# An accuracy improvement in Egorov's theorem.

• Volume: 51, Issue: 1, page 77-120
• ISSN: 0214-1493

top

## Abstract

top
We prove that the theorem of Egorov, on the canonical transformation of symbols of pseudodifferential operators conjugated by Fourier integral operators, can be sharpened. The main result is that the statement of Egorov's theorem remains true if, instead of just considering the principal symbols in Sm/Sm-1 for the pseudodifferential operators, one uses refined principal symbols in Sm/Sm-2, which for classical operators correspond simply to the principal plus the subprincipal symbol, and can generally be regarded as the first two terms of its Weyl symbol expansion: we call it the principal Weyl symbol of the pseudodifferential operator. Particular unitary Fourier integral operators, associated to the graph of the canonical transformation, have to be used in the conjugation for the higher accuracy to hold, leading to microlocal representations by oscillatory integrals with specific symbols that are given explicitly in terms of the generating function that locally describes the graph of the transformation. The motivation for the result is based on the optimal symplectic invariance properties of the Weyl correspondence in Rn and its symmetry for real symbols.

## How to cite

top

Silva, Jorge Drumond. "An accuracy improvement in Egorov's theorem.." Publicacions Matemàtiques 51.1 (2007): 77-120. <http://eudml.org/doc/41888>.

@article{Silva2007,
abstract = {We prove that the theorem of Egorov, on the canonical transformation of symbols of pseudodifferential operators conjugated by Fourier integral operators, can be sharpened. The main result is that the statement of Egorov's theorem remains true if, instead of just considering the principal symbols in Sm/Sm-1 for the pseudodifferential operators, one uses refined principal symbols in Sm/Sm-2, which for classical operators correspond simply to the principal plus the subprincipal symbol, and can generally be regarded as the first two terms of its Weyl symbol expansion: we call it the principal Weyl symbol of the pseudodifferential operator. Particular unitary Fourier integral operators, associated to the graph of the canonical transformation, have to be used in the conjugation for the higher accuracy to hold, leading to microlocal representations by oscillatory integrals with specific symbols that are given explicitly in terms of the generating function that locally describes the graph of the transformation. The motivation for the result is based on the optimal symplectic invariance properties of the Weyl correspondence in Rn and its symmetry for real symbols.},
author = {Silva, Jorge Drumond},
journal = {Publicacions Matemàtiques},
keywords = {Operadores pseudodiferenciales; Símbolos; Operadores integrales; pseudo-differential operator; Fourier integral operator; canonical transformation},
language = {eng},
number = {1},
pages = {77-120},
title = {An accuracy improvement in Egorov's theorem.},
url = {http://eudml.org/doc/41888},
volume = {51},
year = {2007},
}

TY - JOUR
AU - Silva, Jorge Drumond
TI - An accuracy improvement in Egorov's theorem.
JO - Publicacions Matemàtiques
PY - 2007
VL - 51
IS - 1
SP - 77
EP - 120
AB - We prove that the theorem of Egorov, on the canonical transformation of symbols of pseudodifferential operators conjugated by Fourier integral operators, can be sharpened. The main result is that the statement of Egorov's theorem remains true if, instead of just considering the principal symbols in Sm/Sm-1 for the pseudodifferential operators, one uses refined principal symbols in Sm/Sm-2, which for classical operators correspond simply to the principal plus the subprincipal symbol, and can generally be regarded as the first two terms of its Weyl symbol expansion: we call it the principal Weyl symbol of the pseudodifferential operator. Particular unitary Fourier integral operators, associated to the graph of the canonical transformation, have to be used in the conjugation for the higher accuracy to hold, leading to microlocal representations by oscillatory integrals with specific symbols that are given explicitly in terms of the generating function that locally describes the graph of the transformation. The motivation for the result is based on the optimal symplectic invariance properties of the Weyl correspondence in Rn and its symmetry for real symbols.
LA - eng
KW - Operadores pseudodiferenciales; Símbolos; Operadores integrales; pseudo-differential operator; Fourier integral operator; canonical transformation
UR - http://eudml.org/doc/41888
ER -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.