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 |
Acceso al artículo completo |