Pseudo-differential operators with isotropic symbols, Wick and anti-Wick operators, and hypoellipticity

Abstract

We study the link between pseudo-differential operators and Wick operators via the Bargmann transform. We deduce a formula for the symbol of the Wick operator in terms of the short-time Fourier transform of the Weyl symbol. This gives characterizations of Wick symbols of pseudo-differential operators of Shubin type and of infinite order, and results on composition. We prove a series expansion of Wick operators in anti-Wick operators which leads to a sharp Gårding inequality and transition of hypoellipticity between Wick and and Shubin symbols. Finally we show continuity results for anti-Wick operators, and estimates for the Wick symbols of anti-Wick operators.

51 pages. This is the second version. Several new results have been added compared to the first version. Especially the part concerning hypoellipticity is new. The title is modified as well