Second-order elliptic operators and heat kernels on Lie groups

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

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

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 ${{\mathbf{R}}^d}$.