QUANTITATIVE RECTIFIABILITY AND LIPSCHITZ MAPPINGS

The classical notion of rectifiability of sets in R(n) is qualitative in nature, and in this paper we are concerned with quantitative versions of it. This issue arises in connection with L(p) estimates for singular integral operators on sets in R(n) . We give a criterion for one reasonably natural quantitative rectifiability condition to hold, and we use it to give a new proof of a theorem in [D3]. We also give some results on the geometric properties of a certain class of sets in R(n) which can be viewed as generalized hypersurfaces. Along the way we shall encounter some questions concerning the behavior of Lipschitz functions, with regard to approximation by affine functions in particular. We shall also discuss an amusing variation of the classical Lipschitz and bilipschitz conditions, which allow some singularities forbidden by the classical conditions while still forcing good behavior on substantial sets.