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.