Multisummability of Formal Solutions of Some Linear Partial Differential Equations

Let û(t, x)=∑n=1∞un(x)tn be a formal power series in one variable t satisfying formally P(t, x,∂t,∂x)u(t, x)=f(t, x),(t, x) × d. The purpose of the present paper is to introduce a class of linear partial differential operators and show that û(t, x) is multisummable provided P(t, x,∂t,∂x) belongs to this class