Weighted Strongly Elliptic Operators on Lie Groups

Abstract

Let (H, G, U) be a continuous representation of a Lie group G by bounded operators g ↦ U(g) on the Banach space X and let (X, g, dU) denote the representation of the Lie algebra g obtained by differentiation. If a1, …, ad′ is a Lie algebra basis of g, Ai = dU(ai) and Aα = Ai1 …Aik whenever α = (i1, …, ik) we consider the operators [formula] where the cα are complex coefficients satisfying a weighted strongly elliptic condition in which different directions may have different weights. This condition is such that the class of operators considered encompasses all the standard strongly elliptic operators. We prove that the closure H̄ of each such operator H generates a holomorphic semigroup S with holomorphy sector which contains a non-empty subsector determined by the coefficients and independent of the representation. Moreover, the semigroup S has a smooth representation independent kernel and we derive bounds on the kernel and all its derivatives. Finally we establish elliptic regularity properties for the operators and their powers and characterize the analytic and Gevrey vectors. As a corollary we derive optimal growth bounds for the eigenfunctions of the anharmonic oscillators P2m + Q2n.