Pseudodifferential Operators

Differential Operators#

Constant coefficient differential operators#

Fourier transform gives an isomorphism

\[\mathcal{F}: \{u\in \mathcal{S}'(\mathbb{R}^n):\text{ supp}(u)\subset\{0\}\} \cong \mathbb{C}[\xi] = \{\text{polynomials in }\xi\}.\]

This can be seen from \(\mathcal{F}(\delta) = 1\) and hence \(\mathcal{F} (D^\alpha \delta(x)) = \xi^\alpha, \forall \alpha \in \mathbb{N}^n_0\) and that the space of distributions with wupport the point \(0\) is just \(\{u\in\mathcal{S}'(\mathbb{R}^n):\text{supp}(u)\subset \{0\}\} = \{ u = \sum_{\text{finite}} c_\alpha D^\alpha \delta\}\).

The isomorphism can be seen in a different light. Consider partial differential operators with constant coefficients, 

\[P(D):\mathcal{S}(\mathbb{R}^n) \to \mathcal{S}(\mathbb{R}^n): D\mapsto \sum c_\alpha D^\alpha\]

the identity becomes, thenl 

\[\mathcal{F}(P(D)\phi)(\xi) = P(\xi)\mathcal{F}(\phi)(\xi),\forall \phi\in\mathcal{S}(\mathbb{R}^n)\]

and the same formula holds for all \(\phi\in\mathcal{S}'(\mathbb{R}^n)\). Put \(\hat{u}(\xi) = (\mathcal{F}(u))(\xi)\), this can then be written as 

\[\widehat{P(D)u(\xi)} = P(\xi)\hat{u}(\xi),\quad P(\xi)=\sum c_\alpha \xi^\alpha.\]

We call the polynomial \(P\) that determines \(P(D)\) uniquely the full characteristic polynomial of \(P(D)\).

Variable constant differential operators#

Fourier transform \(P(D) u(x)\) and back, we get an iterated integral 

\[P(D)u(x) = (2\pi)^{-n}\int\int e^{i(x-y)\cdot\xi}P(\xi)u(y) dy d\xi\]

. Consider the space \(C^\infty_\infty(\mathbb{R}^n)\) of \(C^\infty\) functions with all derivative bounded on \(\mathbb{R}^n\), \(\mathcal{S}(\mathbb{R}^n\subset C^\infty_\infty(\mathbb{R}^n)\) (\(\mathcal{S}(\mathbb{R}^n)\) is much smaller, in particular, \(1\in C^\infty_\infty(\mathbb{R}^n)\)). By Leibniz's formula 

\[D^\alpha(uv) = \sum_{\beta\leq\alpha}\binom{\alpha}{\beta} D^\beta u \cdot D^{\alpha-\beta}v\]

 it follows that \(\mathcal{S}(\mathbb{R}^n)\) is a module over \(C^\infty_\infty(\mathbb{R}^n)\). If follows that given 

\[P(x,D) = \sum_{|\alpha|\leq m}p_\alpha(x) D^\alpha, p_\alpha \in C^\infty_\infty (\mathbb{R}^n)\]

 then \(P(x,D): \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}(\mathbb{R}^n)\). Thus we have 

\[P(x,D)\phi = (2\pi)^{-n}\int e^{i(x-y)\cdot \xi}P(x,\xi)\phi(y) dy d\xi\]

 where again this is an iterated integral with \(P(x,\xi)\) being the full characteristic polynomial.

Pseudodifferential Operators on Euclidean Spaces#

The point of pseudodifferential operators is the generalization of the notion of a differential operator. We have for a differential operator \(P(x,D)\phi = (2\pi)^{-n}\int e^{i(x-y)\cdot \xi}P(x,\xi)\phi(y) dy d\xi,\) replacing \(P(x,\xi)\) from a polynomial in \(\xi\) to depend on the ‘incoming’ variables \(y\) (in which case the symbol is called an amplitude) 

\[A(x,D) u = (2\pi)^{-n}\int e^{i(x-y)\cdot \xi}a(x,y,\xi) u(y) dy d\xi, u\in\mathcal{S}(\mathbb{R}^n).\]

The integral can be interpreted as an oscillatory integral, and thereby defines an operator on \(\mathcal{S}(\mathbb{R}^n)\).

Symbols#

Definition. The space \(S^m_\infty(\mathbb{R}^p;\mathbb{R}^n)\) of symbols of order \(m\) with coefficients in \(C^\infty_\infty(\mathbb{R}^p)\) consists of those functions \(a\in C^\infty(\mathbb{R}^p\times \mathbb{R}^n)\) satisfying all the estimates 

\[|D_z^\alpha D_\xi^\beta a(z,\xi)|\leq C_{\alpha,\beta}(1+|\xi|)^{m-|\beta|}\]

 on \(\mathbb{R}^p\times \mathbb{R}^n, \forall \alpha \in \mathbb{N}^p_0, \beta\in\mathbb{N}^n_0\).

Definition. The space \(S^m_\infty(\Omega;\mathbb{R}^n)\) where \(\Omega \subset \mathbb{R}^p\) and \(\Omega \subset \text{closure}(\text{int}(\Omega))\) consists of those \(a\in C^\infty_\infty(\text{int}(\Omega)\times\mathbb{R}^n)\) satisfying 

\[A(x,D) u = (2\pi)^{-n}\int e^{i(x-y)\cdot \xi}a(x,y,\xi) u(y) dy d\xi, u\in\mathcal{S}(\mathbb{R}^n)\]

 on \(\mathbb{R}^p\times \mathbb{R}^n, \forall \alpha \in \mathbb{N}^p_0, \beta\in\mathbb{N}^n_0\) for \((z,\xi) \in \text{int}(\Omega)\times \mathbb{R}^n\).

The estimates can be used to define the following norms: 

\[||a||_{N,m} = \underset{z\in\text{int}(\Omega),\xi\in\mathbb{R}^n}{\text{sup}} \underset{|\alpha|+|\beta| \leq N}{\text{max}} (1+|\xi|)^{-m + |\beta|} |D^\alpha_z D^\beta_\xi a(z,\xi)|<\infty\]

that endows \(S^m_\infty(\Omega;\mathbb{R}^n)\) with the structure of a Frechet space. The topology is then given by the metric 

\[d(a,b) = \sum_{N\geq 0}2^{-N}\frac{||a-b||_{N,m}}{1+||a-b||_{N,m}},\quad a,b\in S^m_\infty(\Omega;\mathbb{R}^n).\]

The rational behind the definition is as follows. A polynomial \(p\) in \(\xi\) of degree at most \(m\) satisfies a bound 

\[|p(\xi)| \leq C(1+|\xi|)^m, \forall \xi \in \mathbb{R}^n.\]

 Now successive derivative \(D^\alpha_\xi p(\xi)\) are polynomials of degree \(m-|\alpha|\) for any multiindex \(\alpha\) we have the family of estimates 

\[|D^\alpha_\xi p(\xi)|\leq C_\alpha(1+|\xi|)^{m-|\alpha|}\quad \forall \xi \in\mathbb{R}^n,\alpha \in \mathbb{N}^n_0.\]

The choice of these estimates is not necessary but it has several virtues: large enough to cover most of the straightforward things we want to do and small enough to work easily.

Pseudodifferential operators#