Hypoellipticity and Local Solvability in Gevrey Classes

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].…”

Anisotropic Operator Symbols Arising From Multivariate Jump Processes

“…Furthermore, for the closely related analysis of hypo-and multi-quasi-elliptic operators we refer to [1,3,4,6,22,25,42,41]. Even though the focus of this work lies on classical Sobolev-and hence L 2 -based results, note that a great number of L p -boundedness results for (different classes of) anisotropic integral operators can be found in [10,16,29,40,46] and the references there.…”

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

“…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 condition of hypoellipticity of the operator P on an open set Ω ensures that its transposed operator t P is locally solvable, as it was proved in [7] in the C ∞ case and in [2] in the frame of Gevrey classes. On the other hand, under the assumptions that P and t P are both hypoelliptic in an open set Ω and surjective on C ∞ (Ω), Malgrange [17] showed that P has a two-sided fundamental kernel, thereby obtaining that the inhomogeneous equation P u = f admits a linear continuous solution operator on the space of C ∞ functions with compact support.…”

“…This research continues work of A. Corli, L. Rodino, A. Morando and the first author. In particular, certain estimates from [2] are essential. The paper concludes with a few examples of partial differential operators with variable coefficients to which our results can be applied.…”