WKB analysis to global solvability and hypoellipticity.

*(English)*Zbl 0842.35021In the paper under consideration global regularity and solvability of differential and pseudodifferential operators on the two-dimensional torus \(T^2\) are investigated. Their properties are studied and explained by using the methods of the WKB analysis. The WKB formal solutions, i.e. the formal power series with respect to a large parameter whose coefficients may have poles of arbitrary large orders, enable the authors to give sharp results on global hypoellipticity and solvability.

More precisely, a necessary and sufficient condition is given for the global hypoellipticity on \(T^2\) of an \(m \times m\) system. Siegel conditions play an important role here, and it is shown that they can be invariantly defined under realizations and changes of variables for the corresponding WKB solutions. A single equation with characteristics of variable multiplicity is studied, and under the assumption that a smooth formal solution of Riccati equation exists, a sharp result on hypoellipticity is proved.

More precisely, a necessary and sufficient condition is given for the global hypoellipticity on \(T^2\) of an \(m \times m\) system. Siegel conditions play an important role here, and it is shown that they can be invariantly defined under realizations and changes of variables for the corresponding WKB solutions. A single equation with characteristics of variable multiplicity is studied, and under the assumption that a smooth formal solution of Riccati equation exists, a sharp result on hypoellipticity is proved.

Reviewer: P.Popivanov (Sofia)