Abstract
In this paper we develop a theory of Besov and Triebel–Lizorkin spaces on general noncompact connected Lie groups endowed with a sub-Riemannian structure. Such spaces are defined by means of hypoelliptic sub-Laplacians with drift, and endowed with a measure whose density with respect to a right Haar measure is a continuous positive character of the group. We prove several equivalent characterizations of their norms, we establish comparison results also involving Sobolev spaces of recent introduction, and investigate their complex interpolation and algebra properties.
Similar content being viewed by others
References
Agrachev, A., Boscain, U., Gauthier, J.-P., Rossi, F.: The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups. J. Funct. Anal. 256(8), 2621–2655 (2009)
Badr, N., Bernicot, F., Russ, E.: Algebra properties for Sobolev spaces - applications to semilinear PDEs on manifolds. J. Anal. Math. 118(2), 509–544 (2012)
Bergh, J., Löfström, J.: Interpolation Spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No 223. Springer, Berlin (1976)
Bruno, T., Peloso, M.M., Tabacco, A., Vallarino, M.: Sobolev spaces on Lie groups: embedding theorems and algebra properties. J. Funct. Anal. 276(10), 3014–3050 (2019)
Bruno, T., Peloso, M.M., Vallarino, M.: Potential spaces on Lie groups, arXiv:1903.06415
Coulhon, T., Russ, E., Tardivel-Nachef, V.: Sobolev algebras on Lie groups and Riemannian manifolds. Am. J. Math. 123(2), 283–342 (2001)
Feneuil, J.: Algebra properties for Besov spaces on unimodular Lie groups. Colloq. Math. 154(2), 205–240 (2018)
Folland, G.B.: Real Analysis. Modern Techniques and Their Applications. Pure and Applied Mathematics (New York), 2nd edn. Wiley, New York (1999)
Guivarc’h, Y.: Croissance polynomiale et périodes des fonctions harmoniques. Bull. Soc. Math. France 101, 333–379 (1973)
Furioli, G., Melzi, C., Veneruso, A.: Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth. Math. Nachr. 279(9–10), 1028–1040 (2006)
Gallagher, I., Sire, Y.: Besov algebras on Lie groups of polynomial growth. Studia Math. 212(2), 119–139 (2012)
Garcia-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland, Amsterdam (1985)
Gogatishvili, A., Koskela, P., Shanmugalingam, N.: Interpolation properties of Besov spaces defined on metric spaces. Math. Nachr. 283(2), 215–231 (2010)
Gogatishvili, A., Koskela, P., Zhou, Y.: Characterizations of Besov and Triebel-Lizorkin spaces on metric measure spaces. Forum Math. 25(4), 787–819 (2013)
Han, Y., Müller, D., Yang, D.: “A theory of Besov and Triebel–Lizorkin spaces on metric measure spaces modeled on Carnot–Carathéodory spaces”, Abstr. Appl. Anal. 2008, Art. ID 893409, 250 pp
Hebisch, W., Mauceri, G., Meda, S.: Spectral multipliers for Sub-Laplacians with drift on Lie groups. Math. Z. 251(4), 899–927 (2005)
Hewitt Ross, E., Kenneth, A.: Abstract Harmonic Analysis. Vol. I. Structure of Topological Groups, Integration Theory, Group Representations. Grundlehren der Mathematischen Wissenschaften, 2nd edn. Springer, Berlin (1979)
Kerkyacharian, G., Petrushev, P.: Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces. Trans. Am. Math. Soc. 367(1), 121–189 (2015)
Koskela, P., Yang, D., Zhou, Y.: A characterization of Hajlasz-Sobolev and Triebel-Lizorkin spaces via grand Littlewood-Paley functions. J. Funct. Anal. 258(8), 2637–2661 (2010)
Meda, S.: On the Littlewood-Paley-Stein \(g\)-function. Trans. Am. Math. Soc. 347(6), 2201–2212 (1995)
Müller, D., Yang, D.: A difference characterization of Besov and Triebel-Lizorkin spaces on RD-space. Forum Math. 21(2), 259–298 (2009)
Peloso, M.M., Vallarino, M.: Sobolev algebras on nonunimodular Lie groups. Calc. Var. Partial Differ. Equ. 57(6), 57:150 (2018)
Hong, Q., Hu, G.: Continuous characterizations of inhomogeneous Besov and Triebel-Lizorkin spaces associated to non-negative self-adjoint operators. arXiv:1902.05686
Ruzhansky, M., Yessirkegenov, N.: Hardy, Hardy–Sobolev, Hardy–Littlewood–Sobolev and Caffarelli–Kohn–Nirenberg inequalities on general Lie groups. arXiv:1810.08845
Stein, E.M.: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Princeton University Press, Princeton (1970)
Triebel, H.: Characterizations of Besov-Hardy-Sobolev spaces via harmonic functions, temperatures, and related means. J. Approx. Theory 35(3), 275–297 (1982)
Triebel, H.: Characterizations of Besov-Hardy-Sobolev spaces: a unified approach. J. Approx. Theory 52(2), 162–203 (1988)
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam (1978)
Triebel, H.: Spaces of Besov-Hardy-Sobolev type on complete Riemannian manifolds. Ark. Mat. 24(2), 299–337 (1986)
Triebel, H.: “Function Spaces on Lie Groups and on Analytic Manifolds. Function Spaces and Applications” (Lund, 1986), 384–396, Lecture Notes in Math., 1302. Springer, Berlin (1988)
Triebel, H.: Characterizations of function spaces on a complete Riemannian manifold with bounded geometry. Math. Nachr. 130, 321–346 (1987)
Triebel, H.: Function spaces on Lie groups, the Riemannian approach. J. Lond. Math. Soc. (2) 35(2), 327–338 (1987)
Triebel, H.: How to measure smoothness of distributions on Riemannian symmetric manifolds and Lie groups? Z. Anal. Anwendungen 7(5), 471–480 (1988)
Triebel, H.: Theory of Function Spaces Reprint of 1983 Edition. Modern Birkhäuser Classics. Birkhäuser, Springer Basel AG, Basel (2010)
Varopoulos, NTh: Analysis on Lie groups. J. Funct. Anal. 76(2), 346–410 (1988)
Yang, D., Zhou, Y.: New properties of Besov and Triebel-Lizorkin spaces on RD-spaces. Manus. Math. 134(1–2), 59–90 (2011)
Acknowledgements
The authors wish to thank Andrea Carbonaro for useful conversations.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Loukas Grafakos.
In memory of Elias M. Stein.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
All authors are partially supported by the grant PRIN 2015 Real and Complex Manifolds: Geometry, Topology and Harmonic Analysis, and are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
Rights and permissions
About this article
Cite this article
Bruno, T., Peloso, M.M. & Vallarino, M. Besov and Triebel–Lizorkin spaces on Lie groups. Math. Ann. 377, 335–377 (2020). https://doi.org/10.1007/s00208-019-01927-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-019-01927-z