Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces

In this paper we give global characterisations of Gevrey–Roumieu and Gevrey–Beurling spaces of ultradifferentiable functions on compact Lie groups in terms of the representation theory of the group and the spectrum of the Laplace–Beltrami operator. Furthermore, we characterise their duals, the spaces of corresponding ultradistributions. For the latter, the proof is based on first obtaining the characterisation of their α-duals in the sense of Köthe and the theory of sequence spaces. We also give the corresponding characterisations on compact homogeneous spaces.     

FBI transforms in Gevrey classes

In this work we develop the FBI Transform tools in Gevrey classes. Our goal is to extend to a Gevrey-s obstacle withs < 3 the localization of poles result obtained by Sjöstrand [10] in the analytic class. In that work, the author proved that the pole-free zone is controlled by a constantC _0,a (which was only implicit in Bardos-Lebeau-Rauch [1]), improving the constantC _0,∞ of the results of Hargé-Lebeau [13] and Sjöstrand-Zworski [13] valid in C^∞ The works [3], [13] and [10] feature an adapted complex scaling for convex obstacles, but in [10] there is the addition of a small complex “G³ deformation”. The study of such Gevrey deformations for operators with symbols in Gevrey classes is the central point of this work.     

Almost periodic pseudodifferential operators and Gevrey classes

We study almost periodic pseudodifferential operators acting on almost periodic functions ${G_{\rm ap}^s(\mathbb {R}^d)}$ of Gevrey regularity index s ≥ 1. We prove that almost periodic operators with symbols of H?rmander type ${S_{\rho,\delta}^m}$ satisfying an s-Gevrey condition are continuous on ${G_{\rm ap}^s(\mathbb {R}^d)}$ provided 0 < ρ ≤ 1, δ?=?0 and s ρ ≥ 1. A calculus is developed for symbols and operators using a notion of regularizing operator adapted to almost periodic Gevrey functions and its duality. We apply the results to show a regularity result in this context for a class of hypoelliptic operators.     

Fourier integral operators and inhomogeneous Gevrey classes

Summary Fourier integral operators with inhomogeneous amplitude and phase junction are studied in the frame of Gevrey classes. Applications are given to propagation of singularities for a pseudodifferential equation.     

Pseudodifferential operators of infinite order and Gevrey classes

Summary A class of pseufodifferential operators of infinite order acting on spaces of Gevrey functions and their duals is defined. For the corresponding symbols the rules of the classical symbolic calculus are proved. In particular for operators satisfying an “hipoellipticity condition” a result of propagation of Gevrey regularity, is obtained by proving the existence of a parametrix.

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Riassunto Viene definita una classe di operatori pseudodifferenziali di ordine infinito che agiscono su spazi di funzioni Gevrey e sui loro duali. Per i corrispondenti simboli si provano le regole del calcolo simbolico classico. In particolare per gli operatori che soddisfano una “condizione di ipoellitticità” si ottiene un risultato di propapagazione di regolarità Gevrey, provando l'esistenza di una parametrice.

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Work supported by Ministero della Pubblica Istruzione and G.N.A.F.A. C.N.R., of Italy.     

Gevrey vectors in involutive tube structures and Gevrey regularity for the solutions of certain classes of semilinear systems

In this work, we introduce the notion of s-Gevrey vectors in locally integrable structures of tube type. Under the hypothesis of analytic hypoellipticity, we study the Gevrey regularity of such vectors and also show how this notion can be applied to the study of the Gevrey regularity of solutions of certain classes of semilinear equations.     

On the zeros of analytic functions belonging to Gevrey classes

For functionsf(z) ? 0, holomorphic in the unit disk u, infinitely differentiable in u, and belonging to a given Gevrey class on ?u, we establish sufficient conditions characterizing the sets K _f ^∞ = (z: ¦z¦ = 1,f ^(k) (z) = 0,k = 0, 1, 2, ... }. These conditions are close to the necessary condition due to L. Carleson and substantially more precise than the conditions given byA.-M. Chollet (see [1, 2]).     

On relationship between classes of (\Psi, \overline\upbeta)-differentiable functions and Gevrey classes

We investigate the relationship between the classes of -differentiable functions introduced by Stepanets and the well-known Gevrey classes. In particular, we establish necessary and sufficient conditions for periodic functions to belong to the Gevrey classes ℐ_α formulated in terms of their -derivatives. A. I. Stepanets (Deceased.) Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 140–144, January, 2009.     

Global solvability and global hypoellipticity in Gevrey classes for vector fields on the torus

Let $L=?/?t+{\sum }_{j=1}^N(a_j+i{b}_j)(t)?/?x_j$ be a vector field defined on the torus ${\mathbb{T}}^{N+1}?{\mathbb{R}}^{N+1}/2\pi {\mathbb{Z}}^{N+1}$, where $a_j$, $b_j$ are real-valued functions and belonging to the Gevrey class $G^s({\mathbb{T}}^1)$, $s>1$, for $j=1,\dots ,N$. We present a complete characterization for the s-global solvability and s-global hypoellipticity of L. Our results are linked to Diophantine properties of the coefficients and, also, connectedness of certain sublevel sets.     

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

Let P be a linear partial differential operator with coefficients in the Gevrey class G^s(Tⁿ) where Tⁿ is the n-dimensional torus and s?1. We prove that if P is s-globally hypoelliptic in Tⁿ then its transposed operator tP is s-globally solvable in Tⁿ, thus extending to the Gevrey classes the well-known analogous result in the corresponding C^∞ class.     

On the global solvability in Gevrey classes on the n-dimensional torus

Let P be a linear partial differential operator with coefficients in the Gevrey class G^s(Tⁿ), where Tⁿ is the n-dimensional torus and s?1. We prove a necessary condition for the s-global solvability of P on Tⁿ. We also apply this result to give a complete characterization for the s-global solvability for a class of formally self-adjoint operators with nonconstant coefficients.     

Moment sequences for ultradistributions

Supported by DAAD, West Germany     

Representation of quasianalytic ultradistributions

We give the following representation theorem for a class containing quasianalytic ultradistributions and all the non-quasianalytic ultradistributions: Every ultradistribution in this class can be written as