A general multiplier theorem
HTML articles powered by AMS MathViewer
- by Stefano Meda PDF
- Proc. Amer. Math. Soc. 110 (1990), 639-647 Request permission
Abstract:
We prove a "multiplier" result for functions of the infinitesimal generator $\mathcal {L}$ of a symmetric semigroup, which generalizes some previous results by E. M. Stein and M. G. Cowling. As an application, we develop a functional calculus for $\mathcal {L}$ in the case when the ${L^p}$-operator norm of ${\mathcal {L}^{iu}}$ has polynomial growth at infinity. In particular, we prove a "multiplier" result of Marcinkiewicz type for functions of $\mathcal {L}$.References
- 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) Internat. Ser. Numer. Math., Vol. 40, Birkhäuser, Basel, 1978, pp. 53–67. MR 0500219
- Michael G. Cowling, Harmonic analysis on semigroups, Ann. of Math. (2) 117 (1983), no. 2, 267–283. MR 690846, DOI 10.2307/2007077
- Leonede De Michele and Giancarlo Mauceri, $L^{p}$ multipliers on the Heisenberg group, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 355–359. MR 545325
- 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
- Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
- 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
- Lars Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93–140. MR 121655, DOI 10.1007/BF02547187
- 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
- Giancarlo Mauceri, Maximal operators and Riesz means on stratified groups, Symposia Mathematica, Vol. XXIX (Cortona, 1984) Sympos. Math., XXIX, Academic Press, New York, 1987, pp. 47–62. MR 951178
- Jaak Peetre, New thoughts on Besov spaces, Duke University Mathematics Series, No. 1, Duke University, Mathematics Department, Durham, N.C., 1976. MR 0461123
- E. M. Stein, On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430–466. MR 112932, DOI 10.1090/S0002-9947-1958-0112932-2
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 —, Topics in harmonic analysis related to the Littlewood-Paley theory, Annals of Math. Studies, no. 63, Princeton, NJ, 1970.
- Nicolas Th. Varopoulos, Aspects of probabilistic Littlewood-Paley theory, J. Functional Analysis 38 (1980), no. 1, 25–60. MR 583240, DOI 10.1016/0022-1236(80)90055-5
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 110 (1990), 639-647
- MSC: Primary 42A45; Secondary 47D03
- DOI: https://doi.org/10.1090/S0002-9939-1990-1028046-7
- MathSciNet review: 1028046