On bounded pseudodifferential operators in a high-dimensional setting

TL;DR

This paper extends boundedness results of Weyl pseudodifferential operators to high-dimensional spaces, providing conditions for uniform bounds independent of dimension by decomposing symbols and analyzing hybrid operators.

Contribution

It introduces a novel decomposition approach for symbols, combining Weyl and anti-Wick operators, to establish dimension-independent boundedness criteria.

Findings

Established uniform L^2 boundedness conditions for high-dimensional Weyl operators.

Developed a decomposition method for symbols into hybrid operators.

Applied classical methods like coherent states to derive bounds.

Abstract

This work is concerned with extending the results of Calder\' on and Vaillancourt proving the boundedness of Weyl pseudo differential operators Op_h^{weyl} (F) in L^2(\R^n). We state conditions under which the norm of such operators has an upper bound independent of n. To this aim, we apply a decomposition of the identity to the symbol F, thus obtaining a sum of operators of a hybrid type, each of them behaving as a Weyl operator with respect to some of the variables and as an anti-Wick operator with respect to the other ones. Then we establish upper bounds for these auxiliary operators, using suitably adapted classical methods like coherent states.