Comptes Rendus
Analyse harmonique
Integrability properties of quasi-regular representations of NA groups
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1125-1134.

Let G=NA, where N is a graded Lie group and A= + acts on N via homogeneous dilations. The quasi-regular representation π=ind A G (1) of G can be realised to act on L 2 (N). It is shown that for a class of analysing vectors the associated wavelet transform defines an isometry from L 2 (N) into L 2 (G) and that the integral kernel of the corresponding orthogonal projector has polynomial off-diagonal decay. The obtained reproducing formula is instrumental for obtaining decomposition theorems for function spaces on nilpotent groups.

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DOI : 10.5802/crmath.372
Classification : 22E15, 22E27, 43A80, 44A35
Jordy Timo van Velthoven 1

1 Delft University of Technology, Mekelweg 4, Building 36, 2628 CD Delft, The Netherlands
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Jordy Timo van Velthoven. Integrability properties of quasi-regular representations of $NA$ groups. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1125-1134. doi : 10.5802/crmath.372. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.372/

[1] Hajer Bahouri; Clotilde Fermanian-Kammerer; Isabelle Gallagher Refined inequalities on graded Lie groups, C. R. Math. Acad. Sci. Paris, Volume 350 (2012) no. 7-8, pp. 393-397 | DOI | MR | Zbl

[2] Joaquim Bruna; Julia Cufí; Hartmut Führ; M. Lara Miró Characterizing abelian admissible groups, J. Geom. Anal., Volume 25 (2015) no. 2, pp. 1045-1074 | DOI | MR | Zbl

[3] Tommaso Bruno Homogeneous algebras via heat kernel estimates (2021) (https://arxiv.org/abs/2102.11613, to appear in Trans. Am. Math. Soc.)

[4] Mattia Calzi; Fulvio Ricci Functional calculus on non-homogeneous operators on nilpotent groups, Ann. Mat. Pura Appl., Volume 200 (2021) no. 4, pp. 1517-1571 | DOI | MR | Zbl

[5] Duván Cardona; Michael Ruzhansky Multipliers for Besov spaces on graded Lie groups., C. R. Math. Acad. Sci. Paris, Volume 355 (2017) no. 4, pp. 400-405 | DOI | MR | Zbl

[6] Jens G. Christensen; Azita Mayeli; Gestur Ólafsson Coorbit description and atomic decomposition of Besov spaces, Numer. Funct. Anal. Optim., Volume 33 (2012) no. 7-9, pp. 847-871 | DOI | MR | Zbl

[7] Bradley N. Currey Admissibility for a class of quasiregular representations, Can. J. Math., Volume 59 (2007) no. 5, pp. 917-942 | DOI | MR | Zbl

[8] Bradley N. Currey; Hartmut Führ; Keith F. Taylor Integrable wavelet transforms with abelian dilation groups, J. Lie Theory, Volume 26 (2016) no. 2, pp. 567-595 | MR | Zbl

[9] Bradley N. Currey; Vignon Oussa Admissibility for monomial representations of exponential Lie groups, J. Lie Theory, Volume 22 (2012) no. 2, pp. 481-487 | MR | Zbl

[10] Michel Duflo; Calvin C. Moore On the regular representation of a nonunimodular locally compact group, J. Funct. Anal., Volume 21 (1976), pp. 209-243 | DOI | MR | Zbl

[11] Pierre Eymard; Marianne Terp La transformation de Fourier et son inverse sur le groupe des ax+b d’un corps local, Analyse harmonique sur les groupes de Lie II (Lecture Notes in Mathematics), Volume 739, Springer, 1979, pp. 207-248 | DOI | MR | Zbl

[12] Hans G. Feichtinger; Karlheinz H. Gröchenig Banach spaces related to integrable group representations and their atomic decompositions. I, J. Funct. Anal., Volume 86 (1989) no. 2, pp. 307-340 | DOI | MR | Zbl

[13] Veronique Fischer; Michael Ruzhansky Progress in Mathematics, 314, Birkhäuser/Springer, 2016, xiii+557 pages | DOI | Zbl

[14] Veronique Fischer; Michael Ruzhansky Sobolev spaces on graded Lie groups, Ann. Inst. Fourier, Volume 67 (2017) no. 4, pp. 1671-1723 | DOI | Numdam | MR | Zbl

[15] Gerald B. Folland Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., Volume 13 (1975), pp. 161-207 | DOI | MR | Zbl

[16] Gerald B. Folland Lipschitz classes and Poisson integrals on stratified groups, Stud. Math., Volume 66 (1979), pp. 37-55 | DOI | MR | Zbl

[17] Gerald B. Folland; Elias M. Stein Hardy spaces on homogeneous groups, Mathematical Notes, 28, Princeton University Press, 1982 | Zbl

[18] Michael Frazier; Björn Jawerth; Guido L. Weiss Littlewood-Paley theory and the study of function spaces, Regional Conference Series in Mathematics, 79, American Mathematical Society, 1991, vii+132 pages | DOI | Zbl

