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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
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 $\mathfrak{X}$.

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Additional Information

Alberto G. Setti
Affiliation: Dipartimento di Matematica, Università di Milano, via Saldini 50, 20133 Milano, Italia

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