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

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

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.\]

Contributors and Attributions

• Prof. Dr. Erich Miersemann (Universität Leipzig)

• Integrated by Justin Marshall.