^{n+m})\), there is a 1-1 correspondence between the space of continuous linear operators \(\mathcal{S}(\mathbb{R}^m)\to\mathcal{S}'(\mathbb{R}^n)\) and \(\mathcal{S}'(\mathbb{R}^{n+m})\) given by \(K\to O_K\).

Observe that \(O_K(\psi)(\phi) = (O_K \psi)(\phi) = \langle K, \phi\boxtimes \psi\rangle.\) Here \(K\) is a distributional integral kernel, and integral kernels of this kind are called Schwartz kernels.