Analytic capacity, Calderón-Zygmund operators, and rectifiability

Título inglés Analytic capacity, Calderón-Zygmund operators, and rectifiability
Título español Capacidad analítica, operadores de Calderón-Zygmund y rectificabilidad.
Autor/es David, Guy
Organización Dep. Mat. Univ. Paris-Sud., Orsay, Francia
Revista 0214-1493
Publicación 1999, 43 (1): 3-25, 62 Ref.
Tipo de documento articulo
Idioma Inglés
Resumen inglés For K Ì C compact, we say that K has vanishing analytic capacity (or γ(K) = 0) when all bounded analytic functions on CK are constant. We would like to characterize γ(K) = 0 geometrically. Easily, γ(K) > 0 when K has Hausdorff dimension larger than 1, and γ(K) = 0 when dim(K) < 1. Thus only the case when dim(K) = 1 is interesting. So far there is no characterization of γ(K) = 0 in general, but the special case when the Hausdorff measure H1(K) is finite was recently settled. In this case, γ(K) = 0 if and only if K is unrectifiable (or Besicovitch irregular), i.e., if H1(K ∩ Γ) = 0 for all C1-curves Γ, as was conjectured by Vitushkin. In the present text, we try to explain the structure of the proof of this result, and present the necessary techniques. These include the introduction to Menger curvature in this context (by M. Melnikov and co-authors), and the important use of geometric measure theory (results on quantitative rectifiability), but we insist most on the role of Calderón-Zygmund operators and T(b)-Theorems.
Clasificación UNESCO 120217
Palabras clave español Funciones analíticas ; Dimensión de Hausdorff ; Teoría de la medida ; Curvatura ; Operadores integrales
Código MathReviews MR1697514
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