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 \[ a\prime {t^{ - d/2}}{e^{ - b\prime |g{h^{ - 1}}{|^2}/t}}{e^{ - \omega ’t}} \leq {K_t}(g;h) \leq a{t^{ - d/2}}{e^{ - b|g{h^{ - 1}}{|^2}/t}}{e^{\omega t}},\] 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 ${{\mathbf {R}}^d}$.