# zbMATH — the first resource for mathematics

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.

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