Abstract
We consider the subgroup G of \(SL(2,\mathbb {R})\) consisting of shearing and dilations, and we study the decay at infinity of the matrix coefficients of the metaplectic representation restricted to G. We prove weak-type estimates for such coefficients, which are uniform for functions in the modulation space \(M^1\). This work represents a continuation of a project aiming at studying weak-type and Strichartz estimates for unitary representations of non-compact Lie groups.
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Change history
06 October 2021
The original version of the book was inadvertently published with incorrect author name in chapter 3. The author’s name “Alessandra Cauli” name has been replaced with a revised name as “Francesca M. Gasparini”. The chapter and book have been updated with the changes.
References
F. M. Gasparini, F. Nicola, A. Tabacco Strichartz estimates for the metaplectic representation, Rev. Mat. Iberoamer., to appear.
E. Cordero, F. De Mari, K. Nowak, A. Tabacco, Analytic features of reproducing groups for the metaplectic representation, J. Fourier Anal. Appl. 12, no. 2, 157–180, (2006).
E. Cordero, F. De Mari, K. Nowak, A. Tabacco, Reproducing groups for the metaplectic representation, Pseudo-differential operators and related topics, 227–244, Oper. Theory Adv. Appl., 164, Birkhäuser, Basel, (2006).
E. Cordero, F. De Mari, K. Nowak, A. Tabacco, Dimensional upper bounds for admissible subgroups for the metaplectic representation, Math. Nachr. 283, no. 7, 982–993 (2010).
E. Cordero, A. Tabacco, Triangular subgroups of Sp(d,R) and reproducing formulae, J. Funct. Anal. 264, no. 9, 2034–2058, (2013).
E. Cordero, K. Gröchenig, F. Nicola, L. Rodino, Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class, J. Math. Phys. 55, no. 8, 081506 (2014).
E. Cordero, F. Nicola, Some new Strichartz estimates for the Schrödinger equation, J. Differential Equations 245, no. 7, 1945-1974, (2008).
E. Cordero, F. Nicola, Strichartz estimates in Wiener amalgam space and applications to the Schrödinger equation, Math. Nachr. 281, no. 1, 25-41, (2008).
E. Cordero, F. Nicola, Metaplectic representation on Wiener amalgam space and applications to the Schrödinger equation, J. Funct. Anal. 254, no.2, 506-534, (2008).
M. Cowling, Sur les coefficients des représentations unitaires des groupes de Lie simples. Analyse harmonique sur les groupes de Lie (Nancy-Strasbourg, France, 1976–78), II, Lecture Notes in Math. 739, Springer, Berlin, 132–178, (1979).
M. Cowling, Herz’s “principe de majoration" and the Kunze-Stein phenomenon. Harmonic analysis and number theory (Montreal, PQ, 1996), 73–88, CMS Conf. Proc., 21, Amer. Math. Soc., (1997).
M. Cowling, U. Haagerup, R. Howe, Almost \(L^2\)matrix coefficient, J. Reine Angew. Math. 387, 97–110 (1988).
F. De Mari, K. Nowak Analysis of the affine transformations of the time-frequency plane, Bull. Austral. Math. Soc., Vol. 63, 195–218, (2001).
M. A. de Gosson, Symplectic Methods in Harmonic Analysis and in Mathematical Physics, Birkhäuser Basel, (2011).
M. A. de Gosson, The Wigner Transform, Advanced Textbooks in Mathematics, World Scientific, (2017).
K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser Basel, (2001).
T. Tao, Nonlinear dispersive equations: local and global analysis, American Mathematical Society, (2006).
Acknowledgements
I am very indebted to Professors Fabio Nicola and Elena Cordero for discussions and remarks which improved this paper in an essential way.
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Gasparini, F.M. (2019). Weak-Type Estimates for the Metaplectic Representation Restricted to the Shearing and Dilation Subgroup of \(SL(2,\mathbb {R})\). In: Boggiatto, P., et al. Landscapes of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-05210-2_3
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DOI: https://doi.org/10.1007/978-3-030-05210-2_3
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