[19] Hartmut Führ Abstract harmonic analysis of continuous wavelet transforms, Lect. Notes Math., 1863, Springer, 2005, x+193 pages | DOI | Zbl

[20] Hartmut Führ Generalized Calderón conditions and regular orbit spaces, Colloq. Math., Volume 120 (2010) no. 1, pp. 103-126 | DOI | Zbl

[21] Hartmut Führ Coorbit spaces and wavelet coefficient decay over general dilation groups, Trans. Am. Math. Soc., Volume 367 (2015) no. 10, pp. 7373-7401 | DOI | MR | Zbl

[22] Hartmut Führ; Azita Mayeli Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization, J. Funct. Spaces Appl., Volume 2012 (2012), 523586, 41 pages | MR | Zbl

[23] Hartmut Führ; Jordy Timo van Velthoven Coorbit spaces associated to integrably admissible dilation groups, J. Anal. Math., Volume 144 (2021) no. 1, pp. 351-395 | DOI | MR | Zbl

[24] Giulia Furioli; Camillo Melzi; Alessandro Veneruso Littlewood–Paley decompositions and Besov spaces on Lie groups of polynomial growth, Math. Nachr., Volume 279 (2006) no. 9-10, pp. 1028-1040 | DOI | MR | Zbl

[25] Daryl Geller; Azita Mayeli Continuous wavelets and frames on stratified Lie groups. I, J. Fourier Anal. Appl., Volume 12 (2006) no. 5, pp. 543-579 | DOI | MR | Zbl

[26] John E. Gilbert; Yong Sheng Han; Jeffrey A. Hogan; Joseph D. Lakey; David Weiland; Guido L. Weiss Smooth molecular decompositions of functions and singular integral operators, Mem. Am. Math. Soc., 742, American Mathematical Society, 2002, 74 pages | Zbl

[27] Karlheinz H. Gröchenig Describing functions: Atomic decompositions versus frames, Monatsh. Math., Volume 112 (1991) no. 1, pp. 1-42 | DOI | MR | Zbl

[28] Karlheinz H. Gröchenig; Eberhard Kaniuth; Keith F. Taylor Compact open sets in duals and projections in L 1 -algebras of certain semi-direct product groups, Math. Proc. Camb. Philos. Soc., Volume 111 (1992) no. 3, pp. 545-556 | DOI | MR | Zbl

[29] Matthias Holschneider Wavelets. An analysis tool, Oxford Math. Monogr., Clarendon Press, 1995, xiii+423 pages | Zbl

[30] Guorong Hu Homogeneous Triebel-Lizorkin spaces on stratified Lie groups, J. Funct. Spaces Appl., Volume 2013 (2013), 475103, 16 pages | MR | Zbl

[31] Andrzej Hulanicki A functional calculus for Rockland operators on nilpotent Lie groups, Stud. Math., Volume 78 (1984), pp. 253-266 | DOI | MR | Zbl

[32] Eberhard Kaniuth; Keith F. Taylor Minimal projections in L 1 -algebras and open points in the dual spaces of semi-direct product groups, J. Lond. Math. Soc., Volume 53 (1996) no. 1, pp. 141-157 | DOI | MR | Zbl

[33] Steven Krantz Lipschitz spaces on stratified groups, Trans. Am. Math. Soc., Volume 269 (1982), pp. 39-66 | DOI | MR | Zbl

[34] Richard S. Laugesen; Nik Weaver; Guido L. Weiss; Edward N. Wilson A characterization of the higher dimensional groups associated with continuous wavelets, J. Geom. Anal., Volume 12 (2002) no. 1, pp. 89-102 | DOI | MR | Zbl

[35] Ronald L. Lipsman Harmonic analysis on exponential solvable homogeneous spaces: The algebraic or symmetric cases, Pac. J. Math., Volume 140 (1989) no. 1, pp. 117-147 | DOI | MR | Zbl

[36] Alexander Nagel; Fulvio Ricci; Elias M. Stein Harmonic analysis and fundamental solutions on nilpotent Lie groups, Analysis and partial differential equations (Lecture Notes in Pure and Applied Mathematics), Volume 122, Marcel Dekker, 1990, pp. 249-275 | MR | Zbl

[37] Vignon Oussa Admissibility for quasiregular representations of exponential solvable Lie groups, Colloq. Math., Volume 131 (2013) no. 2, pp. 241-264 | DOI | MR | Zbl

[38] José Luis Romero; Jordy Timo van Velthoven; Felix Voigtlaender On dual molecules and convolution-dominated operators, J. Funct. Anal., Volume 280 (2021) no. 10, 108963, 57 pages | MR | Zbl

[39] Koichi Saka Besov spaces and Sobolev spaces on a nilpotent Lie group, Tôhoku Math. J., Volume 31 (1979), pp. 383-437 | MR | Zbl

[40] Eckart Schulz; Keith F. Taylor Extensions of the Heisenberg group and wavelet analysis in the plane, Spline functions and the theory of wavelets, American Mathematical Society, 1999, pp. 217-225 | DOI | Zbl

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