$L^p$ 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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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 $\mathfrak {X}$.
- M. G. Cowling, S. Meda and A. G. Setti, On spherical analysis on groups of isometries of homogeneous trees, (preprint).
- M. G. Cowling, S. Meda and A. G. Setti, Estimates for functions of the Laplace operator on homogeneous trees, (preprint).
- A. Erdélyi, Asymptotic Expansions, Dover, 1956.
- 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
- 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
- S. Giulini, Estimates for the complex time heat operator on real hyperbolic spaces, (preprint).
- Lars Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93–140. MR 121655, DOI https://doi.org/10.1007/BF02547187
- 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
- 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, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. MR 0435697
- Tadeusz Pytlik, Radial convolutors on free groups, Studia Math. 78 (1984), no. 2, 179–183. MR 766714, DOI https://doi.org/10.4064/sm-78-2-179-183
- G. N. Watson, A Treatise on the Theory of Bessel Functions, second edition, Cambridge University Press, Cambridge, 1944; reprints, 1966, 1995. ; MR 96i:33010
Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 43A85, 35K05, 39A12
Retrieve articles in all journals with MSC (1991): 43A85, 35K05, 39A12
Additional Information
Alberto G. Setti
Affiliation:
Dipartimento di Matematica, Università di Milano, via Saldini 50, 20133 Milano, Italia
MR Author ID:
289546
Email:
setti@dsdipa.mat.unimi.it
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