Abstract
We reconsider the quantization of symbols defined on the product between a nilpotent Lie algebra and its dual. To keep track of the non-commutative group background, the Lie algebra is endowed with the Baker-Campbell-Hausdorff product, making it via the exponential diffeomorphism a copy of its unique connected simply connected nilpotent Lie group. Using harmonic analysis tools, we emphasize the role of a Weyl system, of the associated Fourier-Wigner transformation and, at the level of symbols, of an important family of exponential functions. Such notions also serve to introduce a family of phase-space shifts. These are used to define and briefly study a new class of coorbit spaces of symbols and its relationship with coorbit spaces of vectors, defined via the Fourier-Wigner transform.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
L. J. Corwin and F. P. Greenleaf: Representations of Nilpotent Lie Groups and Applications, Cambridge Univ. Press, 1990.
M. Christ, D. Geller, P. Glowacki, D. Polin, Pseudodifferential Operators on Groups with Dilations, Duke Math. J., 68, 31–65, (1992).
E. Cordero, K. Gröchenig, F. Nicola and L. Rodino: Wiener Algebras of Fourier Integral Operators, J. Math. Pures Appl., 99, 219–233, (2013).
E. Cordero, F. Nicola and L. Rodino: Time-Frequency Analysis of Fourier Integral Operators, Comm. Pure Appl. Anal., 9(1), 1–21, (2010).
E. Cordero and L. Rodino: Time-Frequency Analysis: Function Spaces and Applications, Note di Matematica, 31, 173–189, (2011).
E. Cordero, J. Toft and P. Wahlberg: Sharp Results for the Weyl Product on Modulation Spaces, J. Funct. Anal., 267(8), 3016–3057, (2014).
A. S. Dynin: An Algebra of Pseudodifferential Operators on the Heisenberg Group: Symbolic Calculus, Dokl. Akad. Nauk SSSR, 227, 508–512, (1976).
H. G. Feichtinger: On a New Segal Algebra, Monatsh. Mat. 92(4), 269–289, (1981).
H. G. Feichtinger: Modulation Spaces on Locally Compact Abelian Groups, Proc. of “International Conference on Wavelets and Applications”, Chenai, India, 99–140, 1983.
H. G. Feichtinger and K. Gröchenig: Banch Spaces Associated to Integrable Group Representations and Their Atomic Decompositions I, J. Funct. Anal. 86, 307–340, (1989).
V. Fischer and M. Ruzhansky: A Pseudo-differential Calculus on Graded Nilpotent Groups, in Fourier Analysis, pp. 107–132, Trends in Mathematics, Birkhäuser, 2014.
V. Fischer and M. Ruzhansky: Quantization on Nilpotent Lie Groups, Progress in Mathematics, 314, Birkhäuser, 2016.
V. Fischer, D. Rottensteiner and M. Ruzhansky: Heisenberg-Modulation Spaces at the Crossroads of Coorbit Theory and Decomposition SpaceTheory, Preprint ArXiV.
P. Głowacki: A Symbolic Calculus andL 2-Boundedness on Nilpotent Lie Groups, J. Funct. Anal. 206, 233–251, (2004).
P. Głowacki: The Melin Calculus for General Homogeneous Groups, Ark. Mat., 45(1), 31–48, (2007).
P. Głowacki: Invertibility of Convolution Operators on Homogeneous Groups, Rev. Mat. Iberoam. 28(1), 141–156, (2012).
K. Gröchenig: Foundations of Time-Frequency Analysis, Birkhäuser Boston Inc., Boston, MA, 2001.
K. Gröchenig: Time-Frequency Analysis of Sjöstrand Class, Revista Mat. Iberoam. 22(2), 703–724, (2006).
K. Gröchenig: Composition and Spectral Invariance of Pseudodifferential Operators on Modulation Spaces, J. Anal. Math. 98, 65–82, (2006).
K. Gröchenig: A Pedestrian Approach to Pseudodifferential Operators, In: C. Heil editor, Harmonic Analysis and Applications, Birkhäuser, Boston, 2006.
