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.
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. 745–801DOI 10.2977/PRIMS/1195166574