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Journal of Lie Theory 23 (2013), No. 3, 655--668
Copyright Heldermann Verlag 2013

Upper Bound for the Heat Kernel on Higher-Rank NA Groups

Richard Penney
Dept. of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907, U.S.A.

Roman Urban
Institute of Mathematics, Wroclaw University, Plac Grunwaldzki 2/4, 50-384 Wroclaw, Poland


\def\R{{\Bbb R}} Let $S$ be a semi-direct product $S=N\rtimes A$ where $N$ is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and $A$ is isomorphic with $\R^k,$ $k>1$. We consider a class of second order left-invariant differential operators ${\cal L}_\alpha$, $\alpha\in\R^k$, on $S$. We obtain an upper bound for the heat kernel for ${\cal L}_\alpha$.

Keywords: Heat kernel, left invariant differential operators, meta-abelian nilpotent Lie groups, solvable Lie groups, homogeneous groups, higher rank $NA$ groups, Brownian motion, exponential functionals of Brownian motion.

MSC: 43A85, 31B05, 22E25, 22E30, 60J25, 60J60

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