Abstract
We show global normal forms of first order real principal type C∞ and Gevrey pseudodifferential operators on the torus T2 under generalized small divisor conditions on the principal symbol. We introduce a family of logarithmic Hausdorff dimensions associated to the Gevrey classes in order to examine the corresponding exceptional sets. We study the global hypoellipticity of such operators through inhomogenous Diophantine conditions for the normal forms.
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References
A. Bergamasco,Perturbation of globally hypoelliptic operators, J. Differential Equations,114:2 (1994), pp. 513–526.
A. Bergamasco—G. Zani,Global hypoellipticity of a class of second-order operators, Canad. Math. Bull.,37 (1994), pp. 301–305.
A. D. Brjuno,Local Methods in Nonlinear Differential Equations, Springer, Berlin, Heidelberg (1989).
J. Bovey—M. M. Dodson,The Hausdorff dimension of systems of linear forms, Acta Arith.,XLV (1986), pp. 337–358.
H. Dickinson—S. L. Velani,Hausdorff measure and linear forms, preprint, Methematica Gottingensis Heft,23 (1995).
M. M. Dodson—J. A. G. Vickers,Exceptional sets in Kolmogorov-Arnol’d-Moser theory, J. Phys. A: Math. Gen.,19 (1986), pp. 349–374.
M. M. Dodson—J. Pöschel—J. A. G. Vickers,The Hausdorff diemnsion of small divisors for lower-dimensional KAM-tori, Proc. R. Soc. LondonA,439 (1992), pp. 359–371.
K. Falconer,Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, New York (1990).
T. Gramchev—P. Popivanov—M. Yoshino,Global properties in spaces of generalized functions on the torus for second order differential operators with variable coefficients, Rend. Sem. Mat. Univ. Pol. Torino, 51:2 (1993), pp. 144–174.
T. Gramchev—M. Yoshino,WKB analysis to global solvability and hypoellipticity, Publ. RIMS Kyoto Univ., 31(3) (1995), pp. 443–464.
S. Greenfield—N. Wallach,Global hypoellipticity and Liouville numbers, Proc. Amer. Math. Soc.,31 (1972), pp. 112–114.
L. Hörmander,The Analysis of Linear Partial Differential Operators, vol. I–IV, Springer-Verlag, Berlin- Heidelberg- New York (1985).
J. Hounie,Globally hypoelliptic and globally solvable first order evolution equations, Trans. A. M. S.,252 (1979), pp. 233–248.
J. Hounie,Globally hypoelliptic vector fields on compact surfaces, Comm. Partial Differential Equations,7 (1982), pp. 343–370.
V. Jarnik,Diophantische Approximationen und Hausdorfsches Mass, Mat. Sb.,36 (1929), pp. 371–382.
L. Rodino,Linear partial differential operators in Gevrey spaces, World Scientific, Singapore-New Jersey-London-Hong Kong (1985).
W. Schmidt,Diophantine approximations, Lect. Notese in Mathematics,785, Springer-Verlag, Berlin, Heidelberg, New York (1980).
J.-C. Yoccoz,Recent developments in dynamics, Proc. of the International Congress of Mathematicians, Zürich, Switzerland, August 1994, Birkhäuser (1995), pp. 246–265.
M. Yoshino,A class of globally hypoelliptic operators on the torus, Math. Z.,201 (1989), pp. 1–11.
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Research supported by the EPSRC.
Supported by Volkswagen-Stiftung (RiP-program at Oberwolfach) and partially supported by GNAFA of the CNR, Italy and by research grant MM-410/94 with MES, Bulgaria.
Supported by the Vokswagen-Stiftung (RiP-program at Oberwolfach) and partially supported by Grant-in-Aid for Scientific Research (No. 07640250), Ministry of Education, Science and Culture, Japan and by Chuo University special research fund, Tokyo, Japan.
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Dickinson, D., Gramchev, T. & Yoshino, M. First order pseudodifferential operators on the torus: Normal forms, diophantine phenomena and global hypoellipticity. Ann. Univ. Ferrara 41 (Suppl 1), 51–64 (1996). https://doi.org/10.1007/BF02825255
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DOI: https://doi.org/10.1007/BF02825255