Subelliptic operators on Lie groups: regularity

Abstract

AbstractLet (ℋ,G, U) be a continuous representation of the Lie groupGby bounded operatorsg↦U(g)on the Banach space ℋ and let (ℋ,g, dU) denote the representation of the Lie algebragobtained by differentiation. Ifa1,…, ad′is a Lie algebra basis ofgandAi= dU(ai)then we examine elliptic regularity properties of the subelliptic operatorswhere (cij) is a real-valued strictly positive-definite matrix andc0, c1,…, cd′∈ C. We first introduce a family of Lipschitz subspaces ℋγ, γ > 0, of ℋ which interpolate between theCn-subspaces of the representation and for which the parameter γ is a continuous measure of differentiability. Secondly, we give a variety of characterizations of the spaces in terms of the semigroup generated by the closureofHand the group representation. Thirdly, for sufficiently large values of Rec0the fractional powers of the closure ofHare defined, and we prove that D()γ⊆γ′, for γ′ < 2γ/rwhereris the rank of the basis. Further we establish that 2γ/ris the optimal regularity value and it is attained for unitary representations or for the representations obtained by restrictingUto ℋγ. Many other regularity properties are obtained.