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 collage this ebook offers with advanced research (analytic capacity), geometric degree concept (rectifiable and uniformly rectifiable units) and harmonic research (boundedness of singular vital 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, in simple terms the Ahlfors-regular case) and a dialogue of X. Tolsa's answer of the Painlevé challenge.

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**Example text**

The Hardy-Littlewood maximal operator Mµ related to µ is deﬁned by 1 µ(B(x, R)) R>0 |f (y)|dµ(y). Mµ f (x) = sup B(x,R) The Hardy-Littlewood maximal operator is not of course a singular integral operator, but is a very useful tool in the Calder´on-Zygmund theory. We will illustrate this by giving a version of Cotlar inequality at the end of this section. Theorem 46. Let µ be a doubling positive Radon measure in Rn . Then, Mµ is bounded on Lp (µ) for all 1 < p < ∞ and is of weak type (1, 1). Recall that an operator T is of weak type (1, 1) (with respect to a positive Radon measure µ) if for all f ∈ L1 (µ) and all λ > 0, µ({x ∈ Rn ; |T f (x)| > λ}) ≤ C ||f ||L1 (µ) .

Meyer [74] and G. David [23]) and has also two more surprising applications: - The resolution of the Beltrami equation for quasi-conformal mappings [3]. - The Atiyah-Singer index theorem [5]. 5. The T 1 and the T b theorems Let µ be a doubling positive Radon measure on Rn . We say that f ∈ BM O(µ) if 1 ||f ||BM O(µ) = sup |f − fB |dµ < +∞ µ(B) B B⊂C balls 1 f dµ. Then, ||f ||BM O(µ) is a semi-norm on BM O(µ), but is µ(B) B a norm on BM O(µ)/{constantf unctions} (that we call also BM O(µ)). Note that L∞ (µ) ⊂ BM O(µ).

We say that f ∈ BM O(µ) if 1 ||f ||BM O(µ) = sup |f − fB |dµ < +∞ µ(B) B B⊂C balls 1 f dµ. Then, ||f ||BM O(µ) is a semi-norm on BM O(µ), but is µ(B) B a norm on BM O(µ)/{constantf unctions} (that we call also BM O(µ)). Note that L∞ (µ) ⊂ BM O(µ). where fB = 60 4. THE CAUCHY SINGULAR INTEGRAL OPERATOR ON AHLFORS REGULAR SETS Theorem 49 (T 1 Theorem of David-Journ´e). Let µ be a doubling positive Radon measure on Rn with polynomial growth and let T be a singular integral operator (associated to an antisymmetric standard kernel).

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

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