# Subcoercive and Subelliptic Operators on Lie Groups: Variable Coefficients

### A. F.M. ter Elst

Australian National University, Canberra, Australia### Derek W. Robinson

Australian National University, Canberra, Australia

## Abstract

Let $a_{1},…,a_{d_{′}}$ be an algebraic basis of rank $r$ in a Lie algebra $g$ of a connected Lie group $G$ and let $A_{τ}$ be the left differential operator in the direction $a_{i}$ on the $L_{p}$-spaces with respect to the left, or right, Haar measure, where $p∈[1,∞]$. We consider $m$-th order operators

with complex variable bounded coefficients $c_{α}$ which are subcoercive of step $r$, i.e., for all $g∈G$ the form obtained by fixing the $c_{α}$ at $g$ is subcoercive of step $r$ and the ellipticity constant is bounded from below uniformly by a positive constant. If the principal coefficients are $m$-times differentiate in $L_{∞}$ in the directions of $a_{1},…,a_{d_{′}}$we prove that the closure of $H$ generates a consistent interpolation semigroup $S$ which has a kernel. We show that $S$ is holomorphic on a non-empty $p$-independent sector and if $H$ is formally self-adjoint then the holomorphy angle is $π/2$. We also derive ‘Gaussian’ type bounds for the kernel and its derivatives up to order $m−1$.

## Cite this article

A. F.M. ter Elst, Derek W. Robinson, Subcoercive and Subelliptic Operators on Lie Groups: Variable Coefficients. Publ. Res. Inst. Math. Sci. 29 (1993), no. 5, pp. 745–801

DOI 10.2977/PRIMS/1195166574