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Chord-arc surfaces with small constant. I. (English) Zbl 0733.42015
There is a well-known generalization of complex analysis to higher- dimensional Euclidean spaces called Clifford analysis. In particular there is an analogue of the Cauchy kernel and the usual integral formulas. Given a hypersurface which is reasonably nice, one can define a pair of Hardy spaces of square-integrable functions on the surface which arise as the boundary values of the Clifford holomorphic function on one of the components of the complement of the hypersurface. For hyperplanes these Hardy spaces are orthogonal. This paper provides geometric characterizations of the hypersurfaces for which they are almost orthogonal. [See also the author’s paper in Proc. Am. Math. Soc. 112, No.2, 403-412 (1991; preceding review).] Such results were known already for curves in the plane, but the higher-dimensional situation is quite different, because of the absence of Riemann mappings and arclength parameterizations, as well as other differences in the geometry.
Although it would be natural to try to give characterizations of these surfaces in terms of harmonic measure (in analogy with some known results for conformal welding of curves, for instance), the author doesn’t seem to provide any.
Reviewer: St.Semmes

MSC:
42B99 Harmonic analysis in several variables
53A05 Surfaces in Euclidean and related spaces
30G35 Functions of hypercomplex variables and generalized variables
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