Abstract
We study multipliers associated to the Hermite operator \(H=-\varDelta + |x|^2\) on modulation spaces \(M^{p,q}(\mathbb R^d)\). We prove that the operator m(H) is bounded on \(M^{p,q}(\mathbb R^d)\) under standard conditions on m, for suitable choice of p and q. As an application, we point out that the solutions to the free wave and Schrödinger equations associated to H with initial data in a modulation space will remain in the same modulation space for all times. We also point out that Riesz transforms associated to H are bounded on some modulation spaces.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bényi, A., Gröchenig, K., Okoudjou, K.A., Rogers, L.G.: Unimodular fourier multipliers for modulation spaces. J. Funct. Anal. 246(2), 366–384 (2007)
Cordero, E., Nicola, F.: Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation. J. Funct. Anal. 254(2), 506–534 (2008)
Feichtinger, H.G.: Modulation spaces on locally compact abelian groups, Technical Report, University of Vienna (1983)
Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser Boston Inc., Boston, MA (2001)
Jao, C.: The energy-critical quantum harmonic oscillator. Commun. Partial Differ. Equ. 41(1), 79–133 (2016)
Hebisch, W.: Multiplier theorem on generalized Heisenberg groups. Colloq. Math. 65(2), 231–239 (1993)
Hebisch, W., Zienkiewicz, J.: Multiplier theorem on generalized Heisenberg groups. II. Colloq. Math. 69(1), 29–36 (1995)
Kato, K., Kobayashi, M., Ito, S.: Remarks on Wiener Amalgam Space Type Estimates for Schrödinger Equation, Harmonic Analysis and Nonlinear Partial Differential Equations, pp. 41–48. RIMS Kokyuroku Bessatsu, B33, Res. Inst. Math. Sci. (RIMS ), Kyoto (2012)
Herz, C., Rivière, N.: Estimates for translation invariant operators on spaces with mixed norms. Studia Math. 44, 511–515 (1972)
Müller, D., Ricci, F., Stein, E.: Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups. II. Math. Z. 221(2), 267–291 (1996)
Müller, D., Stein, E.M.: On spectral multipliers for Heisenberg and related groups. J. Math. Pures Appl. 73(4), 413–440 (1994)
Ruzhansky, M., Sugimoto, M., Wang, B.: Modulation spaces and nonlinear evolution equations. In: Evolution Equations of Hyperbolic and Schrödinger Type. Progress in Mathematics, vol. 301, pp. 267–283. Birkhäuser/Springer Basel AG, Basel (2012)
Sanjay, P.K., Thangavelu, S.: Revisiting Riesz transforms on Heisenberg groups. Rev. Mat. Iberoam. 28(4), 1091–1108 (2012)
Schonbek, T.P.: \(L^p-\)multipliers: a new proof of an old theorem. Proc. Am. Math. Soc. 102(2), 361–364 (1988)
Thangavelu, S.: Lectures on Hermite and Laguerre Expansions. Mathematical Notes, vol. 42. Princeton University Press, Princeton (1993)
Thangavelu, S.: An Introduction to the Uncertainty Principle. Hardy’s theorem on Lie groups. Progress in Mathematics, vol. 217, Birkhäuser., Boston, MA (2004)
Thangavelu, S.: Harmonic Analysis on the Heisenberg Group. Progress in Mathematics, vol. 159. Birkhäuser, Boston, MA (1998)
Thangavelu, S.: Multipliers for Hermite expansions. Rev. Mat. Iberoam. 3(1), 1–24 (1987)
Wang, B.X., Zhao, L., Guo, B.: Isometric decomposition operators, function spaces \(E_{p, q}^{\lambda }\) and applications to nonlinear evolution equations. J. Funct. Anal. 233(1), 1–39 (2006)
Wang, B.X., Zhaohui, H., Chengchun, H., Zihua, G.: Harmonic Analysis Method for Nonlinear Evolution Equations I. World Scientific Publishing Co., Pte. Lt (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Bhimani, D.G., Balhara, R., Thangavelu, S. (2019). Hermite Multipliers on Modulation Spaces. In: Delgado, J., Ruzhansky, M. (eds) Analysis and Partial Differential Equations: Perspectives from Developing Countries. Springer Proceedings in Mathematics & Statistics, vol 275. Springer, Cham. https://doi.org/10.1007/978-3-030-05657-5_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-05657-5_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-05656-8
Online ISBN: 978-3-030-05657-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)