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

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.

Show description

Read Online or Download Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral PDF

Best functional analysis books

Foundations of Differential Calculus Euler

The optimistic reaction to the ebook of Blanton's English translations of Euler's "Introduction to research of the countless" proven the relevance of this 240 yr previous paintings and inspired Blanton to translate Euler's "Foundations of Differential Calculus" besides. the present publication constitutes simply the 1st nine out of 27 chapters.

Introduction to Calculus and Analysis, Vol. 1

From the Preface: (. .. ) The booklet is addressed to scholars on numerous degrees, to mathematicians, scientists, engineers. It doesn't fake to make the topic effortless via glossing over problems, yet fairly attempts to aid the certainly reader by way of throwing gentle at the interconnections and reasons of the complete.

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

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.

Download PDF sample

Rated 4.35 of 5 – based on 14 votes