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Mathematics LibreTexts

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

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

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