# Measure-preserving quality within mappings.

Revista Matemática Iberoamericana (2000)

- Volume: 16, Issue: 2, page 363-458
- ISSN: 0213-2230

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topSemmes, Stephen. "Measure-preserving quality within mappings.." Revista Matemática Iberoamericana 16.2 (2000): 363-458. <http://eudml.org/doc/39603>.

@article{Semmes2000,

abstract = {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], [19], [10]. In general, this has not occurred.Here we shall present a variation of David's methods 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.},

author = {Semmes, Stephen},

journal = {Revista Matemática Iberoamericana},

keywords = {Aplicación lipschitziana; Integrales singulares; Espacios métricos; Lipschitz maps; bilipschitz maps; Carleson sets; singular integral operators},

language = {eng},

number = {2},

pages = {363-458},

title = {Measure-preserving quality within mappings.},

url = {http://eudml.org/doc/39603},

volume = {16},

year = {2000},

}

TY - JOUR

AU - Semmes, Stephen

TI - Measure-preserving quality within mappings.

JO - Revista Matemática Iberoamericana

PY - 2000

VL - 16

IS - 2

SP - 363

EP - 458

AB - 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], [19], [10]. In general, this has not occurred.Here we shall present a variation of David's methods 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.

LA - eng

KW - Aplicación lipschitziana; Integrales singulares; Espacios métricos; Lipschitz maps; bilipschitz maps; Carleson sets; singular integral operators

UR - http://eudml.org/doc/39603

ER -

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