Pseudo-Differential Operators

Abstract

We will now turn to a class of operators which — roughly speaking (details below) — are locally presentable in the form $$\left( {Pu} \right)\left( x \right)\,: = \int {ei < x,\varepsilon > p\left( {x,\varepsilon } \right)} \mathop u\limits^ \wedge \left( \varepsilon \right)d\varepsilon$$ where $$ \hat u\left( x \right): = \int {e^{ - i\left\langle {x,\xi } \right\rangle } u\left( \xi \right)d\xi } $$ is the Fourier transform of u (see the crash course above in I.8), p is the “amplitude” of the operator and $< x, \xi > = x_1 \xi _1 + ... + x_n \xi _n$ its “phase function”.