By Hervé M. Pajot

Based on a graduate path given by way of the writer at Yale college this e-book bargains with advanced research (analytic capacity), geometric degree thought (rectifiable and uniformly rectifiable units) and harmonic research (boundedness of singular critical operators on Ahlfors-regular sets). specifically, those notes include an outline of Peter Jones' geometric touring salesman theorem, the facts of the equivalence among uniform rectifiability and boundedness of the Cauchy operator on Ahlfors-regular units, the full proofs of the Denjoy conjecture and the Vitushkin conjecture (for the latter, simply the Ahlfors-regular case) and a dialogue of X. Tolsa's resolution of the Painlevé challenge.

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**Example text**

Meyer [69]. There are now a lot of diﬀerent proofs of this famous theorem (see, for instance, [18], [20], [22], [49], [76]. The proof that we present is due to M. Melnikov and J. Verdera [72]. First, recall that Γ satisﬁes the local curvature condition (see theorem 30). Note that, by the T 1 theorem, it is enough to prove that, for any interval I ⊂ R, I |CΓε (χI (x))|2 dL1 (x) ≤ C|I| where C > 0 does not depend on I and where |I| denotes the length of I. But, I |CΓε (χI (x))|2 dL1 (x) = I CΓε (χI (x))CΓε (χI (x))dL1 (x) dL1 (x)dL1 (y)dL1 (z) = I Iε Iε (γ(y) − γ(x))(γ(z) − γ(x)) where Iε (x) = {t ∈ I; |t − x| > ε}.

If E is uniformly rectiﬁable, then E satisﬁes the local curvature condition. Remark. The estimate of c2 (µ) in theorem 31 can be improved if you assume that the curve Γ has some extra properties. For instance, if the measure µ is supported by an Ahlfors regular curve, then c2 (µ) ≤ Cµ(C) where C depends on the regularity constant of the curve. In general, this estimate is not true (see [88]). Proof. We start with three technical lemmas. Lemma 34. Let x, y, z and z be four points in the complex plane C such that (i) x, z, z are collinear, and z ∈ [x, z ]; (ii) |y − z | ≤ |y − z|.

Theorem 31. Let Γ be a rectiﬁable curve in C and let µ be a positive compactly supported measure with suppµ ⊂ Γ. Assume that µ has linear growth: µ(B) ≤ diamB for every ball B ⊂ C. ) t In particular, by P. Jones’traveling salesman theorem, this implies that c2 (µ) ≤ Cl(Γ) < +∞. From this and Frostman’s lemma, we get Corollary 32. Let E be a compact set in the complex plane C. Assume that H 1 (E) > 0 and that E is contained in a rectiﬁable curve Γ. Then, there exists a positive Radon measure µ compactly supported on E with linear growth and ﬁnite Menger curvature.