# Basic Notions of Phase Space Analysis

• Nicolas Lerner
Part of the Pseudo-Differential Operators book series (PDO, volume 3)

## Abstract

A differential operator of order m on &#x211D; n can be written as
$$a(x,D) = \sum\limits_{\left| \alpha \right| \leqslant m} {a_a (x)D_x^\alpha } ,$$
where we have used the notation (4.1.4) for the multi-indices. Its symbol is a polynomial in the variable ζ and is defined as
$$a(x,\xi ) = \sum\limits_{\left| \alpha \right| \leqslant m} {a_a (x)\xi ^\alpha } , \xi ^\alpha = \xi _1^{\alpha _1 } \ldots \xi _n^{\alpha _n } .$$
We have the formula
$$(a(x,D)u)(x) = \int_{\mathbb{R}^n } {e^{2i\pi x \cdot \xi } a(x,\xi )\hat u(\xi )d\xi } ,$$
(1.1)
where û is the Fourier transform as defined in (4.1.1). It is possible to generalize the previous formula to the case where a is a tempered distribution on ℝ2n.

## Keywords

Open Subset Compact Subset Basic Notion Principal Symbol Fourier Multiplier
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