# zbMATH — the first resource for mathematics

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
Gevrey classes
Full Text:
##### References:
 [1] Albanese, A.A.; Zanghirati, L., Global hypoellipticity and global solvability in Gevrey classes on the n-dimensional torus, J. differential equations, 199, 256-268, (2004) · Zbl 1063.35059 [2] Albanese, A.A.; Popivanov, P., On the global solvability in Gevrey classes on the n-dimensional torus, J. math. anal. appl., 297, 2, 659-672, (2004) · Zbl 1058.35055 [3] Amano, K., The global hypoellipticity of a class of degenerate elliptic – parabolic operators, Proc. Japan acad., 60, 4, 312-314, (1984) · Zbl 0574.35019 [4] Bergamasco, A.P., Perturbations of globally hypoelliptic operators, J. differential equations, 114, 513-526, (1994) · Zbl 0815.35009 [5] Bergamasco, A.P., Remarks about global analytic hypoellipticity, Trans. amer. math. soc., 351, 10, 4113-4126, (1999) · Zbl 0932.35046 [6] Bergamasco, A.P.; Cordaro, P.D.; Petronilho, G., Global solvability for certain classes of underdetermined systems of vector fields, Math. Z., 223, 2, 261-274, (1996) · Zbl 0863.58062 [7] Bergamasco, A.P.; Cordaro, P.D.; Petronilho, G., Global solvability for a class of complex vector fields on the two-torus, Comm. partial differential equations, 29, 5-6, 785-819, (2004) · Zbl 1065.35088 [8] Bergamasco, A.P.; Dattori da Silva, P.L., Solvability in the large for a class of vector fields on the torus, J. math. pures appl., 86, 427-447, (2006) · Zbl 1157.35304 [9] Bergamasco, A.P.; Nunes, W.V.L.; Zani, S.L., Global analytic hypoellipticity and pseudoperiodic functions, VI workshop on partial differential equations, part I, Rio de Janeiro, 1999, Mat. contemp., 18, 43-57, (2000) · Zbl 0979.35036 [10] Bergamasco, A.P.; Nunes, W.V.L.; Zani, S.L., Global properties of a class of overdetermined systems, J. funct. anal., 200, 1, 31-64, (2003) · Zbl 1034.32024 [11] Bergamasco, A.P.; Petronilho, G., Closedness of the range for vector fields on the torus, J. differential equations, 154, 132-139, (1999) · Zbl 0926.35030 [12] Bergamasco, A.P.; Zani, S.L., Prescribing analytic singularities for solutions of a class of vector fields on the torus, Trans. amer. math. soc., 357, 10, 4159-4174, (2005) · Zbl 1077.35004 [13] Bove, A.; Treves, F., On the Gevrey hypo-ellipticity of sums of squares of vector fields, Ann. inst. Fourier (Grenoble), 54, 5, 1443-1475, (2004) · Zbl 1073.35067 [14] Cardoso, F.; Hounie, J., Global solvability of an abstract complex, Proc. amer. math. soc., 65, 1, 117-124, (1977) · Zbl 0335.58015 [15] Chen, W.; Chi, M.Y., Hypoelliptic vector fields and almost periodic motions on the torus $$T^n$$, Comm. partial differential equations, 25, 1-2, 337-354, (2000) · Zbl 0945.35007 [16] Christ, M., Global analytic hypoellipticity in the presence of symmetry, Math. res. lett., 1, 559-563, (1994) · Zbl 0841.35027 [17] Cordaro, P.D.; Himonas, A.A., Global analytic hypoellipticity of a class of degenerate elliptic operators on the torus, Math. res. lett., 1, 501-510, (1994) · Zbl 0836.35036 [18] Cordaro, P.D.; Himonas, A.A., Global analytic regularity for sums of squares of vector fields, Trans. amer. math. soc., 350, 4993-5001, (1998) · Zbl 0914.35087 [19] Cordaro, P.D.; Trépreau, J.