Analysis of joint spectral multipliers on Lie groups of polynomial growth

Martini, Alessio

doi:10.5802/aif.2721

Analysis of joint spectral multipliers on Lie groups of polynomial growth
[Analyse de multiplicateurs spectraux conjoints sur des groupes de Lie à croissance polynomiale]

Martini, Alessio ¹

¹ Scuola Normale Superiore Piazza dei Cavalieri, 7 56126 Pisa (Italy)

Annales de l'Institut Fourier, Tome 62 (2012) no. 4, pp. 1215-1263.

Résumé
Abstract

On étudie la bornitude $L^{p}$ ( $1 < p < \infty$ ) des opérateurs de la forme $m (L_{1}, \dots, L_{n})$ pour un système commutatif $L_{1}, \dots, L_{n}$ d’opérateurs différentiels autoadjoints invariants à gauche sur un groupe de Lie $G$ à croissance polynomiale, qui engendrent une algèbre contenant un opérateur sous-coercif pondéré. En particulier, quand $G$ est un groupe homogène et $L_{1}, \dots, L_{n}$ sont homogènes, on prouve des analogues des theorèmes de multiplicateurs de Mihlin-Hörmander et Marcinkiewicz.

We study the problem of $L^{p}$ -boundedness ( $1 < p < \infty$ ) of operators of the form $m (L_{1}, \dots, L_{n})$ for a commuting system of self-adjoint left-invariant differential operators $L_{1}, \dots, L_{n}$ on a Lie group $G$ of polynomial growth, which generate an algebra containing a weighted subcoercive operator. In particular, when $G$ is a homogeneous group and $L_{1}, \dots, L_{n}$ are homogeneous, we prove analogues of the Mihlin-Hörmander and Marcinkiewicz multiplier theorems.

MR Zbl

DOI : 10.5802/aif.2721

Classification : 43A22, 22E30, 42B15
Keywords: spectral multipliers, joint functional calculus, differential operators, Lie groups, polynomial growth, singular integral operators
Mots-clés : multiplicateurs spectraux, calcul fonctionnel conjoint, opérateurs différentiels, groupes de Lie, croissance polynomiale, opérateurs intégraux singuliers

Affiliations des auteurs :

Martini, Alessio ¹

¹ Scuola Normale Superiore Piazza dei Cavalieri, 7 56126 Pisa (Italy)

@article{AIF_2012__62_4_1215_0,
     author = {Martini, Alessio},
     title = {Analysis of joint spectral multipliers on {Lie} groups of polynomial growth},
     journal = {Annales de l'Institut Fourier},
     pages = {1215--1263},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {4},
     year = {2012},
     doi = {10.5802/aif.2721},
     mrnumber = {3025742},
     zbl = {1255.43003},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2721/}
}

TY  - JOUR
AU  - Martini, Alessio
TI  - Analysis of joint spectral multipliers on Lie groups of polynomial growth
JO  - Annales de l'Institut Fourier
PY  - 2012
SP  - 1215
EP  - 1263
VL  - 62
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2721/
DO  - 10.5802/aif.2721
LA  - en
ID  - AIF_2012__62_4_1215_0
ER  -

%0 Journal Article
%A Martini, Alessio
%T Analysis of joint spectral multipliers on Lie groups of polynomial growth
%J Annales de l'Institut Fourier
%D 2012
%P 1215-1263
%V 62
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2721/
%R 10.5802/aif.2721
%G en
%F AIF_2012__62_4_1215_0

Martini, Alessio. Analysis of joint spectral multipliers on Lie groups of polynomial growth. Annales de l'Institut Fourier, Tome 62 (2012) no. 4, pp. 1215-1263. doi : 10.5802/aif.2721. https://aif.centre-mersenne.org/articles/10.5802/aif.2721/

Bibliographie
Cité par

[1] Alexopoulos, G. Spectral multipliers on Lie groups of polynomial growth, Proc. Amer. Math. Soc., Volume 120 (1994) no. 3, pp. 973-979 | DOI | MR | Zbl

[2] Bergh, Jöran; Löfström, Jörgen Interpolation spaces. An introduction, Springer-Verlag, Berlin, 1976 (Grundlehren der Mathematischen Wissenschaften, No. 223) | MR | Zbl

