Opérateurs intégraux singuliers sur certaines courbes du plan complexe
Opérateurs d'intégrale singulière sur les surfaces régulières
Morceaux de graphes lipschitziens et integrales singulières sur une surface.
Unrectifiable 1-sets have vanishing analytic capacity.
We complete the proof of a conjecture of Vitushkin that says that if E is a compact set in the complex plane with finite 1-dimensional Hausdorff measure, then E has vanishing analytic capacity (i.e., all bounded anlytic functions on the complement of E are constant) if and only if E is purely unrectifiable (i.e., the intersection of E with any curve of finite length has zero 1-dimensional Hausdorff measure). As in a previous paper with P. Mattila, the proof relies on a rectifiability criterion using...
Analytic capacity, Calderón-Zygmund operators, and rectifiability
For K ⊂ 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 H(K) is finite was recently settled. In this...
Hausdorff dimension of uniformly non flat sets with topology.
The local regularity of soap films after Jean Taylor
The following text is a minor modification of the transparencies that were used in the conference; please excuse the often telegraphic style. The main goal of the series of lectures is a presentation (with some proofs) of Jean Taylor’s celebrated theorem on the regularity of almost minimal sets of dimension in , and a few more recent extensions or perspectives. Some of the results presented below are work of, or with T. De Pauw, V. Feuvrier A. Lemenant, and T. Toro. The...
Hölder regularity of two-dimensional almost-minimal sets in
We give a different and probably more elementary proof of a good part of Jean Taylor’s regularity theorem for Almgren almost-minimal sets of dimension in . We use this opportunity to settle some details about almost-minimal sets, extend a part of Taylor’s result to almost-minimal sets of dimension in , and give the expected characterization of the closed sets of dimension in that are minimal, in the sense that for every closed set such that there is a bounded set so that out...
Courbes corde-arc et espaces de Hardy généralisés
Étant donné une courbe de Jordan rectifiable du plan complexe admettant le paramétrage par la longueur d’arc , on étudie les relations entre la géométrie de et la position dans des deux espaces de Hardy associés à . Plus précisément, on montre que si est la somme presque-orthogonale des espaces de Hardy, la courbe satisfait à une condition de type corde-arc, c’est-à-dire que pour tout et tout de , . Ce résultat est une sorte de réciproque à la généralisation du théorème de Calderón...
A Non-Probabilistic Proof of the Assouad Embedding Theorem with Bounds on the Dimension
We give a non-probabilistic proof of a theorem of Naor and Neiman that asserts that if (E, d) is a doubling metric space, there is an integer N > 0, depending only on the metric doubling constant, such that for each exponent α ∈ (1/2; 1), one can find a bilipschitz mapping F = (E; dα ) ⃗ ℝ RN.
Surfaces quasiminimales de codimension 1 : un morceau de démonstration
Uniform rectifiability and singular sets
Removable sets for Lipschitz harmonic functions in the plane.
The main motivation for this work comes from the century-old Painlevé problem: try to characterize geometrically removable sets for bounded analytic functions in C.
Harmonic analysis and the geometry of subsets of R.
This subject has several natural points of view, but we shall start with the one that corresponds to the following question: Is there something like Littlewood-Paley theory which is useful for analyzing the geometry of subsets of R, in much the same way that traditional Littlewood-Paley theory is good for analyzing functions and operators?
Regular mappings between dimensions
The notions of Lipschitz and bilipschitz mappings provide classes of mappings connected to the geometry of metric spaces in certain ways. A notion between these two is given by regular mappings (reviewed in Section 1), in which some non-bilipschitz behavior is allowed, but with limitations on this, and in a quantitative way. In this paper we look at a class of mappings called (s, t)-. These mappings are the same as ordinary regular mappings when s = t, but otherwise they behave somewhat like projections....
Monotonicity and separation for the Mumford–Shah problem
Lusin-type Theorems for Cheeger Derivatives on Metric Measure Spaces
A theorem of Lusin states that every Borel function onRis equal almost everywhere to the derivative of a continuous function. This result was later generalized to Rn in works of Alberti and Moonens-Pfeffer. In this note, we prove direct analogs of these results on a large class of metric measure spaces, those with doubling measures and Poincaré inequalities, which admit a form of differentiation by a famous theorem of Cheeger.
Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation.
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