Abstract
We study the uniform regularity and the decay at infinity for anisotropic tensor products of Shubin-type differenential operators as well as for degenerate harmonic oscillators. As applications of our general results we obtain new theorems for global hypoellipticity for classes of degenerate operators in inductive and projective Gelfand-Shilov spaces.
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References
A. Avantaggiati, S-spaces by means of the behaviour of Hermite-Fourier coefficients, Boll. Un. Mat. Ital. 6(4-A) (1985), 487–495.
M. Cappiello, T. Gramchev and L. Rodino, Exponential decay and regularity for SG-elliptic operators with polynomial coefficients, J. Funct. Anal. 237 (2006), 634–654.
A. Dasgupta and M. W. Wong, Essential self-adjointness and global hypoellipticity of the twisted Laplacian, Rend. Sem. Mat. Univ. Pol. Torino 66 (2008), 75–85.
I. M. Gelfand and G. E. Shilov, Generalized Functions II, Academic Press, 1968.
T. Gramchev, S. Pilipović and L. Rodino, Classes of degenerate elliptic operators in Gelfand-Shilov spaces, in New Developments in Pseudo-Differential Operators, Operator Theory: Advances and Applications 189 Birkhäuser, 2009, 15–31.
H. Holden, B. Øksendal, J. Ubøe and T. Zhang, Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach, Probability and its Applications, Birkhäuser, 1996.
L. Hörmander, The Analysis of Linear Partial Differential Operators, Springer, Part III, 1985; Part IV, 1985.
M. Langenburch, Hermite functions and weighted spaces of generalized functions, Manuscripta Math. 119 (2006), 269–285.
B. S. Mitjagin, Nuclearity and other properties of spaces of type S, in American Mathematical Society Translations Ser. 2 93, 1970, 45–59.
Z. Lozanov Crvenković, D. Perisić and M. Tasković, Gelfand-Shilov spaces, structural and kernel theorems, preprint.
S. Pilipović, Generalization of Zemanian spaces of generalized functions which have orthonormal series expansions, SIAM J. Math. Anal. 17 (1986), 477–484.
S. Pilipović, Tempered ultradistributions, Boll. Un. Mat. Ital., VII. Ser., B 2(2) (1988), 235–251.
M. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, 1987.
M. W. Wong, Weyl transforms, the heat kernel and Green function of a degenerate elliptic operator, Ann. Global Anal. Geom. 28 (2005), 271–283.
M. W. Wong, Weyltransforms and a degenerate elliptic partial differential equation, Proc. R. Soc. Lond. Ser. A. 461 (2005), 3863–3870.
M. W. Wong, The heat equation for the Hermite operator on the Heisenberg group, Hokkaido Math. J. 34 (2005), 393–404.
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© 2009 Birkhäuser Verlag Basel/Switzerland
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Gramchev, T., Pilipović, S., Rodino, L. (2009). Global Regularity and Stability in S-Spaces for Classes of Degenerate Shubin Operators. In: Schulze, BW., Wong, M.W. (eds) Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations. Operator Theory: Advances and Applications, vol 205. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0198-6_4
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DOI: https://doi.org/10.1007/978-3-0346-0198-6_4
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