G- and G^∞-hypoellipticity of partial differential operators with constant Colombeau coefficients

Abstract

Abstract. We provide a deep investigation of the notions of G-and G ∞ -hypoellipticity for partial differential operators with constant Colombeau coefficients. This involves generalized polynomials and fundamental solutions in the dual of a Colombeau algebra. Sufficient conditions and necessary conditions for G-and G ∞ -hypoellipticity are given.

Sufficient conditions of local solvability for partial differential operators in the Colombeau context

“…One sees in the proof of Theorem 1.2 (Proposition 3.5 and Theorem 3.3 in [9]) that for each ε the distribution E ε is a fundamental solution of the operator P ε (D). Theorem 1.2 entails the following solvability result.…”

“…The G-elliptic polynomials (see [9,Section 6]) and their corresponding differential operators can be characterized by means of the order relation ≺. We recall that a polynomial P (ξ) with coefficients in C is G-elliptic (or equivalently the operator…”

“…Their properties will inspire the more general model introduced in Section 4. In the sequel we often refer to the work on generalized hypoelliptic and elliptic symbols in [8,9,10].…”

“…Differential operators with constant Colombeau coefficients, i.e. coefficients in the ring C of complex generalized numbers, have been studied by various authors [7,9,20]. In particular a notion of fundamental solution has been introduced in [7] as a functional in the dual L(G c (R n ), C) providing, by means of a generalized version of the Malgrange-Ehrenpreis theorem, a straightforward result of solvability in the Colombeau context.…”

“…In particular a notion of fundamental solution has been introduced in [7] as a functional in the dual L(G c (R n ), C) providing, by means of a generalized version of the Malgrange-Ehrenpreis theorem, a straightforward result of solvability in the Colombeau context. In detail, a solution to the equation P (D)u = v, P (D) = |α|≤m c α D α with c α ∈ C has been obtained via convolution of the right hand side v with a fundamental solution E and certain regularity qualities of the operator P (D), the Gand G ∞ -hypoellipticity for instance, have been proven to be equivalent to some structural properties of its fundamental solutions [9,Theorems 3.6,4.2].…”