[3] Berkson, Earl; Paluszyński, Maciej; Weiss, Guido Transference couples and their applications to convolution operators and maximal operators, Interaction between functional analysis, harmonic analysis, and probability (Columbia, MO, 1994) (Lecture Notes in Pure and Appl. Math.), Volume 175, Dekker, New York, 1996, pp. 69-84 | MR | Zbl

[4] Christ, Michael $L^{p}$ bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc., Volume 328 (1991) no. 1, pp. 73-81 | DOI | MR | Zbl

[5] Christ, Michael The strong maximal function on a nilpotent group, Trans. Amer. Math. Soc., Volume 331 (1992) no. 1, pp. 1-13 | DOI | MR | Zbl

[6] Coifman, Ronald R.; Weiss, Guido Transference methods in analysis, American Mathematical Society, Providence, R.I., 1976 (Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 31) | MR | Zbl

[7] Cowling, Michael Herz’s “principe de majoration” and the Kunze-Stein phenomenon, Harmonic analysis and number theory (Montreal, PQ, 1996) (CMS Conf. Proc.), Volume 21, Amer. Math. Soc., Providence, RI, 1997, pp. 73-88 | MR | Zbl

[8] Cowling, Michael; Sikora, Adam A spectral multiplier theorem for a sublaplacian on $SU (2)$ , Math. Z., Volume 238 (2001) no. 1, pp. 1-36 | DOI | MR | Zbl

[9] Duong, Xuan Thinh; Ouhabaz, El Maati; Sikora, Adam Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal., Volume 196 (2002) no. 2, pp. 443-485 | DOI | MR | Zbl

[10] ter Elst, A. F. M.; Robinson, Derek W. Weighted subcoercive operators on Lie groups, J. Funct. Anal., Volume 157 (1998) no. 1, pp. 88-163 | DOI | MR | Zbl

[11] Fischer, Véronique; Ricci, Fulvio Gelfand transforms of $SO (3)$ -invariant Schwartz functions on the free group $N_{3, 2}$ , Ann. Inst. Fourier (Grenoble), Volume 59 (2009) no. 6, pp. 2143-2168 | DOI | Numdam | MR | Zbl

[12] Folland, G. B.; Stein, Elias M. Hardy spaces on homogeneous groups, Mathematical Notes, 28, Princeton University Press, Princeton, N.J., 1982 | MR | Zbl

[13] Fraser, A. J. Marcinkiewicz multipliers on the Heisenberg group, Princeton University (1997) (Ph. D. Thesis) | MR

[14] Fraser, A. J. Convolution kernels of $(n + 1)$ -fold Marcinkiewicz multipliers on the Heisenberg group, Bull. Austral. Math. Soc., Volume 64 (2001) no. 3, pp. 353-376 | DOI | MR | Zbl

[15] Fraser, A. J. An $(n + 1)$ -fold Marcinkiewicz multiplier theorem on the Heisenberg group, Bull. Austral. Math. Soc., Volume 63 (2001) no. 1, pp. 35-58 | DOI | MR | Zbl

[16] Grafakos, Loukas Classical Fourier analysis, Graduate Texts in Mathematics, 249, Springer, New York, 2008 | MR | Zbl

[17] Hebisch, Waldemar Multiplier theorem on generalized Heisenberg groups, Colloq. Math., Volume 65 (1993) no. 2, pp. 231-239 | MR | Zbl

[18] Hebisch, Waldemar Functional calculus for slowly decaying kernels, 1995 (Preprint. Available on the web http://www.math.uni.wroc.pl/~hebisch/)

[19] Hebisch, Waldemar; Sikora, Adam A smooth subadditive homogeneous norm on a homogeneous group, Studia Math., Volume 96 (1990) no. 3, pp. 231-236 | MR | Zbl

[20] Hebisch, Waldemar; Zienkiewicz, Jacek Multiplier theorem on generalized Heisenberg groups. II, Colloq. Math., Volume 69 (1995) no. 1, pp. 29-36 | MR | Zbl

[21] Helffer, B. Conditions nécessaires d’hypoanalyticité pour des opérateurs invariants à gauche homogènes sur un groupe nilpotent gradué, J. Differential Equations, Volume 44 (1982) no. 3, pp. 460-481 | DOI | MR | Zbl

