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Keywords:
stratified Lie group; Hölder-Zygmund space; Littlewood-Paley decomposition
Summary:
We give a characterization of the Hölder-Zygmund spaces $\mathcal {C}^{\sigma }(G)$ ($0< \sigma <\infty $) on a stratified Lie group $G$ in terms of Littlewood-Paley type decompositions, in analogy to the well-known characterization of the Euclidean case. Such decompositions are defined via the spectral measure of a sub-Laplacian on $G$, in place of the Fourier transform in the classical setting. Our approach mainly relies on almost orthogonality estimates and can be used to study other function spaces such as Besov and Triebel-Lizorkin spaces on stratified Lie groups.
References:
[1] Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie Groups and Potential Theory for Their Sub-Laplacians. Springer Monographs in Mathematics, Springer, Berlin (2007). DOI 10.1007/978-3-540-71897-0 | MR 2363343 | Zbl 1128.43001
[2] Christ, M.: $L^{p}$ bounds for spectral multipliers on nilpotent groups. Trans. Am. Math. Soc. 328 (1991), 73-81. DOI 10.2307/2001877 | MR 1104196 | Zbl 0739.42010
[3] Folland, G. B.: Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13 (1975), 161-207. DOI 10.1007/BF02386204 | MR 0494315 | Zbl 0312.35026
[4] Folland, G. B.: Lipschitz classes and Poisson integrals on stratified groups. Stud. Math. 66 (1979), 37-55. DOI 10.4064/sm-66-1-37-55 | MR 0562450 | Zbl 0439.43005
[5] Folland, G. B., Stein, E. M.: Hardy Spaces on Homogeneous Groups. Mathematical Notes 28, Princeton University Press, Princeton (1982). MR 0657581 | Zbl 0508.42025
[6] Führ, H., Mayeli, A.: Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization. J. Funct. Spaces Appl. 2012 (2012), Article ID. 523586, 41 pages. DOI 10.1155/2012/523586 | MR 2923803 | Zbl 1255.46016
[7] Furioli, G., Melzi, C., Veneruso, A.: Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth. Math. Nachr. 279 (2006), 1028-1040. DOI 10.1002/mana.200510409 | MR 2242964 | Zbl 1101.22006
[8] Giulini, S.: Approximation and Besov spaces on stratified groups. Proc. Am. Math. Soc. 96 (1986), 569-578. DOI 10.2307/2046306 | MR 0826483 | Zbl 0605.41013
[9] Grafakos, L.: Modern Fourier Analysis. Graduate Texts in Mathematics 250, Springer, New York (2009). DOI 10.1007/978-0-387-09434-2 | MR 2463316 | Zbl 1158.42001
[10] Hu, G.: Maximal Hardy spaces associated to nonnegative self-adjoint operators. Bull. Aust. Math. Soc. 91 (2015), 286-302. DOI 10.1017/S0004972714001105 | MR 3314148 | Zbl 1316.42024
[11] Hulanicki, A.: A functional calculus for Rockland operators on nilpotent Lie groups. Stud. Math. 78 (1984), 253-266. DOI 10.4064/sm-78-3-253-266 | MR 0782662 | Zbl 0595.43007
[12] Kerkyacharian, G., Petrushev, P.: Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces. Trans. Am. Math. Soc. 367 (2015), 121-189. DOI 10.1090/S0002-9947-2014-05993-X | MR 3271256 | Zbl 1321.58017
[13] Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics, McGraw-Hill, New York (1991). MR 1157815 | Zbl 0867.46001
[14] Saka, K.: Besov spaces and Sobolev spaces on a nilpotent Lie group. Tohoku Math. J., II. Ser. 31 (1979), 383-437. DOI 10.2748/tmj/1178229728 | MR 0558675 | Zbl 0429.43004
[15] Triebel, H.: Theory of Function Spaces. Monographs in Mathematics 78, Birkhäuser, Basel (1983). DOI 10.1007/978-3-0346-0416-1 | MR 0781540 | Zbl 0546.46027
[16] Varopoulos, N., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge Tracts in Mathematics 100, Cambridge University Press, Cambridge (1992). DOI 10.1017/CBO9780511662485 | MR 1218884 | Zbl 1179.22009
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