# Revista Matemática Iberoamericana

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**Volume 16, Issue 2, 2000, pp. 363–458**

**DOI: 10.4171/RMI/279**

Published online: 2000-08-31

Measure-preserving quality within mappings

Stephen Semmes^{[1]}(1) Rice University, Houston, USA

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 Davids 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 occurred. 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. In general this has not occurred Here we shall present a variation on Davids method which breaks down into simpler pieces We shall also take advantage of some com ponents of the work of Peter Jones Jones approach uses some LittlewoodPaley theory and one of the important features of Davids method was to avoid this operating in a more directly geometric way which could be applied more broadly To some extent the present or ganization gives a reconciliation between the two and between Davids stoppingtime argument and techniques related to Carleson measures and Carlesons Corona construction Some special cases of Davids results concerning projections of subsets of Euclidean spaces of codimension or mappings dened on Euclidean spaces rather than sets or met ric spaces of less simple nature have been given alternate and much simpler proofs as in In general this has not occurred Here we shall present a variation on Davids method which breaks down into simpler pieces We shall also take advantage of some com ponents of the work of Peter Jones Jones approach uses some LittlewoodPaley theory and one of the important features of Davids method was to avoid this operating in a more directly geometric way which could be applied more broadly To some extent the present or ganization gives a reconciliation between the two and between Davids stoppingtime argument and techniques related to Carleson measures and Carlesons Corona construction

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Semmes Stephen: Measure-preserving quality within mappings. *Rev. Mat. Iberoam.* 16 (2000), 363-458. doi: 10.4171/RMI/279