[22] Helffer, B.; Nourrigat, J. Caracterisation des opérateurs hypoelliptiques homogènes invariants à gauche sur un groupe de Lie nilpotent gradué, Comm. Partial Differential Equations, Volume 4 (1979) no. 8, pp. 899-958 | DOI | MR | Zbl

[23] Hulanicki, Andrzej A functional calculus for Rockland operators on nilpotent Lie groups, Studia Math., Volume 78 (1984) no. 3, pp. 253-266 | MR | Zbl

[24] Ludwig, Jean Dual topology of diamond groups, J. Reine Angew. Math., Volume 467 (1995), pp. 67-87 | DOI | MR | Zbl

[25] Martini, Alessio Algebras of differential operators on Lie groups and spectral multipliers, Scuola Normale Superiore (2010) (Ph. D. Thesis)

[26] Martini, Alessio Spectral theory for commutative algebras of differential operators on Lie groups, J. Funct. Anal., Volume 260 (2011) no. 9, pp. 2767-2814 | DOI | MR | Zbl

[27] Mauceri, Giancarlo Zonal multipliers on the Heisenberg group, Pacific J. Math., Volume 95 (1981) no. 1, pp. 143-159 http://projecteuclid.org/getRecord?id=euclid.pjm/1102735536 | DOI | MR | Zbl

[28] Mauceri, Giancarlo; Meda, Stefano Vector-valued multipliers on stratified groups, Rev. Mat. Iberoamericana, Volume 6 (1990) no. 3-4, pp. 141-154 | DOI | MR | Zbl

[29] Métivier, Guy Hypoellipticité analytique sur des groupes nilpotents de rang $2$ , Duke Math. J., Volume 47 (1980) no. 1, pp. 195-221 http://projecteuclid.org/getRecord?id=euclid.dmj/1077313871 | DOI | MR | Zbl

[30] Müller, D.; Stein, Elias M. On spectral multipliers for Heisenberg and related groups, J. Math. Pures Appl. (9), Volume 73 (1994) no. 4, pp. 413-440 | MR | Zbl

[31] Müller, Detlef; Ricci, Fulvio; Stein, Elias M. Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups. I, Invent. Math., Volume 119 (1995) no. 2, pp. 199-233 | DOI | MR | Zbl

[32] Müller, Detlef; Ricci, Fulvio; Stein, Elias M. Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups. II, Math. Z., Volume 221 (1996) no. 2, pp. 267-291 | DOI | MR | Zbl

[33] Müller, Detlef; Seeger, Andreas Singular spherical maximal operators on a class of two step nilpotent Lie groups, Israel J. Math., Volume 141 (2004), pp. 315-340 | DOI | MR | Zbl

[34] Nielsen, Ole A. Unitary representations and coadjoint orbits of low-dimensional nilpotent Lie groups, Queen’s Papers in Pure and Applied Mathematics, 63, Queen’s University, Kingston, ON, 1983 | MR | Zbl

[35] Rockland, Charles Hypoellipticity on the Heisenberg group-representation-theoretic criteria, Trans. Amer. Math. Soc., Volume 240 (1978), pp. 1-52 | DOI | MR | Zbl

[36] Rudin, Walter Real and complex analysis, McGraw-Hill Book Co., New York, 1974 (McGraw-Hill Series in Higher Mathematics) | MR | Zbl

[37] Runst, Thomas; Sickel, Winfried Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, de Gruyter Series in Nonlinear Analysis and Applications, 3, Walter de Gruyter & Co., Berlin, 1996 | MR | Zbl

[38] Schmeisser, Hans-Jürgen Recent developments in the theory of function spaces with dominating mixed smoothness, Nonlinear Analysis, Function Spaces and Applications. Proceedings of the Spring School held in Prague, May 30-June 6, 2006, Volume 8, Czech Academy of Sciences, Mathematical Institute, Praha, 2007, pp. 145-204 | MR

[39] Schmeisser, Hans-Jürgen; Sickel, Winfried Spaces of functions of mixed smoothness and approximation from hyperbolic crosses, J. Approx. Theory, Volume 128 (2004) no. 2, pp. 115-150 | DOI | MR | Zbl

[40] Schmeisser, Hans-Jürgen; Triebel, Hans Topics in Fourier analysis and function spaces, Mathematik und ihre Anwendungen in Physik und Technik [Mathematics and its Applications in Physics and Technology], 42, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987 | MR | Zbl

