Boundary Behavior of Harmonic Forms on a Rank One Symmetric Space
Transactions of the American Mathematical Society
We study the boundary behavior of 1-forms on a rank-one symmetric space M satisfying the equations du = 0 = Su; the role of boundary is played by a nilpotent (Iwasawa) group Not isometries of M. For forms satisfying certain Hp integrability conditions, we obtain the existence of boundary values in an appropriate sense, characterize these boundary values by means of fractional and singular integral operators on the group N~, and exhibit explicit isomorphisms between Hp spaces of forms on M and
... e ordinary IP spaces of functions on the group N. Let M be a Riemannian manifold and let 5 be the adjoint of the exterior differential d on M. The equation du> = 0 = 5w can be considered as a generalization of the classical Cauchy-Riemann equations. Their solutions were studied by Stein and Weiss in the case M -R" X R+ with the euclidean metric (conjugate systems of harmonic functions) , by Korányi and Vági in the case when M is a euclidean ball in R" , and by Coifman and Weiss in the case M = G X R+, G being a compact Lie group with the bi-invariant metric  . In this paper we consider the case when M is a noncompact symmetric space of rank one, define H^ spaces of 1-forms satisfying the above equations and study their boundary behavior. The role of boundary is played by a nilpotent group N of isometries of M, which is the Cartan-conjugate of A" in a fixed Iwasawa decomposition G = KAN of the connected group of isometries of M . A right action of the solvable group S = A W induces a decomposition of the tangent bundle of M along the A and 77-directions ("vertical" and "horizontal" directions, by analogy with the upper half plane). It is shown that if a form w is in certain H^ classes, its components along these directions have boundary values and that, moreover, the boundary values of the horizontal components can be obtained from the boundary values of the vertical Received by the editors February 2, 1976.