Linear Partial Differential Operators in Gevrey Spaces

On Gevrey solvability and regularity

“…For results on interesting open problems of local and global G s , 1 ≤ s ≤ ∞, solvability we refer the reader to the following papers as well as the references therein: Albanese, Corli, and Rodino [1], Bergamasco, Cordaro, and Petronilho [7], Cardoso [10], Cardoso and Hounie [11], Gramchev, Popivanov, and Yoshino [20]- [22], Gramchev and Yoshino [23], Hounie [37], Petronilho [43]- [46], Rodino [47].…”

“…Local and global G s , 1 ≤ s ≤ ∞, hypoellipticity has been studied by many authors, including Albanese, Corli, and Rodino [1], Amano [3], Baouendi and Goulaouic [4], Bell and Mohammed [5], Bergamasco, Cordaro, and Malagutti [6], Bove and Tartakoff [8], [9], Christ [12]- [14], Cordaro and Himonas [15], [16], Dickinson, Gramchev, and Yoshino [17], Fedii [18], Fujiwara and Omori [19], Gramchev,Popivanov,and Yoshino [20]- [22], Greenfield and Wallach [24], Hanges and Himonas [25], [26], Helffer [27], Himonas [28], [29], Himonas and Petronilho [30]- [33], Himonas, Petronilho, and dos Santos [34], Hörmander [35], [36], Kohn [38], Pham The Lai and Robert [39], Metivier [40], Oleinik and Radkevic [41], Omori and Kobayashi [42], Rodino [47], Rothschild and Stein [48], Tartakoff [49].…”

Hypoellipticity and Local Solvability in Gevrey Classes

“…We emphasize that there are in fact many different but equivalent ways to define these spaces; see [9], [8], [16]. We also remark that, if H and K are compact sets with H ⊆ K, then the inclusion map [1,Lemma 4.6]).…”

“…These operators are s-locally solvable and non-s-hypoelliptic for 1 < s < m/(m − 1) (see [16]). Several results are known in Gevrey classes with index s ≥ m/(m − 1), under suitable assumptions on Q m−1 , (cf.…”

Rapidly convergent iteration method for simultaneous normal forms of commuting maps

“…We refer to [14] for basic properties of the Gevrey classes and applications to PDE. In the sequel we change the order of notation for T , T etc.…”