Singular integrals on symmetric spaces, II
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- by Alexandru D. Ionescu PDF
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Abstract:
We extend some of our earlier results on boundedness of singular integrals on symmetric spaces of real rank one to arbitrary noncompact symmetric spaces. Our main theorem is a transference principle for operators defined by $\mathbb {K}$-bi-invariant kernels with certain large scale cancellation properties. As an application we prove $L^p$ boundedness of operators defined by Fourier multipliers that satisfy singular differential inequalities of the Hörmander–Michlin type.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
- J.-P. Anker and L. Ji, Heat kernel and Green function estimates on noncompact symmetric spaces, Geom. Funct. Anal. 9 (1999), no. 6, 1035–1091. MR 1736928, DOI 10.1007/s000390050107
- J.-Ph. Anker and L. Ji, Heat kernel and Green function estimates on noncompact symmetric spaces II, Preprint (1999).
- 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
- 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
- Ronald R. Coifman and Guido Weiss, Transference methods in analysis, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 31, American Mathematical Society, Providence, R.I., 1976. MR 0481928
- Michael Cowling, The Kunze-Stein phenomenon, Ann. of Math. (2) 107 (1978), no. 2, 209–234. MR 507240, DOI 10.2307/1971142
- Michael Cowling, Saverio Giulini, and Stefano Meda, $L^p$-$L^q$ estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. I, Duke Math. J. 72 (1993), no. 1, 109–150. MR 1242882, DOI 10.1215/S0012-7094-93-07206-7
- Saverio Giulini, Giancarlo Mauceri, and Stefano Meda, $L^p$ multipliers on noncompact symmetric spaces, J. Reine Angew. Math. 482 (1997), 151–175. MR 1427660, DOI 10.1515/crll.1997.482.151
- Harish-Chandra, Spherical functions on a semisimple Lie group. I, Amer. J. Math. 80 (1958), 241–310. MR 94407, DOI 10.2307/2372786
- 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
- Sigurdur Helgason, Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, vol. 39, American Mathematical Society, Providence, RI, 1994. MR 1280714, DOI 10.1090/surv/039
- Carl Herz, Sur le phénomène de Kunze-Stein, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A491–A493 (French). MR 281022
- Alexandru D. Ionescu, Singular integrals on symmetric spaces of real rank one, Duke Math. J. 114 (2002), no. 1, 101–122. MR 1915037, DOI 10.1215/S0012-7094-02-11415-X
- R. A. Kunze and E. M. Stein, Uniformly bounded representations and harmonic analysis of the $2\times 2$ real unimodular group, Amer. J. Math. 82 (1960), 1–62. MR 163988, DOI 10.2307/2372876
- Lars-Ȧke Lindahl, Fatou’s theorem for symmetric spaces, Ark. Mat. 10 (1972), 33–47. MR 383010, DOI 10.1007/BF02384800
- N. Lohoué and Th. Rychener, Some function spaces on symmetric spaces related to convolution operators, J. Funct. Anal. 55 (1984), no. 2, 200–219. MR 733916, DOI 10.1016/0022-1236(84)90010-7
- 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 and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Jan-Olov Strömberg, Weak type $L^{1}$ estimates for maximal functions on noncompact symmetric spaces, Ann. of Math. (2) 114 (1981), no. 1, 115–126. MR 625348, DOI 10.2307/1971380
- 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
Additional Information
- Alexandru D. Ionescu
- Affiliation: Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- Address at time of publication: University of Wisconsin – Madison, Madison, Wisconsin 53706
- MR Author ID: 660963
- Email: aionescu@math.mit.edu, ionescu@math.wisc.edu
- Received by editor(s): September 12, 2001
- Published electronically: April 25, 2003
- Additional Notes: The author was supported in part by the National Science Foundation under NSF Grant No. 0100021
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 3359-3378
- MSC (2000): Primary 22E46, 43A85
- DOI: https://doi.org/10.1090/S0002-9947-03-03312-9
- MathSciNet review: 1974692