Non-isotropic Gevrey classes in domains of finite type in \(\mathbb{C}^ 2\) (Q1317305)

The paper develops the theory of Grevrey classes for domains of finite type in \(\mathbb{C}^ 2\). The nonisotropic Gevrey class \(G_{NI}^{1 + \alpha}\) consists of those functions in the Gevrey class \(G^{1 + \alpha}\) which have additional Gevrey regularity in the complex tangential directions. In particular, holomorphic functions in \(G^{1 + \alpha}\) are in \(G_{NI}^{1 + \alpha}\). After developing the necessary machinery of adapted coordinates, pseudoballs, and regions of approach, the author proves a Taylor theorem for functions in \(G_{NI}^{1 + \alpha}\). Then he examines certain domains of finite type having a polynomial defining function: for such domains he obtains formulations of the definition of \(G_{NI}^{1 + \alpha}\) and Taylor's theorem using globally defined vector fields on the boundary. The author also constructs special partitions of unity by functions in \(G_{NI}^{1 + \alpha}\) and uses these to extend a Whitney jet on a compact set in the boundary to a function in \(G_{NI}^{1 + \alpha}\). The author refers to a second paper [Math. Z. 212, No. 4, 555-580 (1993; Zbl 0790.32008)] in which he uses these results to study Gevrey interpolation in pseudoconvex domains of finite type in \(\mathbb{C}^ 2\).