Spectral multipliers on Lie groups of polynomial growth

Alexopoulos, G.

doi:10.1090/S0002-9939-1994-1172944-4

Spectral multipliers on Lie groups of polynomial growth
HTML articles powered by AMS MathViewer

by G. Alexopoulos PDF

Proc. Amer. Math. Soc. 120 (1994), 973-979 Request permission

Abstract:

Let $L$ be a left invariant sub-Laplacian on a connected Lie group $G$ of polynomial volume growth, and let $\{ {E_\lambda },\lambda \geqslant 0\}$ be the spectral resolution of $L$ and $m$ a bounded Borel measurable function on $[0,\infty )$. In this article we give a sufficient condition on $m$ for the operator $m(L) = \smallint _0^\infty m(\lambda )d{E_\lambda }$ to extend to an operator bounded on ${L^p}(G),\;1 < p < \infty$, and also from ${L^1}(G)$ to weak-${L^1}(G)$.

References

Jean-Philippe Anker, $\textbf {L}_p$ Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Ann. of Math. (2) 132 (1990), no. 3, 597–628. MR 1078270, DOI 10.2307/1971430
Jean-Philippe Anker and Noël Lohoué, Multiplicateurs sur certains espaces symétriques, Amer. J. Math. 108 (1986), no. 6, 1303–1353 (French). MR 868894, DOI 10.2307/2374528
Michael Christ, $L^p$ bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc. 328 (1991), no. 1, 73–81. MR 1104196, DOI 10.1090/S0002-9947-1991-1104196-7
J. L. Clerc and E. M. Stein, $L^{p}$-multipliers for noncompact symmetric spaces, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 3911–3912. MR 367561, DOI 10.1073/pnas.71.10.3911
Leonede De Michele and Giancarlo Mauceri, $H^p$ multipliers on stratified groups, Ann. Mat. Pura Appl. (4) 148 (1987), 353–366. MR 932770, DOI 10.1007/BF01774295
G. B. Folland and Elias M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR 657581
Yves Guivarc’h, Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France 101 (1973), 333–379 (French). MR 369608
Lars Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93–140. MR 121655, DOI 10.1007/BF02547187

Hypoelliptic second order differential operators

119

Andrzej Hulanicki and Joe W. Jenkins, Almost everywhere summability on nilmanifolds, Trans. Amer. Math. Soc. 278 (1983), no. 2, 703–715. MR 701519, DOI 10.1090/S0002-9947-1983-0701519-0
G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York-Chicago, Ill.-Toronto, Ont., 1966. MR 0213785
Giancarlo Mauceri and Stefano Meda, Vector-valued multipliers on stratified groups, Rev. Mat. Iberoamericana 6 (1990), no. 3-4, 141–154. MR 1125759, DOI 10.4171/RMI/100
Richard Melrose, Propagation for the wave group of a positive subelliptic second-order differential operator, Hyperbolic equations and related topics (Katata/Kyoto, 1984) Academic Press, Boston, MA, 1986, pp. 181–192. MR 925249
S. G. Mikhlin, Multidimensional singular integrals and integral equations, Pergamon Press, Oxford-New York-Paris, 1965. Translated from the Russian by W. J. A. Whyte; Translation edited by I. N. Sneddon. MR 0185399
Alexander Nagel, Elias M. Stein, and Stephen Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147. MR 793239, DOI 10.1007/BF02392539

Constructive function theory, vol

uniform approximation

Laurent Saloff-Coste, Analyse sur les groupes de Lie à croissance polynômiale, Ark. Mat. 28 (1990), no. 2, 315–331 (French, with English summary). MR 1084020, DOI 10.1007/BF02387385
Robert J. Stanton and Peter A. Tomas, Expansions for spherical functions on noncompact symmetric spaces, Acta Math. 140 (1978), no. 3-4, 251–276. MR 511124, DOI 10.1007/BF02392309
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Elias M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory. , Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. MR 0252961
Michael E. Taylor, $L^p$-estimates on functions of the Laplace operator, Duke Math. J. 58 (1989), no. 3, 773–793. MR 1016445, DOI 10.1215/S0012-7094-89-05836-5
N. Th. Varopoulos, Analysis on Lie groups, J. Funct. Anal. 76 (1988), no. 2, 346–410. MR 924464, DOI 10.1016/0022-1236(88)90041-9
N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and geometry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press, Cambridge, 1992. MR 1218884
Kôsaku Yosida, Functional analysis, 5th ed., Grundlehren der Mathematischen Wissenschaften, Band 123, Springer-Verlag, Berlin-New York, 1978. MR 0500055