Singular Integrals and rectifiable sets in $\mathbb {R}^n$

Singular Integrals and rectifiable sets in $\mathbb {R}^n$

Singular Integrals and rectifiable sets in $\mathbb {R}^n$
  • Année : 1991
  • Tome : 193
  • Format : Électronique, Papier
  • Langue de l'ouvrage :
  • Class. Math. : 49F20, 42B40
  • Nb. de pages : 170
  • ISSN : 0303-1179
  • DOI : 10.24033/ast.68

This monograph is concerned with quantitative versions of the notion of rectifiability. Recall that a $d$-dimensional subset of $\mathbf {R}^n$ is called rectifiable if it is contained in the union of countable family of $d$-dimensional $C^1$ submanifolds, except possibly for a set of Hausdorff measure zero. This is clearly a qualitative condition, i.e., there are no bounds involved. Our main result provides the equivalencebetween several conditions which can be viewed as providing a natural definition for quantitative rectifiability. As amusing feature of our methods is the role played by singular integral operators, which provide a bridge for passing between various geometrical conditions. We also obtain a higher-dimensinal version of Peter Jones' travelling salesman theorem.

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