J Austral Math Soc Ser A 57 pp179--229, 1994.

Subelliptic Operators on Lie Groups: Regularity

A. F. M. Ter Elst and Derek W. Robinson

Abstract

Let (X, G, U) be a continuous regresentation of the 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, ..., a_d¢ is a Lie algebra basis of g and A_i = dU(a_i) then we examine elliptic regularity properties of the subelliptic operators

H = - d¢ å i,j=1  cijAiAj + d¢ å i=1  ciAi + c0I

where (c_ij) is a real-valued strictly positive-definite matrix and c₀, c₁, ..., c_d¢ Î C. We first introduce a family of Lipschitz subspaces X_g , g > 0, of X, which interpolate between the Cⁿ-subspaces of the representation and for which the parameter g is a continuous measure of differentiability. Secondly, we give a variety of characterization of the spaces in terms of the semigroup generated by the closure ¾H of H and the group representation. Thirdly, for sufficiently large values of Re c₀ the fractional powers of the closure of H are defined, and we prove that D(¾H^g ) Í X_g¢ for g¢ < 2g/r where r is the rank of the basis. Further we establish that 2g/r is the optimal regularity value and it is attained for unitary representations or for the representations obtained by restricting U to X_g. Many other regularity properties are obtained.

1991 AMS Subject Classification: 43A65, 41A05, 22E45

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Authors

A. F. M. Ter Elst

Derek W. Robinson

Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, ACT 0200, Australia.