K. Gröchenig and C. Heil: Modulation Spaces and Pseudodifferential Operators, Integral Equations Operator Theory, 34, 439–457, (1999).
K. Gröchenig and Z. Rzeszotnik: Banach Algebras of Pseudodifferential Operators and Their Almost Diagonalization, Ann. Inst. Fourier. 58(6), 2279–2314, (2008).
K. Gröchenig and T. Strohmer: Pseudodifferential Operators on Locally Compact Abelian Groups and Sjöstrand’s Symbol Class, J. Reine Angew. Math. 613, 121–146, (2007).
A. Holst, J. Toft and P. Wahlberg: Weyl Product Algebras and Modulation Spaces, J. Funct. Anal. 251, 463–491, (2007).
R. Howe: Quantum Mechanics and Partial Differential Operators, J. Funct. Anal. 38, 188–254, (1980).
R. Howe: The Role of the Heisenberg Group in Harmonic Analysis, Bull. Amer. Math. Soc. 3(2), 821–843, (1980).
D. Manchon: Formule de Weyl pour les groupes de Lie nilpotente, J. Reine Angew. Mat. 418, 77–129, (1991).
D. Manchon: Calcul symbolyque sur les groupes de Lie nilpotentes et applications, J. Funct. Anal. 102 (2), 206–251, (1991).
M. Măntoiu: Coorbit Spaces of Symbols for Square Integrable Families of Operators, Math. Reports, 18(1), 63–83 (2016).
M. Măntoiu: Berezin-Type Operators on the Cotangent Bundle of a Nilpotent Group, submitted.
M. Măntoiu and R. Purice: On Fréchet-Hilbert Algebras, Archiv der Math. 103(2), 157–166, (2014).
M. Măntoiu and M. Ruzhansky: Pseudo-differential Operators, Wigner Transform and Weyl Systems on Type I Locally Compact Groups, Doc. Math., 22 , 1539–1592, (2017).
M. Măntoiu and M. Ruzhansky: Quantizations on Nilpotent Lie Groups and Algebras Having Flat Coadjoint Orbits, J. Geometric Analysis, (2019).
A. Melin: Parametrix Constructions for Right Invariant Differential Operators on Nilpotent Groups, Ann. Global Anal. Geom. 1(1), 79–130, (1983).
K. Miller: Invariant Pseudodifferential Operators on Two Step Nilpotent Lie Groups, Michigan Math. J. 29, 315–328, (1982).
K. Miller: Inverses and Parametrices for Right-Invariant Pseudodifferential Operators on Two-Step Nilpotent Lie Groups, Trans. of the AMS, 280 (2), 721–736, (1983).
M. Ruzhansky and V. Turunen: Pseudodifferential Operators and Symmetries, Pseudo-Differential Operators: Theory and Applications 2, Birkhäuser Verlag, 2010.
J. Sjöstrand: An Algebra of Pseudodifferential Operators, Math. Res. Lett. 1(2), 185–192, (1994).
J. Toft: Continuity Properties for Modulation Spaces, with Applications to Pseudo-differential Calculus. I, J. Funct. Anal. 207(2), 399–426, (2004).
J. Toft: Continuity Properties for Modulation Spaces, with Applications to Pseudo-differential Calculus. II, Annals of Global Analysis and Geometry, 26, 73–106, (2004).
Acknowledgements
The author has been supported by the Fondecyt Project 1160359.
He is grateful for having the opportunity to participate in the Conference MicroLocal and Time Frequency Analysis 2018 in honor of Luigi Rodino on the occasion of his 70th Birthday.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Măntoiu, M. (2020). Quantization and Coorbit Spaces for Nilpotent Groups. In: Boggiatto, P., et al. Advances in Microlocal and Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-36138-9_20
Download citation
DOI: https://doi.org/10.1007/978-3-030-36138-9_20
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-36137-2
Online ISBN: 978-3-030-36138-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)