Global hypoellipticity and global solvability in Gevrey classes on the n-dimensional torus

Abstract

Let P be a linear partial differential operator with coefficients in the Gevrey class G s ðT n Þ where T n is the n-dimensional torus and sX1: We prove that if P is s-globally hypoelliptic in T n then its transposed operator t P is s-globally solvable in T n ; thus extending to the Gevrey classes the well-known analogous result in the corresponding C N class. r 2004 Elsevier Inc. All rights reserved.

Global analytic and Gevrey solvability of sublaplacians under Diophantine conditions

“…This result extends to analytic and Gevrey classes the one of Petronilho [11] where the C ∞ -global solvability for such a class of operators was characterized and to any dimensions of one of [14]. On the other hand, since there is a connection between the global analytic and Gevrey solvability of any partial differential operator P and the global analytic and Gevrey hypoellipticity of t P as it was shown in [15] (or see [16]), this paper can be regarded as a continuation of Himonas [6] where a study of the global analytic and Gevrey hypoellipticity for this class of operators was presented.…”

“…Clearly, it holds that P is s-globally solvable in G s (T m+n ) if and only if P S and P S c are s-globally solvable in G s S (T m+n ) and in G s S c (T m+n ) respectively. Proceeding exactly as in the proof of [15, Theorem 2.1] with minor changes (or see [16] where such a result was announced), we obtain that Theorem 3 Let S ⊂ Z n and s ≥ 1. If P S is s-globally hypoelliptic in T m+n , then t (P S ) is s-globally solvable in E s,S (T m+n ).…”

“…Indeed, the topology here is defined in a similar way to the topology in the classical Gevrey spaces on the torus (see (1)) and the product of two functions from G s (T n ), 0 < s < 1, is in the same class. This yields that the main results of [14,15] can be applied to cover the case 0 < s < 1. On the other hand, by using Fourier-expansion one shows that representation (2) holds (see e.g.…”

“…By integrating (15) first with respect to t on T m and then with respect to s on I ξ , we obtain that…”

On Gevrey solvability and regularity

“…We would like to point out that in the C ∞ case the proof of Proposition 2.1 follows from a variation of [36, Theorem 26.1.7] while Albanese and Zanghirati [2] have proved it in the Gevrey case, s ≥ 1.…”