-M., On the solvability of linear partial differential equations in spaces of hyperfunctions, Ark. mat., 36, 1, 41-71, (1998) · Zbl 0910.46028 [20] Dickinson, D.; Gramchev, T.; Yoshino, M., Perturbations of vector fields on tori: resonant normal forms and Diophantine phenomena, Proc. edinb. math. soc., 45, 731-759, (2002) · Zbl 1032.37010 [21] Fujiwara, D.; Omori, H., An example of a globally hypoelliptic operator, Hokkaido math. J., 12, 293-297, (1983) · Zbl 0548.35020 [22] Gramchev, T.; Popivanov, P.; Yoshino, M., Some note on Gevrey hypoellipticity and solvability on torus, J. math. soc. Japan, 43, 3, 501-514, (1991) · Zbl 0739.35015 [23] Gramchev, T.; Popivanov, P.; Yoshino, M., Global solvability and hypoellipticity on the torus for a class of differential operators with variable coefficients, Proc. Japan acad., 68, 53-57, (1992) · Zbl 0805.35023 [24] Gramchev, T.; Popivanov, P.; Yoshino, M., Global properties in spaces of generalized functions on the torus for second order differential operators with variable coefficients, Rend. sem. mat. univ. politec. Torino, 51, 2, 145-172, (1993) · Zbl 0824.35027 [25] Gramchev, T.; Yoshino, M., WKB analysis to global solvability and hypoellipticity, Publ. res. inst. math. sci., 31, 3, 443-464, (1995) · Zbl 0842.35021 [26] Greenfield, S.J.; Wallach, N.R., Global hypoellipticity and Liouville numbers, Proc. amer. math. soc., 31, 112-114, (1972) · Zbl 0229.35023 [27] Helffer, B., Conditions nécessaires d’hypoanalyticité pour des opérateurs invariants à gauche homogènes sur un groupe nilpotent gradué, J. differential equations, 44, 460-481, (1982) · Zbl 0458.35019 [28] Himonas, A.A., On degenerate elliptic operators of infinite type, Math. Z., 220, 449-460, (1995) · Zbl 0843.35020 [29] Himonas, A.A., Global analytic and Gevrey hypoellipticity of sublaplacians under Diophantine conditions, Proc. amer. math. soc., 129, 7, 2061-2067, (2000) · Zbl 0984.35050 [30] Himonas, A.A.; Petronilho, G., On global hypoellipticity of degenerate elliptic operators, Math. Z., 230, 241-257, (1999) · Zbl 0931.35034 [31] Himonas, A.A.; Petronilho, G., Global hypoellipticity and simultaneous approximability, J. funct. anal., 170, 356-365, (2000) · Zbl 0943.35013 [32] Himonas, A.A.; Petronilho, G., Propagation of regularity and global hypoellipticity, Michigan math. J., 50, 471-481, (2002) · Zbl 1028.35049 [33] Himonas, A.A.; Petronilho, G., On $$C^\infty$$ and Gevrey regularity of sub-Laplacians, Trans. amer. math. soc., 358, 11, 4809-4820, (2006) · Zbl 1109.35027 [34] Himonas, A.A.; Petronilho, G.; dos Santos, L.A.C., Regularity of a class of sub-Laplacians on the 3-dimensional torus, J. funct. anal., 240, 568-591, (2006) · Zbl 1112.58021 [35] Hounie, J., Globally hypoelliptic and globally solvable first order evolution equation, Trans. amer. math. soc., 252, 233-248, (1979) · Zbl 0424.35030 [36] Petronilho, G., Global solvability on the torus for certain classes of operators in the form of a sum of squares of vector fields, J. differential equations, 145, 1, 101-118, (1998) · Zbl 0931.35035 [37] Petronilho, G., Global solvability on the torus for certain classes of formally self-adjoint operators, Manuscripta math., 103, 9-18, (2000) · Zbl 0994.58012 [38] Petronilho, G., Global solvability and simultaneously approximable vectors, J. differential equations, 184, 48-61, (2002) · Zbl 1157.35336 [39] Petronilho, G., Global s-solvability, global s-hypoellipticity and Diophantine phenomena, Indag. math. (N.S.), 16, 1, 67-90, (2005) · Zbl 1065.35100 [40] Petronilho, G., Simultaneous reduction of a family of commuting real vector fields and global hypoellipticity, Israel J. math., 155, 81-92, (2006) · Zbl 1137.35016 [41] Ruzhansky, M.; Turunen, V., On the Fourier analysis of operators on the torus, (), 87-105 · Zbl 1160.35044 [42] Tartakoff, D.S., Global (and local) analyticity for second order operators constructed from rigid vector fields on products of tori, Trans. amer. math. soc., 348, 7, 2577-2583, (1996) · Zbl 0863.58063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.