# 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, ⋯ , *ad'* be an algebraic basis of rank *r* in a Lie algebra *g* of a connected Lie group *G* and let *Aτ* be the left differential operator in the direction *ai* on the *Lp*-spaces with respect to the left, or right, Haar measure, where *p* ∈ [1, ∞]. We consider *m*-th order operators

H= Σ *cαAα*

with complex variable bounded coefficients *cα* which are subcoercive of step *r*, i.e., for all *g* ∈ *G* the form obtained by fixing the *cα* 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, ⋯ , *ad'* 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*—l.