Joan Verdera

A geometric proof of the L2 boundedness of the Cauchy integral on Lipschitz graphs

In this paper we give a new proof of the L2 boundedness of the Cauchy integral on Lipschitz graphs (and chord-arc curves). Our method consists in controlling the Cauchy integral by an appropiate square function measuring the curvature of the graph. The square function is then estimated via a Fourier transform computation. Let Γ = {(x, y) ∈ R2 : y = A(x)} be the graph of a Lipschitz function A defined on the real line. Then A is locally absolutely continuous and A′ is bounded. The Cauchy integral of f ∈ L2(Γ ) is

On theT(1)-theorem for the Cauchy integral

The main goal of this paper is to present an alternative, real variable proof of theT(1)-theorem for the Cauchy integral. We then prove that the estimate from below of analytic capacity in terms of total Menger curvature is a direct consequence of theT(1)-theorem. An example shows that theL∞-BMO estimate for the Cauchy integral does not follow fromL2 boundedness when the underlying measure is not doubling.

The planar Cantor sets of zero analytic capacity and the local ()-Theorem

In this paper we obtain rather precise estimates for the analytic capacity of a big class of planar Cantors sets. In fact, we show that analytic capacity and positive analytic capacity are comparable for these sets. The main tool for the proof is an appropriate version of the T (b)-Theorem.

Removability, capacity and approximation

In this paper we are primarily interested in problems of qualitative approximation by holomorphic functions of one complex variable belonging to some fixed class, that is defined by restricting the growth of the functions (L p , 1 < p ≤ ∞) or by requiring certain smoothness (Lip s or C m ). Part of the approximation problem consists in understanding the removable sets for the class under consideration and its associated capacity.

Boundary Regularity of Rotating Vortex Patches

We show that the boundary of a rotating vortex patch (or V-state, in the terminology of Deem and Zabusky) is C∞, provided the patch is close to the bifurcation circle in the Lipschitz norm. The rotating patch is also convex if it is close to the bifurcation circle in the C2 norm. Our proof is based on Burbea’s approach to V-states.

A weak type inequality for cauchy transforms of finite measures

In this note we present a simple proof of a recent result of Mattila and Melnikov on the existence of lime?0 ?|?-z|>e (? - z)-1dµ(?) for finite Borel measures µ in the plane.

Duality in spaces of finite linear combinations of atoms

In this paper we describe the dual and the completion of the space of finite linear combinations of (p, oo)-atoms, 0 < p ≤ 1. As an application, we show an extension result for operators uniformly bounded on (p, ∞)-atoms, 0 < p < 1, whose analogue for p = 1 is known to be false. Let 0 < p < 1 and let T be a linear operator defined on the space of finite linear combinations of (p, ∞)-atoms, 0 < p < 1, which takes values in a Banach space B. If T is uniformly bounded on (p, ∞)-atoms, then T extends to a bounded operator from H p (ℝ n ) into B.

Convergence of singular integrals with general measures

We show that

The Capacity Associated to Signed Riesz Kernels, and Wolff Potentials

L^2

A new characterization of Sobolev spaces on

-bounded singular integrals in metric spaces with respect to general measures and kernels converge weakly. This implies a kind of average convergence almost everywhere. For measures with zero density we prove the almost everywhere existence of principal values.