$L^p-L^q$ estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. III

Cowling, Michael, Giulini, Saverio, and Meda, Stefano. "$L^p-L^q$ estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. III." Annales de l’institut Fourier 51.4 (2001): 1047-1069. <http://eudml.org/doc/115933>.

@article{Cowling2001,
abstract = {Let $X$ be a symmetric space of the noncompact type, with Laplace–Beltrami operator $- \{\mathcal \{L\}\}$, and let $[b,\infty )$ be the $L^2(X)$-spectrum of $\{\mathcal \{L\}\}$. For $\tau $ in $\{\mathbb \{C\}\}$ such that $\{\rm Re\}\,\tau \ge 0$, let $\{\mathcal \{P\}\}_\tau $ be the operator on $L^2(X)$ defined formally as $\{\rm exp\}\,(-\tau (\{\mathcal \{L\}\} - b)^\{1/2\} )$. In this paper, we obtain $L^p-L^q$ operator norm estimates for $\{\mathcal \{P\}\}_\tau $ for all $\tau $, and show that these are optimal when $\tau $ is small and when $\vert \{\rm arg\}\,\tau \vert $ is bounded below $\pi /2$.},
affiliation = {University of New South Wales, School of Mathematics, Sydney NSW 2052 (Australie); Università di Genova, Dipartimento di Matematica, Via Dodescano 35, 16146 Genova (Italie); Università di Milano-Bicocca, Dipartimento di Statistica, Via Bicocca degli Arcimboldi 8, 20126 Milano (Italie)},
author = {Cowling, Michael, Giulini, Saverio, Meda, Stefano},
journal = {Annales de l’institut Fourier},
keywords = {symmetric space; wave equation; $L^p-L^q$ estimates; estimates; Poisson semigroup; semisimple Lie group; Laplace-Beltrami operator},
language = {eng},
number = {4},
pages = {1047-1069},
publisher = {Association des Annales de l'Institut Fourier},
title = {$L^p-L^q$ estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. III},
url = {http://eudml.org/doc/115933},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Cowling, Michael
AU - Giulini, Saverio
AU - Meda, Stefano
TI - $L^p-L^q$ estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. III
JO - Annales de l’institut Fourier
PY - 2001
PB - Association des Annales de l'Institut Fourier
VL - 51
IS - 4
SP - 1047
EP - 1069
AB - Let $X$ be a symmetric space of the noncompact type, with Laplace–Beltrami operator $- {\mathcal {L}}$, and let $[b,\infty )$ be the $L^2(X)$-spectrum of ${\mathcal {L}}$. For $\tau $ in ${\mathbb {C}}$ such that ${\rm Re}\,\tau \ge 0$, let ${\mathcal {P}}_\tau $ be the operator on $L^2(X)$ defined formally as ${\rm exp}\,(-\tau ({\mathcal {L}} - b)^{1/2} )$. In this paper, we obtain $L^p-L^q$ operator norm estimates for ${\mathcal {P}}_\tau $ for all $\tau $, and show that these are optimal when $\tau $ is small and when $\vert {\rm arg}\,\tau \vert $ is bounded below $\pi /2$.
LA - eng
KW - symmetric space; wave equation; $L^p-L^q$ estimates; estimates; Poisson semigroup; semisimple Lie group; Laplace-Beltrami operator
UR - http://eudml.org/doc/115933
ER -