
# 5.1.1: Pseudodifferential Operators


The properties of Fourier transform lead to a general theory for linear partial differential or integral equations. In this subsection we define

$$D_k=\frac{1}{i}\frac{\partial}{\partial x_k},\ \ k=1,\ldots,n,\] and for each multi-index $$\alpha$$ as in Subsection 3.5.1$$D^\alpha=D_1^{\alpha_1}\ldots D_n^{\alpha_n}.\]

Thus

$$D^\alpha=\frac{1}{i^{|\alpha|}}\frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}\ldots \partial x_n^{\alpha_n}}.\] Let$$p(x,D):=\sum_{|\alpha|\le m}a_\alpha(x)D^\alpha,\]

be a linear partial differential of order $$m$$, where $$a_\alpha$$ are given sufficiently regular functions.

According to Theorem 5.1 and Proposition 5.3, we have, at least for $$u\in{\mathcal{S}}(\mathbb{R}^n)$$,

$$u(x)=(2\pi)^{-n/2}\int_{\mathbb{R}^n}\ e^{ix\cdot\xi}\widehat{u}(\xi)\ d\xi,\] which implies$$D^\alpha u(x)=(2\pi)^{-n/2}\int_{\mathbb{R}^n}\ e^{ix\cdot\xi}\xi^\alpha\widehat{u}(\xi)\ d\xi.\]

Consequently

\label{pseudo1}\tag{5.1.1.1}
p(x,D)u(x)=(2\pi)^{-n/2}\int_{\mathbb{R}^n}\ e^{ix\cdot\xi}p(x,\xi)\widehat{u}(\xi)\ d\xi,

where

$$p(x,\xi)=\sum_{|\alpha|\le m}a_\alpha(x)\xi^\alpha.\] The right hand side of (\ref{pseudo1}) makes sense also for more general functions $$p(x,\xi)$$, not only for polynomials. Definition. The function $$p(x,\xi)$$ is called symbol and$$(Pu)(x):=(2\pi)^{-n/2}\int_{\mathbb{R}^n}\ e^{ix\cdot\xi}p(x,\xi)\widehat{u}(\xi)\ d\xi\]

is said to be pseudodifferential operator.

An important class of symbols for which the right hand side in this definition of a pseudodifferential operator is defined is $$S^m$$ which is the subset of $$p(x,\xi)\in C^\infty(\Omega\times\mathbb{R}^n)$$ such that

$$|D^\beta_xD_\xi^\alpha p(x,\xi)|\le C_{K,\alpha,\beta}(p)\left(1+|\xi|\right)^{m-|\alpha|}\] for each compact $$K\subset\Omega$$. Above we have seen that linear differential operators define a class of pseudodifferential operators. Even integral operators can be written (formally) as pseudodifferential operators. Let$$(Pu)(x)=\int_{\mathbb{R}^n}\ K(x,y)u(y)\ dy\]

be an integral operator. Then

\begin{eqnarray*}
(Pu)(x)&=&{(2\pi)^{-n/2}\int_{\mathbb{R}^n}\ K(x,y)\int_{\mathbb{R}^n}\ e^{ix\cdot\xi}\xi^\alpha\widehat{u}(\xi)}\ d\xi\\
&=&(2\pi)^{-n/2}\int_{\mathbb{R}^n}\ e^{i x\cdot \xi}\left(\int_{\mathbb{R}^n}\ e^{i(y-x)\cdot\xi}K(x,y)\ dy\right)\widehat{u}(\xi).
\end{eqnarray*}

Then the symbol associated to the above integral operator is

p(x,\xi)=\int_{\mathbb{R}^n}\ e^{i(y-x)\cdot\xi}K(x,y)\ dy.\]