Measure-preserving quality within mappings (Q1840666)

In a by now classical paper devoted to singular integral operators on hypersurfaces, G. David proved that a Lipschitz map has a bilipschitz behaviour on sufficiently large subsets of its domain, with uniform bounds depending on the Lipschitz constant and on the size of the image. In this paper this result is revisited and in some sense reconciled with the more local approach used by P. Jones, where the first goal is to find an approximate bilipschitz behaviour at many (in terms of Carleson measure) places and scales. The basic result of the paper, containing both David's and Jones's results, is given in Section 10 of the paper.