On some degenerate pseudodifferential operators (Q2304360)

The authors investigate a class of degenerate pseudodifferential operators with a variable symbol depending on a complex parameter. Pseudodifferential operators are constructed by applying a special integral transform containing a weight, which expresses the degeneration property; see [\textit{A. D. Baev}, Sov. Math., Dokl. 26, 182--185 (1982; Zbl 0528.35044); translation from Dokl. Akad. Nauk SSSR 265, 1044--1046 (1982)]. Theorems on the composition and boundedness of these operators in special weighted spaces are proved. The behaviour of these operators on hyperplanes of degeneration is investigated. Theorems on the commutation of these operators with differentiation operators are established. An adjoint operator is constructed, and an analogue of Gårding's inequality for degenerate pseudodifferential operators is proved.