Global analytic and Gevrey surjectivity of the Mizohata operator \(D_ 2+ix_ 2^{2k}D_ 1\). (English) Zbl 0707.35036
Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 1, No. 1, 37-39 (1990).
Analytic surjectivity of all linear partial differential operators with constant coefficients on \({\mathbb{R}}^ 2\) and existence of a counterexample for the case of operators in \({\mathbb{R}}^ 3\) have been proved. In this paper, the authors prove the surjectivity of the operator \(D_ 2+ix_ 2^{2k}D_ 1\), which is an operator with variable coefficients, from the Gevrey space \({\mathcal E}^{\{s\}}({\mathbb{R}}^ 2)\) into itself and its non-surjectivity from \({\mathcal E}^{\{s\}}({\mathbb{R}}^ 3)\) to \({\mathcal E}^{\{s\}}({\mathbb{R}}^ 3)\).
Reviewer: K.Taniguchi
MSC:
35E20 | General theory of PDEs and systems of PDEs with constant coefficients |
35F05 | Linear first-order PDEs |