Regularity and solvability of pseudo-differential operators with double characteristics (Q2325710)

From the introduction: This paper is a continuation and generalization of the first author's papers [Rend. Semin. Mat., Univ. Politec. Torino 66, No. 4, 321--338 (2008; Zbl 1180.35176); Oper. Theory: Adv. Appl. 260, 171--184 (2017; Zbl 1390.35445)], where classical \(\psi\) do with symplectic and involutive characteristics have been studied. Here, the authors investigate at the first classes of operators with double involutive characteristics as well as operators with non-negative principal symbol which vanishes of sharp order two on the intersection of transversal each to other symplectic and involutive manifolds. To simplify the things, the principal symbol is written into the canonical form of non-negative symmetric quadratic forms in symplectic coordinates in \(T^\ast(\mathbb{R}^n)\). The authors introduce new functional spaces and prove there subelliptic estimates for the operators with symplectic-involutive characteristics. Nonsolvability is also considered. More precisely, microlocal non-solvability is proved for operators having involutive characteristics. Concerning the operators with symplectic-involutive characteristics, it is interesting to point out that the conditions imposed on the subprincipal symbol are the same as the conditions imposed on the subelliptic \(\psi\) do of principal type. However, for the latter these conditions, ensure estimates in Sobolev spaces with sharp loss of regularity \(\frac{k}{k+1}\), \(k \in \mathbb{N}\) (see [\textit{Yu. V. Egorov}, Linear differential equations of principal type. Transl. from the Russian by Dang Prem Kumar. New York: Consultants Bureau (1986; Zbl 0669.35001), [\textit{L. Hörmander}, The analysis of linear partial differential operators. I: Distribution theory and Fourier analysis. Berlin: Springer (1983; Zbl 0521.35001); The analysis of linear partial differential operators. II: Differential operators with constant coefficients. Berlin: Springer (1983; Zbl 0521.35002); The analysis of linear partial differential operators. III: Pseudo-differential operators. Berlin: Springer (1985; Zbl 0601.35001); The analysis of linear partial differential operators. IV: Fourier integral operators. Berlin: Springer (1985; Zbl 0612.35001)]), while for the operators under consideration, they lead to estimates with loss of regularity \(1 + \frac{k}{k+1}\). At the end of the paper, an inverse operator of a model one with involutive characteristics and non-elliptic subprincipal symbol is constructed into explicit form.