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Transactions of the American Mathematical Society

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$L^{p}$ and operator norm estimates
for the complex time heat operator
on homogeneous trees


Author: Alberto G. Setti
Journal: Trans. Amer. Math. Soc. 350 (1998), 743-768
MSC (1991): Primary 43A85, 35K05; Secondary 39A12
DOI: https://doi.org/10.1090/S0002-9947-98-02042-X
MathSciNet review: 1443889
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Abstract: Let $\mathfrak{X}$ be a homogeneous tree of degree greater than or equal to three. In this paper we study the complex time heat operator ${\mathcal{H}}_{\zeta }$ induced by the natural Laplace operator on $\mathfrak{X}$. We prove comparable upper and lower bounds for the $L^{p}$ norms of its convolution kernel $h_{\zeta }$ and derive precise estimates for the $L^{p}\text{--}L^{r}$ operator norms of ${\mathcal{H}}_{\zeta }$ for $\zeta $ belonging to the half plane $\text{Re}\,\zeta \geq 0.$ In particular, when $\zeta $ is purely imaginary, our results yield a description of the mapping properties of the Schrödinger semigroup on $\mathfrak{X}$.


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

  • [CMS1] M. G. Cowling, S. Meda and A. G. Setti, On spherical analysis on groups of isometries of homogeneous trees, (preprint).
  • [CMS2] M. G. Cowling, S. Meda and A. G. Setti, Estimates for functions of the Laplace operator on homogeneous trees, (preprint).
  • [E] A. Erdélyi, Asymptotic Expansions, Dover, 1956. MR 17:1202c
  • [FTN] A. Figà Talamanca and C. Nebbia, Harmonic Analysis and Representation Theory for Groups Acting on Homogeneous Trees, London Math. Society Lecture Note Series, No. 162, Cambridge University Press, 1991. MR 93f:22004
  • [FTP] A. Figà Talamanca and M. Picardello, Harmonic Analysis on Free Groups, Lecture Notes in Pure and Applied Mathematics, No. 87, Marcel Dekker, 1983. MR 85j:43001
  • [G] S. Giulini, Estimates for the complex time heat operator on real hyperbolic spaces, (preprint).
  • [H] L. Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93-140. MR 22:12389
  • [L] N. N. Lebedev, Special Functions and their Applications, Dover, New York, 1972. MR 50:2568
  • [N] C. Nebbia, Groups of isometries of a tree and the Kunze-Stein phenomenon, Pacific J. Math. 133 (1988), 141-149. MR 89h:43005
  • [O] F. W. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974. MR 55:8655
  • [P] T. Pytlik, Radial convolutors on free groups, Studia Math. 78 (1984), 178-183. MR 86j:43001
  • [W] G. N. Watson, A Treatise on the Theory of Bessel Functions, second edition, Cambridge University Press, Cambridge, 1944; reprints, 1966, 1995. MR 6:64a; MR 96i:33010

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Additional Information

Alberto G. Setti
Affiliation: Dipartimento di Matematica, Università di Milano, via Saldini 50, 20133 Milano, Italia
Email: setti@dsdipa.mat.unimi.it

DOI: https://doi.org/10.1090/S0002-9947-98-02042-X
Keywords: Homogeneous trees, complex time heat operator, spherical Fourier analysis
Received by editor(s): June 10, 1996
Additional Notes: Work partially supported by the Italian M.U.R.S.T
Article copyright: © Copyright 1998 American Mathematical Society

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