May the Cauchy transform of a non-trivial finite measure vanish on the support of the measure?
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...
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 lim ∫ (ζ - z)dμ(ζ) for finite Borel measures μ in the plane.
Extensiones finitogeneradas y proyectivas de un álgebra m-convexa.
Convergence of singular integrals with general measures
We show that -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.
BMO harmonic approximation in the plane and spectral synthesis for Hardy-Sobolev spaces.
The spectral synthesis theorem for Sobolev spaces of Hedberg and Wolff [7] has been applied in combination with duality, to problems of L approximation by analytic and harmonic functions. In fact, such applications were one of the main motivations to consider spectral synthesis problems in the Sobolev space setting. In this paper we go the opposite way in the context of the BMO-H duality: we prove a BMO approximation theorem by harmonic functions and then we apply the ideas in its proof to produce...
BMO and Lipschitz approximation by solutions of elliptic equations
We consider the problem of qualitative approximation by solutions of a constant coefficients homogeneous elliptic equation in the Lipschitz and BMO norms. Our method of proof is well-known: we find a sufficient condition for the approximation reducing matters to a weak spectral synthesis problem in an appropriate Lizorkin-Triebel space. A couple of examples, evolving from one due to Hedberg, show that our conditions are sharp.
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