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