Given a Radon measure µ on R d , which may be non doubling, we introduce a space of type BMO with respect to this measure. It is shown that many properties which hold for the classical space BMO(µ) when µ is a doubling measure remain valid for the space of type BMO introduced in this paper, without assuming µ doubling. For instance, Calderón-Zygmund operators which are bounded on L 2 (µ) are also bounded from L ∞ (µ) into the new BMO space. Moreover, this space also satisfies a John-Nirenberg inequality, and its predual is an atomic space H 1 . Using a sharp maximal operator it is shown that operators which are bounded from L ∞ (µ) into the new BMO space and from its predual H 1 into L 1 (µ) must be bounded on L p (µ), 1 < p < ∞. From this result one can obtain a new proof of the T (1) theorem for the Cauchy transform for non doubling measures. Finally, it is proved that commutators of Calderón-Zygmund operators bounded on L 2 (µ) with functions of the new BMO are bounded on L p (µ),

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Let γ(E) be the analytic capacity of a compact set E and let γ+(E) be the capacity of E originated by Cauchy transforms of positive measures. In this paper we prove that γ(E) ≈ γ+(E) with estimates independent of E. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that γ is semiadditive.

Let m be a Radon measure on R d which may be non-doubling. The only condition that m must satisfy is m (B(x, r), r > 0, and for some fixedOne of the main difficulties to be solved is the construction of ''reasonable'' approximations of the identity in order to obtain a Calderó n type reproducing formula. Moreover, it is shown that the T(1) theorem for n-dimensional Calderón-Zygmund operators, without doubling assumptions, can be proved using the Littlewood-Paley type decomposition that is obtained for functions in L 2 (m), as in the classical case of homogeneous spaces.}

We prove that if µ is a d-dimensional Ahlfors-David regular measure in R d+1 , then the boundedness of the d-dimensional Riesz transform in L 2 (µ) implies that the non-BAUP David-Semmes cells form a Carleson family. Combined with earlier results of David and Semmes, this yields the uniform rectifiability of µ.

In this paper we study some questions in connection with uniform rectifiability and the L2 boundedness of Calderón–Zygmund operators (CZOs). We show that uniform rectifiability can be characterized in terms of some new adimensional coefficients that are related to the Jones’ β numbers. We also use these new coefficients to prove that n‐dimensional CZOs with odd kernel of type 𝒞2 are bounded in L2(μ), if μ is an n‐dimensional uniformly rectifiable measure.

Abstract. We show that a Radon measure µ in R d which is absolutely continuous with respect to the n-dimensional Hausdorff measure H n is n-rectifiable if the so called Jones' square function is finite µ-almost everywhere. The converse of this result is proven in a companion paper by the second author, and hence these two results give a classification of all n-rectifiable measures which are absolutely continuous with respect to H n . Further, in this paper we also investigate the relationship between the Jones' square function and the so called Menger curvature of a measure with linear growth, and we show an application to the study of analytic capacity.

Let ϕ : C → C be a bilipschitz map. We prove that if E ⊂ C is compact, and γ(E), α(E) stand for its analytic and continuous analytic capacity respectively, thenwhere C depends only on the bilipschitz constant of ϕ. Further, we show that if µ is a Radon measure on C and the Cauchy transform is bounded on L 2 (µ), then the Cauchy transform is also bounded on L 2 (ϕ µ), where ϕ µ is the image measure of µ by ϕ. To obtain these results, we estimate the curvature of ϕ µ by means of a corona type decomposition.