A generalization of the problem of elliptic iterates
by
Chikh Bouzar
Département de Mathématiques. Université d'Oran. Algérie.
Coauthors: Rachid CHAILI (Université d'Oran)

The aim of this work is to find algebraic necessary and sufficient conditions such that the next inclusion, between spaces of Gevrey vectors of systems of linear partial differential operators,

Gs( \Omega, ( Pj) j=1N) subset Gs'( \Omega, ( Qj) j=1L),

(1)

holds.

If the system ( Q_j) _j=1^L is reduced to the elementary system of differential operators (D₁, .., D_n) we obtain then the classical ``problem of elliptic iterates''.

The general problem (1) is completely solved in the case of systems of differential operators with constant coefficients.

Theorem 1 Let \Omega be an open set of Rⁿ and ( P_j)_j=1^N and ( Q_j) _j=1^L two systems of linear partial differential operators of orders, respectively,  m and r with constant complex coefficients ,  satisfying some conditions ( H) and (C) ,  then

for alls >=  \gamma(P) m ,     Gs( \Omega, (Pj) j=1N) subset Gsh[ m/r]( \Omega, ( Qj) j=1L)     ,

if and only if

existsC > 0, existsh > 0:    æ è 1+ N å j=1  | Pj( \xi) | ö ø h   >= C L å j=1  | Qj( \xi)|,     for all\xi in Rn

By definition G^s( \Omega, ( D_j) _j=1ⁿ) is the classical isotropic Gevrey space G^s( \Omega) . We note G^sq( \Omega) the classical anisotropic Gevrey space, where q=(q₁, .., q_n) in (R₊)ⁿ. We define G^\Gamma(\Omega) the generalized anisotropic Gevrey space, where \Gamma is a Newton polyhedra.

In the case of systems of differential operators with variable coefficients we have final results when the space G^s( \Omega, (Q_j) _j=1^L) is G^sq( \Omega) or G^\Gamma( \Omega) .

(T)

Date received: April 13, 2000