Subcoercive and subelliptic operators on Lie groups : variable coefficients

Let a1, ¿ , ad' be an algebraic basis of rank r in a Lie algebra g of a connected Lie group G and let At 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= S caAa with complex variable bounded coefficients ca which are subcoercive of step r, i.e., for all g ¿ G the form obtained by fixing the ca 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 L8 in the directions of a1, ¿ , 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 p/2. We also derive 'Gaussian' type bounds for the kernel and its derivatives up to order m—l.