[41] Sickel, Winfried; Ullrich, Tino Tensor products of Sobolev-Besov spaces and applications to approximation from the hyperbolic cross, J. Approx. Theory, Volume 161 (2009) no. 2, pp. 748-786 | DOI | MR | Zbl

[42] Sikora, Adam On the $L^{2} \to L^{\infty}$ norms of spectral multipliers of “quasi-homogeneous” operators on homogeneous groups, Trans. Amer. Math. Soc., Volume 351 (1999) no. 9, pp. 3743-3755 | DOI | MR | Zbl

[43] Sikora, Adam Multivariable spectral multipliers and analysis of quasielliptic operators on fractals, Indiana Univ. Math. J., Volume 58 (2009) no. 1, pp. 317-334 | DOI | MR | Zbl

[44] Stein, Elias M. Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970 | MR | Zbl

[45] Stein, Elias M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993 (With the assistance of Timothy S. Murphy. Monographs in Harmonic Analysis, III) | MR | Zbl

[46] Stempak, Krzysztof A weighted multiplier theorem for Rockland operators, Colloq. Math., Volume 51 (1987), pp. 335-344 | MR | Zbl

[47] Triebel, Hans Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam, 1978 | MR | Zbl

[48] Triebel, Hans Spaces of Besov-Hardy-Sobolev type, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1978 (Teubner-Texte zur Mathematik, With German, French and Russian summaries) | MR | Zbl

[49] Triebel, Hans Theory of function spaces, Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983 | DOI | MR | Zbl

[50] Triebel, Hans The structure of functions, Monographs in Mathematics, 97, Birkhäuser Verlag, Basel, 2001 | MR | Zbl

[51] Varopoulos, N. Th.; Saloff-Coste, L.; Coulhon, T. Analysis and geometry on groups, Cambridge Tracts in Mathematics, 100, Cambridge University Press, Cambridge, 1992 | MR | Zbl

[52] Veneruso, Alessandro Marcinkiewicz multipliers on the Heisenberg group, Bull. Austral. Math. Soc., Volume 61 (2000) no. 1, pp. 53-68 | DOI | MR | Zbl

