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2) Lp (µ) ≤C M f Lp (µ) (assuming f dµ = 0 if µ < ∞). 2) are satisﬁed too if µ is non doubling (choosing an appropiate grid of cubes). However, the above deﬁnition of the sharp operator is not useful for our purposes because we do not have the equivalence f ∈ RBMO(µ) ⇐⇒ M f ∈ L∞ (µ). 3) Now we want to introduce another sharp maximal operator suitable for our space RBMO(µ) enjoying properties similar to the ones of the classical sharp operator. 4) M f (x) = sup 1 3 Q x µ( 2 Q) Q |mQ f − mR f | .

8. M. S. Melnikov, J. Verdera. A geometric proof of the L2 boundedness of the Cauchy integral on Lipschitz graphs. Int. Math. Res. Not. 7 (1995), 325–331 9. P. Morse. Perfect blankets. Trans. Amer. Math. Soc. 6 (1947), 418–442 10. F. Nazarov, S. Treil, A. Volberg. Cauchy integral and Calder´on-Zygmund operators on nonhomogeneous spaces. Int. Math. Res. Not. 15 (1997), 703–726 11. F. Nazarov, S. Treil, A. Volberg. Weak type estimates and Cotlar inequalities for Calder´onZygmund operators in nonhomogeneous spaces.

Another was given later in [18]. The proof of the present paper follows the lines of the proof found by Melnikov and Verdera for the L2 boundedness of the Cauchy integral in the (doubling) case where µ is the arc length on a Lipschitz graph [8]. 5), the assumption was Q |Cε χQ |2 dµ ≤ C µ(Q) for any square Q ⊂ C. 5). 3. 4. The Cauchy integral operator is bounded on L2 (µ) if and only if Cε (1) ∈ BMO ρ (µ) (uniformly on ε > 0), for some ρ > 1. Proof. Suppose that Cε 1 ∈ BMO ρ (µ). Let us see that this implies Cε 1 ∈ RBMO(µ).