Measure-preserving quality within mappings

In [6], Guy David introduced some methods for finding controlled behavior in Lipschitz mappings with substantial images (in terms of measure). Under suitable conditions, David produces subsets on which the given mapping is bilipschitz, with uniform bounds for the bilipschitz constant and the size of the subset. This has applications for boundedness of singular integral operators and "uniform rectifiability" of sets, as in [6], [7], [11], [13]. Some special cases of David's results, concerning projections of subsets of Euclidean spaces of codimension 1, or mappings defined on Euclidean spaces (rather than sets or metric spaces of less simple nature), have been given alternate and much simpler proofs, as in [8]. [9], [10]. In general this has not ocurred. Here we shall present a variation on David's method which breaks down into simpler pieces. We shall also take advantage of some components of the work of Peter Jones [19]. Jones' approach uses some Littlewood-Paley theory, and one of the important features of David's method was to avoid this, operating in a more directly geometric way which could be applied more broadly. To some extent, the present organization gives a reconciliation between the two, and between David's stopping-time argument and techniques related to Carleson measures and Carleson's Corona construction