J. Aust. Math. Soc.  78 (2005), 109-147
Heat kernels on homogeneous spaces

C. M. P. A. Smulders
  Department of Mathematics and Comp. Sci.
  Eindhoven University of Technology
  P.O. Box 513
  5600 MB Eindhoven
  The Netherlands
  camsmul@cs.com


Abstract
Let $a_1,\dots,a_d$ be a basis of the Lie algebra $\mathfrak{g}$ of a connected Lie group $G$ and let $M$ be a Lie subgroup of $G$. If $dx$ is a non-zero positive quasi-invariant regular Borel measure on the homogeneous space $X=G/M$ and $S: X\times G\to\mathbb{C}$ is a continuous cocycle, then under a rather weak condition on $dx$ and $S$ there exists in a natural way a (weakly*) continuous representation $U$ of $G$ in $L_p(X;dx)$ for all $p\in [1,\infty]$. Let $A_i$ be the infinitesimal generator with respect to $U$ and the direction $a_i$ for all $i\in\{1,\dots,d\}$ . We consider $n$-th order strongly elliptic operators $H = \sum c_\alpha A^\alpha$ with complex coefficients $c_\alpha$. We show that the semigroup $S$ generated by the closure of $H$ has a reduced heat kernel $\kappa$ and we derive upper bounds for $\kappa$ and all its derivatives.
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