Estimates for functions of the Laplace operator on homogeneous trees
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- by Michael Cowling, Stefano Meda and Alberto G. Setti PDF
- Trans. Amer. Math. Soc. 352 (2000), 4271-4293 Request permission
Abstract:
In this paper, we study the heat equation on a homogeneous graph, relative to the natural (nearest–neighbour) Laplacian. We find pointwise estimates for the heat and resolvent kernels, and the $L^{p}-L^{q}$ mapping properties of the corresponding operators.References
- Isaac Chavel and Edgar A. Feldman, Modified isoperimetric constants, and large time heat diffusion in Riemannian manifolds, Duke Math. J. 64 (1991), no. 3, 473–499. MR 1141283, DOI 10.1215/S0012-7094-91-06425-2
- 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
- Thierry Coulhon, Noyau de la chaleur et discrétisation d’une variété riemannienne, Israel J. Math. 80 (1992), no. 3, 289–300 (French, with English summary). MR 1202573, DOI 10.1007/BF02808072
- Michael G. Cowling, Harmonic analysis on semigroups, Ann. of Math. (2) 117 (1983), no. 2, 267–283. MR 690846, DOI 10.2307/2007077
- Michael Cowling, Square functions in Banach spaces, Miniconference on linear analysis and function spaces (Canberra, 1984) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 9, Austral. Nat. Univ., Canberra, 1985, pp. 177–184. MR 825523
- Michael Cowling, Ian Doust, Alan McIntosh, and Atsushi Yagi, Banach space operators with a bounded $H^\infty$ functional calculus, J. Austral. Math. Soc. Ser. A 60 (1996), no. 1, 51–89. MR 1364554
- Michael Cowling, Garth Gaudry, Saverio Giulini, and Giancarlo Mauceri, Weak type $(1,1)$ estimates for heat kernel maximal functions on Lie groups, Trans. Amer. Math. Soc. 323 (1991), no. 2, 637–649. MR 967310, DOI 10.1090/S0002-9947-1991-0967310-X
- 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
- M. G. Cowling, S. Meda and A. G. Setti, On spherical analysis on groups of isometries of homogeneous trees, preprint.
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- Alessandro Figà-Talamanca and Claudio Nebbia, Harmonic analysis and representation theory for groups acting on homogeneous trees, London Mathematical Society Lecture Note Series, vol. 162, Cambridge University Press, Cambridge, 1991. MR 1152801, DOI 10.1017/CBO9780511662324
- Alessandro Figà-Talamanca and Massimo A. Picardello, Harmonic analysis on free groups, Lecture Notes in Pure and Applied Mathematics, vol. 87, Marcel Dekker, Inc., New York, 1983. MR 710827
- Carl Herz, Sur le phénomène de Kunze-Stein, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A491–A493 (French). MR 281022
- Lars Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93–140. MR 121655, DOI 10.1007/BF02547187
- Masahiko Kanai, Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds, J. Math. Soc. Japan 37 (1985), no. 3, 391–413. MR 792983, DOI 10.2969/jmsj/03730391
- Masahiko Kanai, Rough isometries and the parabolicity of Riemannian manifolds, J. Math. Soc. Japan 38 (1986), no. 2, 227–238. MR 833199, DOI 10.2969/jmsj/03820227
- N. N. Lebedev, Special functions and their applications, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR 0350075
- Stefano Meda, On the Littlewood-Paley-Stein $g$-function, Trans. Amer. Math. Soc. 347 (1995), no. 6, 2201–2212. MR 1264824, DOI 10.1090/S0002-9947-1995-1264824-6
- Claudio Nebbia, Groups of isometries of a tree and the Kunze-Stein phenomenon, Pacific J. Math. 133 (1988), no. 1, 141–149. MR 936361
- F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0435697
- M. M. H. Pang, Heat kernels of graphs, J. London Math. Soc. (2) 47 (1993), no. 1, 50–64. MR 1200977, DOI 10.1112/jlms/s2-47.1.50
- Tadeusz Pytlik, Radial convolutors on free groups, Studia Math. 78 (1984), no. 2, 179–183. MR 766714, DOI 10.4064/sm-78-2-179-183
- 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
- Nicolas Th. Varopoulos, Brownian motion and random walks on manifolds, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 2, 243–269. MR 746500
- Nicholas Th. Varopoulos, Long range estimates for Markov chains, Bull. Sci. Math. (2) 109 (1985), no. 3, 225–252 (English, with French summary). MR 822826
Additional Information
- Michael Cowling
- Affiliation: School of Mathematics, University of New South Wales, Sydney NSW 2052, Australia
- MR Author ID: 52360
- ORCID: 0000-0003-0995-3054
- Email: m.cowling@unsw.edu.au
- Stefano Meda
- Affiliation: Dipartimento di Matematica, Politecnico di Milano, via Bonardi 9, 20133 Milano, Italy
- Address at time of publication: Department of Statistics, University of Milan-Bicocca, Edificio U7 II piano, v. Le Sarca 202, I-20100 Milan, Italy
- Email: stemed@ipmma1.mate.polimi.it
- Alberto G. Setti
- Affiliation: Dipartimento di Matematica, Università di Milano, via Saldini 50, 20133 Milano, Italy
- Address at time of publication: Faculty of Science, Universitá dell’Insubria-1 Como via Lucini 3, I-22100 Como, Italy
- MR Author ID: 289546
- Email: setti@fis.unico.it
- Received by editor(s): October 4, 1996
- Published electronically: April 14, 2000
- Additional Notes: Work partially supported by the Australian Research Council and the Italian M.U.R.S.T
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 4271-4293
- MSC (1991): Primary 43A85; Secondary 20E08, 43A90, 22E35
- DOI: https://doi.org/10.1090/S0002-9947-00-02460-0
- MathSciNet review: 1653343