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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Second-order elliptic operators and heat kernels on Lie groups
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by Ola Bratteli and Derek W. Robinson PDF
Trans. Amer. Math. Soc. 325 (1991), 683-713 Request permission

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 \[ a\prime {t^{ - d/2}}{e^{ - b\prime |g{h^{ - 1}}{|^2}/t}}{e^{ - \omega ’t}} \leq {K_t}(g;h) \leq a{t^{ - d/2}}{e^{ - b|g{h^{ - 1}}{|^2}/t}}{e^{\omega t}},\] where $d$ is the dimension of $G$, $|g{h^{ - 1}}|$ 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
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 325 (1991), 683-713
  • MSC: Primary 22D10; Secondary 35J99, 46L99, 47F05, 58G11
  • DOI: https://doi.org/10.1090/S0002-9947-1991-1041043-6
  • MathSciNet review: 1041043