Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the Littlewood-Paley-Stein $g$-function

Author: Stefano Meda
Journal: Trans. Amer. Math. Soc. 347 (1995), 2201-2212
MSC: Primary 47D06; Secondary 42B25, 43A85
MathSciNet review: 1264824
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider semigroups $({T_t})$, which are contractive on ${L^p}(M)$ for all $p \in [q,q’]$ and $q \in [1,2)$. We give an example (on symmetric spaces of the noncompact type) which shows that the Littlewood-Paley-Stein $g$-function associated to the infinitesimal generator of $({T_t})$ may be unbounded on ${L^q}(M)$ and on ${L^{q’}}(M)$. We prove that variants of the $g$-function are bounded on these Lebesgue spaces.

References [Enhancements On Off] (What's this?)

  • Jean-Philippe Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J. 65 (1992), no. 2, 257–297. MR 1150587, DOI
  • R. R. Coifman, R. Rochberg, and Guido Weiss, Applications of transference: the $L^{p}$ version of von Neumann’s inequality and the Littlewood-Paley-Stein theory, Linear spaces and approximation (Proc. Conf., Math. Res. Inst., Oberwolfach, 1977) Birkhäuser, Basel, 1978, pp. 53–67. Internat. Ser. Numer. Math., Vol. 40. MR 0500219
  • Michael G. Cowling, Harmonic analysis on semigroups, Ann. of Math. (2) 117 (1983), no. 2, 267–283. MR 690846, DOI
  • Alan G. R. McIntosh and Alan J. Pryde (eds.), Miniconference on linear analysis and function spaces, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 9, Australian National University, Centre for Mathematical Analysis, Canberra, 1985. MR 825512
  • M. G. Cowling, I. Doust, A. McIntosh, and A. Yagi, Banach space operators with a bounded ${H_\infty }$ functional calculus, J. Austral. Math. Soc. (to appear). M. G. Cowling, S. Giulini, and S. Meda, ${L^p} - {L^q}$ estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. I, Duke Math. J. (to appear). ---, ${L^p} - {L^q}$ estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. II, preprint.
  • Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
  • Noël Lohoué, Estimation des fonctions de Littlewood-Paley-Stein sur les variétés riemanniennes à courbure non positive, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 4, 505–544 (French). MR 932796
  • 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
  • E. M. Stein, The development of square functions in the work of A. Zygmund, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 359–376. MR 663787, DOI

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47D06, 42B25, 43A85

Retrieve articles in all journals with MSC: 47D06, 42B25, 43A85

Additional Information

Keywords: <IMG WIDTH="16" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$g$">-function, functional calculus, symmetric spaces
Article copyright: © Copyright 1995 American Mathematical Society