Newton polyhedrons and a formal Gevrey space of double indices for linear partial differential operators. (English) Zbl 1140.35575

The author considers linear operators in the variables \((t,x)\in\mathbb{C}^n\times\mathbb{C}^n\), of the form \(L=P(t,D_t)+Q(t,x,D_t,D_x)\), where \(P(t,D_t)=\sum_{|\alpha|\leq m}c_\alpha(tD_t)^\alpha\), with \(tD_t=(t_1D_{t_1},\cdots,t_nD_{t_n})\), and \(Q(t,x,D_t,D_x)\) is an integro-differential operator in the complex domain. The author then defines the Newton polyhedron associated to \(L\), depending on the order of degeneracy with respect to \(t\) and \(x\) of the coefficients of \(Q\).
A precise result is expressed concerning the bijectivity of \(L\) in formal Gevrey spaces with double indices \(G^{(s_t,s_x)}\), with \(s_t\) and \(s_x\) determined in terms of the Newton polyhedron.


35S05 Pseudodifferential operators as generalizations of partial differential operators
35A20 Analyticity in context of PDEs