Subcoercive and Subelliptic Operators on Lie Groups: Variable Coefficients

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, &#8734]. 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.