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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Second-order elliptic operators and heat kernels on Lie groups


Authors: Ola Bratteli and Derek W. Robinson
Journal: Trans. Amer. Math. Soc. 325 (1991), 683-713
MSC: Primary 22D10; Secondary 35J99, 46L99, 47F05, 58G11
MathSciNet review: 1041043
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Abstract: Arendt, Batty, and Robinson proved that each second-order strongly elliptic operator $ C$ associated with left translations on the $ {L_p}$-spaces of a Lie group $ G$ generates an interpolating family of semigroups $ T$, whenever the coefficients of $ C$ are sufficiently smooth. We establish that $ T$ has an integral kernel $ K$ satisfying the bounds

$\displaystyle a\prime{t^{ - d/2}}{e^{ - b\prime\vert g{h^{ - 1}}{\vert^2}/t}}{e... ...t}(g;h) \leq a{t^{ - d/2}}{e^{ - b\vert g{h^{ - 1}}{\vert^2}/t}}{e^{\omega t}},$

where $ d$ is the dimension of $ G$, $ \vert g{h^{ - 1}}\vert$ is the right invariant distance from $ h$ to $ g$, and $ a\prime$, $ b\prime$, $ \omega\prime$, etc. are positive constants. Both bounds are derived by generalization of Nash's arguments for pure second-order operators on $ {{\mathbf{R}}^d}$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1041043-6
PII: S 0002-9947(1991)1041043-6
Article copyright: © Copyright 1991 American Mathematical Society