By Hervé M. Pajot

ISBN-10: 3540000011

ISBN-13: 9783540000013

ISBN-10: 3540360743

ISBN-13: 9783540360742

Based on a graduate direction given through the writer at Yale college this ebook bargains with advanced research (analytic capacity), geometric degree conception (rectifiable and uniformly rectifiable units) and harmonic research (boundedness of singular crucial operators on Ahlfors-regular sets). specifically, 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, merely the Ahlfors-regular case) and a dialogue of X. Tolsa's resolution of the Painlevé challenge.

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

**Example text**

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)).

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

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