Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$L^{p}$ 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
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 $\mathfrak{X}$.


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


Similar Articles

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
Email: setti@dsdipa.mat.unimi.it

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02042-X
PII: S 0002-9947(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