Let m be a Radon measure on C without atoms. In this paper we prove that if the Cauchy transform is bounded in L(m), then all 1-dimensional Calderón-Zygmund operators associated to odd and sufficiently smooth kernels are also bounded in L(m).

In this paper we survey some recent results in connection with the so called Painlevé's problem and the semiadditivity of analytic capacity γ. In particular, we give the detailed proof of the semiadditivity of the capacity γ, and we show almost completely all the arguments for the proof of the comparability between γ and γ.

Given a doubling measure μ on R, it is a classical result of harmonic analysis that Calderón-Zygmund operators which are bounded in L(μ) are also of weak type (1,1). Recently it has been shown that the same result holds if one substitutes the doubling condition on μ by a mild growth condition on μ. In this paper another proof of this result is given. The proof is very close in spirit to the classical argument for doubling measures and it is based on a new Calderón-Zygmund decomposition adapted to...

Let µ be a Borel measure on R which may be non doubling. The only condition that µ must satisfy is µ(B(x, r)) ≤ Cr for all x ∈ R, r > 0 and for some fixed n with 0 < n ≤ d. In this paper we introduce a maximal operator N, which coincides with the maximal Hardy-Littlewood operator if µ(B(x, r)) ≈ r for x ∈ supp(µ), and we show that all n-dimensional Calderón-Zygmund operators are bounded on L(w dµ) if and only if N is bounded on L(w dµ), for a fixed p ∈ (1, ∞). Also, we prove that this...

For $1\le n<d$ integers and $\rho >2$, we prove that an $n$-dimensional Ahlfors-David regular measure $\mu $ in ${\mathbb{R}}^{d}$ is uniformly $n$-rectifiable if and only if the $\rho $-variation for the Riesz transform with respect to $\mu $ is a bounded operator in ${L}^{2}\left(\mu \right)$. This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the ${L}^{2}\left(\mu \right)$ boundedness of the Riesz transform to the uniform rectifiability of $\mu $.

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