Weighted strongly elliptic operators on Lie groups

AbstractLet (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

Elliptic operators on Lie groups

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High order divergence-form elliptic operators on Lie groups

We give a straightforward proof that divergence-form elliptic operators of order m on a d-dimensional Lie group with m ≥ d have Hölder continuous kernels satisfying Gaussian bounds

Riesz transforms and Lie groups of polynomial growth

Let G be a Lie group of polynomial growth. We prove that the second-order Riesz transforms onL2(G; dg) are bounded if, and only if, the group is a direct product of a compact group and a nilpotent group, in which case the transforms of all orders are bounded