By Hervé M. Pajot

ISBN-10: 3540000011

ISBN-13: 9783540000013

ISBN-10: 3540360743

ISBN-13: 9783540360742

Based on a graduate path given through the writer at Yale collage this e-book offers with advanced research (analytic capacity), geometric degree idea (rectifiable and uniformly rectifiable units) and harmonic research (boundedness of singular crucial operators on Ahlfors-regular sets). particularly, those notes include an outline of Peter Jones' geometric touring salesman theorem, the evidence of the equivalence among uniform rectifiability and boundedness of the Cauchy operator on Ahlfors-regular units, the entire proofs of the Denjoy conjecture and the Vitushkin conjecture (for the latter, in basic terms the Ahlfors-regular case) and a dialogue of X. Tolsa's answer of the Painlevé challenge.

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

**Example text**

Observation 2 Let B1 and B3 be two balls of B such that 2B1 ∩ 2B3 = ∅. Let z ∈ B1 . As previously, 1 |x3 − x1 | ≤ |x3 − z| ≤ 3|x3 − x1 |. 3 By lemma 35, c(x1 , x2 , x3 ) ≤ c(z, x2 , x3 ) + 10 |x1 − z| . |z − x2 ||z − x3 | Let y1 ∈ W1 (B1 ). Then, there exists i ∈ {1, 2, 3, 4} such that |x1 −yi | ≤ 10d(yi , Lx2 x3 ). Thus, c(x1 , x2 , x3 ) ≤ 51c(yi , x2 , x3 ). We now prove the estimate c2 (µ) ≤ CdiamE. First, note that, if x, y and z are three points in Suppµ, then there exists a minimal ball B ∈ F (denoted by B(x, y, z)) such that x, y, z ∈ B.

To see this, note that b is deﬁned almost everywhere on R (by the Rademacher theorem), b is para-accretive since Re(b) ≥ 1 (such a function is usually called accretive), and b(y) dL1 (y) = 0. γ(x) − γ(y) 7. CAUCHY SINGULAR OPERATOR AND RECTIFIABILITY 63 2) A classical result of I. Privalov [92] (see also [33] page 244) says that, except a set φ(ξ) of zero measure, the Cauchy integral f (z) = dξ (where Γ is a rectiﬁable Jorξ −z Γ dan curve) has a non tangential limit at z0 if and only if the Cauchy singular operator CΓ exists at z0 in the principal value sense.

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)).

### Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral by Hervé M. Pajot

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