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Extra info for Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral

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N But, ω(xn , yn , xn+1 , yn+1 ) is comparable to β(xn , yn , xn+1 , yn+1 )λn R, and if we set cn = c(xn , yn , xn+1 ) + c(xn , yn , yn+1 ), cn is comparable to β(xn , yn , xn+1 , yn+1 )λ−n R−1 . From this, we get β∞ (Q) ≤ C R ω(xn , yn , xn+1 , yn+1 ) ≤ n C R cn (λn R)2 . n Since we would like to control β∞ (Q) by a triple integral of Menger curvature, a natural idea is to choose the points (xn ) and (yn ) such that c2n ≤ C Bn ˜n B ˜n+1 Bn+1 ∪B c(x, y, z)2 d˜ µ(x)d˜ µ(y)d˜ µ(z). ˜n such that Step n: Assume that we have chosen xn ∈ Bn and yn ∈ B (i) F (xn , yn ) ≤ C1 Gn , (ii) c(xn−1 , yn−1 , xn )2 ≤ C2 Gn−1 and c(xn−1 , yn−1 , yn )2 ≤ C2 Gn−1 .

6. L2 boundedness of the Cauchy singular operator on Lipschitz graphs Let Γ be a Lipschitz graph in C, that is Γ is a subset of C of the form Γ = {x + iA(x), x ∈ R} where A : R → R is Lipschitz. We set γ(x) = x + iA(x) and let ds be the arc length measure on Γ. Let f ∈ L2 (Γ, ds). The Cauchy integral of f along Γ is given formally by f (ξ) 1 ds(ξ). 2iπ Γ z − ξ To give a more precise sense of the Cauchy operator on Γ, define a continuous linear operator on the space of Lipschitz functions on Γ by 1 1 [f (x)g(y) − f (y)g(x)]ds(x)ds(y).

We first recall a basic covering theorem (see [54] page 2). Theorem 47. Let F be a family of balls in a metric space X such that supB∈F diamB < +∞. Then, there exists a subfamily G of F such that - The balls of G are pairwise disjoint; - B∈F B ⊂ B∈G 5B, where 5B is the ball with the same center as B but whose diameter is 5diamB. Fix t > 0. For each x ∈ {y ∈ Rn ; Mµ f (y) > t}, there exists R = R(x) |f (y)|dµ(y) > tµ(B(x, R)). By the previous covering theorem, there such that B(x,R) exists a countable family of pairwise disjoint balls B(x, R(x)), x ∈ G, such that {y ∈ Rn ; Mµ f (y) > t} ⊂ x∈G B(x, 5R(x)).

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