Martini, Alessio; Plewa, Paweł A sharp multiplier theorem for solvable extensions of Heisenberg and related groups, Annali di Matematica Pura ed Applicata (1923 -), Volume 203 (2024) no. 3, p. 1361 | DOI:10.1007/s10231-023-01405-z
Niedorf, Lars An $L^{p}$ -Spectral Multiplier Theorem with Sharp p-Specific Regularity Bound on Heisenberg Type Groups, Journal of Fourier Analysis and Applications, Volume 30 (2024) no. 2 | DOI:10.1007/s00041-024-10075-1
Fang, Jingxuan; Li, Hongbo; Zhao, Jiman Multilinear Spectral Multipliers on Besov and Triebel–Lizorkin Spaces on Lie Groups of Polynomial Growth, The Journal of Geometric Analysis, Volume 33 (2023) no. 12 | DOI:10.1007/s12220-023-01442-3
Casarino, Valentina; Ciatti, Paolo; Martini, Alessio Weighted Spectral Cluster Bounds and a Sharp Multiplier Theorem for Ultraspherical Grushin Operators, International Mathematics Research Notices, Volume 2022 (2022) no. 12, p. 9209 | DOI:10.1093/imrn/rnab007
Chatzakou, Marianna A note on spectral multipliers on Engel and Cartan groups, Proceedings of the American Mathematical Society (2022) | DOI:10.1090/proc/15830
Fang, Jingxuan; Li, Hongbo; Zhao, Jiman Multilinear and Multiparameter Spectral Multipliers on Homogeneous Besov and Triebel–Lizorkin Spaces on Lie Groups of Polynomial Growth, The Journal of Geometric Analysis, Volume 32 (2022) no. 3 | DOI:10.1007/s12220-021-00844-5
Calzi, Mattia; Ricci, Fulvio Functional calculus on non-homogeneous operators on nilpotent groups, Annali di Matematica Pura ed Applicata (1923 -), Volume 200 (2021) no. 4, p. 1517 | DOI:10.1007/s10231-020-01047-5
Bruno, Tommaso; Peloso, Marco M.; Vallarino, Maria Potential Spaces on Lie Groups, Geometric Aspects of Harmonic Analysis, Volume 45 (2021), p. 149 | DOI:10.1007/978-3-030-72058-2_4
Ciatti, Paolo; Wright, James Strongly Singular Integrals on Stratified Groups, Geometric Aspects of Harmonic Analysis, Volume 45 (2021), p. 299 | DOI:10.1007/978-3-030-72058-2_8
Horwich, Adam D.; Martini, Alessio Almost everywhere convergence of Bochner–Riesz means on Heisenberg‐type groups, Journal of the London Mathematical Society, Volume 103 (2021) no. 3, p. 1066 | DOI:10.1112/jlms.12401
Cardona, Duván; Delgado, Julio; Ruzhansky, Michael $L^{p}$ -Bounds for Pseudo-differential Operators on Graded Lie Groups, The Journal of Geometric Analysis, Volume 31 (2021) no. 12, p. 11603 | DOI:10.1007/s12220-021-00694-1
Fang, Jingxuan; Li, Hongbo; Zhao, Jiman Multilinear Spectral Multipliers on Lie Groups of Polynomial Growth, The Journal of Geometric Analysis, Volume 31 (2021) no. 7, p. 7386 | DOI:10.1007/s12220-020-00530-y
Calzi, Mattia Spectral Multipliers on 2-Step Stratified Groups, I, Journal of Fourier Analysis and Applications, Volume 26 (2020) no. 2 | DOI:10.1007/s00041-020-09740-y
Ricci, Fulvio; Wróbel, Błażej Spectral multipliers for functions of fixed K‐type on Lp(SL(2,R)), Mathematische Nachrichten, Volume 293 (2020) no. 3, p. 554 | DOI:10.1002/mana.201800412
Fang, Jingxuan; Zhao, Jiman $H^{p}$ Hp Boundedness of Multilinear Spectral Multipliers on Stratified Groups, The Journal of Geometric Analysis, Volume 30 (2020) no. 1, p. 197 | DOI:10.1007/s12220-018-00142-7
Fang, Jingxuan; Zhao, Jiman Multilinear and multiparameter spectral multipliers on stratified groups, Mathematical Methods in the Applied Sciences, Volume 41 (2018) no. 13, p. 5327 | DOI:10.1002/mma.5082
Chen, Peng; Duong, Xuan Thinh; Li, Ji; Ward, Lesley A.; Yan, Lixin Marcinkiewicz-Type Spectral Multipliers on Hardy and Lebesgue Spaces on Product Spaces of Homogeneous Type, Journal of Fourier Analysis and Applications, Volume 23 (2017) no. 1, p. 21 | DOI:10.1007/s00041-016-9460-3
Wróbel, Błażej Joint Spectral Multipliers for Mixed Systems of Operators, Journal of Fourier Analysis and Applications, Volume 23 (2017) no. 2, p. 245 | DOI:10.1007/s00041-016-9469-7
Martini, Alessio Joint functional calculi and a sharp multiplier theorem for the Kohn Laplacian on spheres, Mathematische Zeitschrift, Volume 286 (2017) no. 3-4, p. 1539 | DOI:10.1007/s00209-016-1813-8
Martini, Alessio; Müller, Detlef Spectral multipliers on 2-step groups: topological versus homogeneous dimension, Geometric and Functional Analysis, Volume 26 (2016) no. 2, p. 680 | DOI:10.1007/s00039-016-0365-8
Müller, Detlef; Seeger, Andreas SharpLpbounds for the wave equation on groups of Heisenberg type, Analysis PDE, Volume 8 (2015) no. 5, p. 1051 | DOI:10.2140/apde.2015.8.1051
Martini, Alessio Spectral multipliers on Heisenberg–Reiter and related groups, Annali di Matematica Pura ed Applicata (1923 -), Volume 194 (2015) no. 4, p. 1135 | DOI:10.1007/s10231-014-0414-6
Choi, Woocheol Maximal functions for multipliers on stratified groups, Mathematische Nachrichten, Volume 288 (2015) no. 10, p. 1098 | DOI:10.1002/mana.201300305
Martini, Alessio; Müller, Detlef Spectral multiplier theorems of Euclidean type on new classes of two-step stratified groups, Proceedings of the London Mathematical Society, Volume 109 (2014) no. 5, p. 1229 | DOI:10.1112/plms/pdu033

Cité par 24 documents. Sources : Crossref

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT