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

By Hervé M. Pajot

In keeping with a graduate path given by means of the writer at Yale college this ebook offers with advanced research (analytic capacity), geometric degree idea (rectifiable and uniformly rectifiable units) and harmonic research (boundedness of singular indispensable operators on Ahlfors-regular sets). specifically, those notes comprise 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 whole 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.

Show description

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

Best functional analysis books

K-Theory: Lecture notes

Those notes are in response to the process lectures I gave at Harvard within the fall of 1964. They represent a self-contained account of vector bundles and K-theory assuming simply the rudiments of point-set topology and linear algebra. one of many positive factors of the remedy is that little need is made from traditional homology or cohomology concept.

Nonlinear functional analysis and its applications. Fixed-point theorems

This is often the fourth of a five-volume exposition of the most rules of nonlinear practical research and its functions to the common sciences, economics, and numerical research. The presentation is self-contained and obtainable to the nonspecialist. themes coated during this quantity comprise purposes to mechanics, elasticity, plasticity, hydrodynamics, thermodynamics, stastical physics, and specified and common relativity together with cosmology.

I: Functional Analysis, Volume 1 (Methods of Modern Mathematical Physics) (vol 1)

This publication is the 1st of a multivolume sequence dedicated to an exposition of useful research equipment in glossy mathematical physics. It describes the basic ideas of practical research and is basically self-contained, even supposing there are occasional references to later volumes. we now have incorporated a couple of functions after we notion that they'd supply motivation for the reader.

A Sequential Introduction to Real Analysis

Actual research presents the basic underpinnings for calculus, arguably the main invaluable and influential mathematical thought ever invented. it's a center topic in any arithmetic measure, and in addition one that many scholars locate hard. A Sequential creation to genuine research supplies a clean tackle actual research through formulating all of the underlying ideas when it comes to convergence of sequences.

Extra resources for Analytic Capacity, Rectifiability, Menger Curvature and Cauchy Integral

Sample text

Ij The trajectories of S are projected onto the geodesics by the canonical projection τ : T Q → Q; the condition x˙ k = vk shows that the trajectories of S are, precisely, the curves t → (c(t), c(t)) ˙ ∈ T Q, derivatives of the geodesics. 6 we see that , ∈ C k , k ≥ 2, implies that S is of class C k−1 . The vector S(vp ) can be also obtained through the horizontal lifting operator Hvp : wp ∈ Tp Q −→ Tvp (T Q) defined as follows. Take the geodesic c(t) characterized by the conditions c(0) = p, c(0) ˙ = wp , and consider the curve V (t) as the parallel transport of vp along c(t), that is, such that V (0) = vp and DVdt(t) = 0.

7); clearly the two definitions ˜ due to the uniqueness of DV on Ω. In this way DV can coincide on Ω ∩ Ω dt dt be extended to the entire manifold Q, using an atlas. Given an affine connection ∇ on a differentiable manifold and a differentiable vector field V = V (t) along a differentiable curve c : t ∈ I → c(t) ∈ Q, one says that V is parallel along c if DV dt = 0. 4. (Parallel translation) Let Q be a C ∞ differentiable manifold with an affine connection ∇, c = c(t) a differentiable curve on Q and Vo ∈ Tc(to ) Q a tangent vector to Q at the point c(to ) of the curve.

In n ∂ fact, in local coordinates (V, y1 , . . , yn ) one can write X = i=1 X i ∂y and i Y = n i=1 ∂ Y i ∂y . We start computing LX i j-component of the vector field dyj (p) X−t∗ = ∂ X−t∗ ∂y i ∂ ∂yi = d dt X−t∗ ∂ ∂yi |t=0 . The at p ∈ Q is ∂ ∂ j (Xt (p)) = dX−t (Xt (p)) ∂yi ∂yi j ∂X−t (Xt (p)) ∂yi so that, LX ∂ ∂yi n = j=1 d dt j ∂X−t (Xt (p)) ∂yi t=0 ∂ = ∂yj n − j=1 ∂X j ∂ , ∂yi ∂yj and n Yi LX (Y ) = LX i=1 n  ∂ ∂yi n LX Y i = i=1 n n ∂ ∂yi ∂X j ∂Y i ∂  = − Xj +Yi ∂yj ∂yi ∂yi i=1 j=1 j=1 + Y i LX ∂ ∂yi =  ∂  , ∂yj so, n Xj LX (Y ) = i,j=1 ∂Y i ∂X i −Yj ∂yj ∂yj ∂ = [X, Y ].

Download PDF sample

Rated 4.45 of 5 – based on 38 votes