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Global \(s\)-solvability and global \(s\)-hypoellipticity for certain perturbations of zero order of systems of constant real vector fields. (English) Zbl 1155.35010
The authors consider on the torus \(\mathbb{T}^2\) a system of first-order operators with real constant coefficients \(L= (L_1,\dots, L_m)\), where \[ L_j= a_j D_x+ b_j D_y,\quad j= 1,\dots, m,\quad (x,y)\in \mathbb{T}^2. \] Global solvability and global hypoellipticity of \(L\) are studied in the frame of the Gevrey classes \(G^s(\mathbb{T}^2)\), \(1\leq s<\infty\), defined as the set of all the functions \(f\in C^\infty(\mathbb{T}^2)\) satisfying for some positive constant \[ C:|D^\alpha_x D^\beta_ y f(x,y)|\leq C^{\alpha+ b\eta+1}(\alpha!)^s(\beta!)^s,\quad (x,y)\in \mathbb{T}^2. \] Precise result are given for the equations \(Lu= f\) and \(Lu= cu_ f\), where \(c\) and \(f\) are vectors of functions in \(G^s(\mathbb{T}^2)\). For related results we address to A. A. Albanese and L. Zanghirati [J. Differ. Equations 199, No. 2, 256–268 (2004; Zbl 1063.35059)]. A. A. Albanese and P. Popivanov [J. Math. Anal. Appl. 297, No. 2, 659–672 (2004; Zbl 1058.35055)], D. Dickinson, T. Gramchev and M. Yoshino [Proc. Edinb. Math. Soc., II. Ser. 45, No.3, 731–759 (2002; Zbl 1032.37010)].

MSC:
35H10 Hypoelliptic equations
35S05 Pseudodifferential operators as generalizations of partial differential operators
Keywords:
Gevrey classes
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