Abstract
In the first section of the paper we shortly review classes of operators with multiple symplectic characteristics for which hypoellipticity and solvability were studied by Grushin, Parenti, Rodino, Mascarello, and others. In the second section we present a new result concerning the second order model operator of Grushin. As well known, hypoellipticity for it depends on discrete conditions on a parameter; here we give explicit formulas for fundamental solutions using hypergeometric functions.
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References
H. Bateman and A. Erdelyi. Higher Transcendental Functions, Vol. I. McGraw-Hill, New York, 1953.
R. Beals. A Note on Fundamental Solutions. Comm. Part. Diff. Equat.,24:369–376, 1999.
R. Beals, P. Greiner and B. Gaveau. Green’s functions for some highly degenerate elliptic operators. J. Funct. Anal., 165:407–429, 1999.
V. V. Grushin. On a Class of Elliptic Pseudo Differential Operators Degenerate on a Submanifold. Math. USSR Sbornik, 13:155–183, 1971.
M. Mascarello and L. Rodino. A Class of Pseudo Differential Operators with Multiple Non-Involutive Characteristics. Ann. Scuola Norm. Sup. Pisa, Cl. Sc., Ser. IV, 8:575–603, 1981.
M. Mascarello and L. Rodino. Partial Differential Equations with Multiple Characteristics. Akademie Verlag-Wiley,Berlin, 1997.
M. Mughetti. A Problem of Transversal Anisotropic Ellipticity. Preprint Dip.Mat. Univ. Bologna Italy, 1999.
C. Parenti and L. Rodino. Parametrices for a Class of Pseudo Differential Operators I, II. Annali Mat. Pura Appl., 125:221–254, 125:255–278, 1980.
N. M. Tri. Remark on Non-Uniform Fundamental and Non-Smooth Solutions of Some Classes of Differential Operators with Double Characteristics. J. Math. Sci. Univ. Tokyo,6:437–452, 1999.
K. Yagdjian. The Cauchy Problem for Hyperbolic Operators. Multiple Characteristics. Micro-Local Approach. Akademie Verlag-Wiley, Berlin, 1997.
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Rodino, L., Mascarello, M., Tri, N.M. (2001). Partial Differential Operators with Multiple Symplectic Characteristics. In: Demuth, M., Schulze, BW. (eds) Partial Differential Equations and Spectral Theory. Operator Theory: Advances and Applications, vol 126. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8231-6_33
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DOI: https://doi.org/10.1007/978-3-0348-8231-6_33
Publisher Name: Birkhäuser, Basel
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