On bounded pseudodifferential operators in a high-dimensional setting

Abstract

This work is concerned with extending the results of Calderón and Vaillancourt, proving the boundedness of Weyl pseudodifferential operators O p h W e y l ( F ) Op_h^{Weyl} (F) in L 2 ( R n ) L^2({\mathbb R}^n) . We state conditions under which the norm of such operators has an upper bound independent of n n . To this aim, we apply a decomposition of the identity to the symbol F 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.