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Some facts about the Wick calculus. (English) Zbl 1180.35596
Rodino, Luigi (ed.) et al., Pseudo-differential operators. Quantization and signals. Lectures given at the C.I.M.E. summer school, Cetraro, Italy, June 19–24, 2006. Berlin: Springer (ISBN 978-3-540-68266-0/pbk). Lecture Notes in Mathematics 1949, 135-174 (2008).
The paper, addressed to a large audience of non-specialists, gives first the basic lines of the pseudo-differential calculus in the Weyl quantization. Attention is then addressed to the Wick quantization. Two applications of the Wick calculus are presented, with, some novelty with respect to known results. Namely, a cheap proof is given of the celebrated subellipticity theorem of Egorov, Treves, Hörmander for operators of principal type. Then, the Fefferman-Phong inequality is recaptured, under minimal assumptions on the regularity of the symbol. In the whole, the paper turns out to be excellent reading for people interested in recent applications of the pseudo-differential calculus to the general theory of the linear partial differential equations.
For the entire collection see [Zbl 1143.35003].

##### MSC:
 35S05 Pseudodifferential operators as generalizations of partial differential operators