Singular Integrals, Analytic Capacity and Rectifiability.
Slowly growing subharmonic function II.
Small values of polynomials: Cartan, Pólya and others.
Some applications of generalized condensers in the theory of analytic functions.
Some homeomorphisms of the sphere conformal off a curve.
Some properties of weighted hyperbolic polynomials.
Sous-ensembles de courbes Ahlfors-régulières et nombres de Jones.
We prove that an Ahlfors-regular set (with dimension one) E ⊂ Rn which verifies a βq-version of P. W. Jones' geometric lemma is included in an Ahlfors-regular curve Γ.This theorem is due to G. David and S. Semmes, we give a more direct proof.
Symmetric and Zygmund measures in several variables
Let be a gauge function satisfying certain mid regularity conditions. A (signed) finite Borel measure is called -Zygmund if there exists a positive constant such that for any pair of adjacent cubes of the same size. Similarly, is called an - symmetric measure if there exists a positive constant such that for any pair of adjacent cubes of the same size, . We characterize Zygmund and symmetric measures in terms of their harmonic extensions. Also, we show that the quadratic condition...
The analytic -capacity and the Cauchy integral
The Dirichlet space: a survey.
The fall of the doubling condition in Calderón-Zygmund theory.
The most important results of standard Calderón-Zygmund theory have recently been extended to very general non-homogeneous contexts. In this survey paper we describe these extensions and their striking applications to removability problems for bounded analytic functions. We also discuss some of the techniques that allow us to dispense with the doubling condition in dealing with singular integrals. Special attention is paid to the Cauchy Integral.[Proceedings of the 6th International Conference on...
The growth of entire and harmonic functions along asymptotic paths.
The homogeneous transfinite diameter of a compact subset of
Let K be a compact subset of . A sequence of nonnegative numbers defined by means of extremal points of K with respect to homogeneous polynomials is proved to be convergent. Its limit is called the homogeneous transfinite diameter of K. A few properties of this diameter are given and its value for some compact subsets of is computed.
The Riesz kernels do not give rise to higher dimensional analogues of the Menger-Melnikov curvature.
Ever since the discovery of the connection between the Menger-Melnikov curvature and the Cauchy kernel in the L2 norm, and its impressive utility in the analytic capacity problem, higher dimensional analogues have been coveted. The lesson from 1-sets was that any such (nontrivial, nonnegative) expression, using the Riesz kernels for m-sets in Rn, even in any Lk norm (k ∈ N), would probably carry nontrivial information on whether the boundedness of these kernels in the appropriate norm implies rectifiability...
The traveling salesman problem and harmonic analysis.
In this paper we propose to discuss some relationships between the classical Traveling Salesman Problem (TSP), Litlewood-Paley theory, and harmonic measure. This circle of ideas is also closely related to the theory of Cauchy integrals on Lipschitz graphs, and this aspect is discussed more fully in the paper of David and Semmes [2] in this proceedings. The main differences between the subjects in [2] and this paper are that the results here are valid for one dimensional sets, whereas [2] treats...
Totally disconnected set non-removable for Lipschitz continous bounded analytic functions.
Two applications of one Robertson's results
Über die Verteilung von gewissen Punktsystemen mit Extremaleigenschaften.
Uniformly perfect sets, Green's function, and fundamental domains.
Univalent functions of given transfinite diameter: A maximum modulus problem.