Résolubilité globale d'opérateurs différentiels invariants sur certains groupes de Lie

Abstract

We study the existence of global fundamental solutions of bi-invariant linear differential operators on the direct product G = H × K, where H and K are Lie groups, K compact. Using the partial Fourier transform on K, we prove that a bi-invariant operator P on G admits a fundamental solution on U × K (with U open subset of H) if and only if its partial Fourier coefficients satisfy a condition of slow growth and each one admits a fundamental solution on U. Hence we deduce an explicit necessary condition for the existence of a global solution for P on G. We also give a sufficient condition for the existence of a fundamental solution of P on G in the case where the group H is solvable and simply connected. Using Rouvière's method, based on L 2 inequalities, we show that a differential operator satisfying this condition has a fundamental solution on every relatively compact open subset of G (i.e., a semiglobal solution). It follows that P has a fundamental solution on G: use of the notion of P-convexity enables us to obtain global solutions from semiglobal solutions. We also prove the existence of a global fundamental solution for every nonzero bi-invariant operator on a simply connected solvable Lie group.