Ultradifferentiable Fundamental Kernels of Linear Partial Differential Operators on Non-quasianalytic Classes of Roumieu Type

Abstract

Let P be a linear partial differential operator with coefficients in the Roumieu class E {ω} (Ω). We prove that if P and its transposed operator t P are {ω}-hypoelliptic in Ω and surjective on the space E {ω} (Ω), then P has a global two-sided ultradifferentiable fundamental kernel in Ω, thus extending to the Roumieu classes the well-known analogous result of B. Malgrange in the C ∞ class. This result is new even for Gevrey classes. §1. Introduction and PreliminariesEhrenpreis [5] and Malgrange [17] proved that every linear partial differential operator with constant coefficients has a fundamental solution; therefore the inhomogeneous equation P u = f admits always C ∞ /E {ω} -solutions for each right hand side f in the class C ∞ /E {ω} respectively, and with compact support. However, it is well-known that this is not longer the case for linear partial differential operators with variable coefficients, i.e., see the Lewy's operator. The