Let $\gamma(E)$ be the analytic capacity of a compact set $E$ and let $\gamma_+(E)$ originated by Cauchy transforms positive measures. In this paper we prove that $\gamma(E)\approx\gamma_+(E)$ with estimates independent $E$. As corollary, characterize removable singularities for bounded functions in terms curvature measures, deduce $\gamma$ is semiadditive, which solves long standing question Vitushkin.

We prove that if μ is a d-dimensional Ahlfors-David regular measure in Rd+1 , then the boundedness of Riesz transform L2(μ) implies non-BAUP David–Semmes cells form Carleson family. Combined with earlier results David and Semmes, this yields uniform rectifiability μ.

Let $\mu$ be a Radon measure on $\mathbb {R}^d$, which may nondoubling. The only condition that must satisfy is the size $\mu (B(x,r))\leq C r^n$, for some fixed $0<n\leq d$. Recently, spaces of type $B\!M\!O(\mu )$ and $H^1(\mu were introduced by author. These new have properties similar to those classical $BMO$ $H^1$ defined doubling measures, they proved useful studying $L^p(\mu boundedness Calderón-Zygmund operators without assuming conditions. In this paper characterization atomic...

Let ϕ : C → be a bilipschitz map.We prove that if E ⊂ is compact, and γ(E), α(E) stand for its analytic continuous capacity respectively, thenwhere depends only on the constant of ϕ.Further, we show µ Radon measure Cauchy transform bounded L 2 (µ), then also (ϕ µ), where image by ϕ.To obtain these results, estimate curvature means corona type decomposition.

In this paper we study some questions in connection with uniform rectifiability and the $L^2$ boundedness of Calderon-Zygmund operators. We show that can be characterized terms new adimensional coefficients which are related to Jones' $\beta$ numbers. also use these prove n-dimensional operators odd kernel type $C^2$ bounded $L^2(\mu)$ if $\mu$ is an uniformly rectifiable measure.

Given a doubling measure µ on R d , it is classical result of harmonic analysis that Calderón-Zygmund operators which are bounded in L 2 (µ) also weak type (1, 1).Recently has been shown the same holds if one substitutes condition by mild growth µ.In this paper another proof given.The very close spirit to argument for measures and based new decomposition adapted non situation.

Let α( E ) be the continuous analytic capacity of a compact set ⊂ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. In this paper we obtain characterization α in terms curvature measures with zero linear density, and deduce that is countably semiadditive. This result has important consequences for theory uniform rational approximation on sets. particular, it implies so-called inner boundary conjecture.

Article Cotlar's inequality without the doubling condition and existence of principal values for Cauchy integral measures was published on September 15, 1998 in journal Journal für die reine und angewandte Mathematik (volume 1998, issue 502).

We show that, given a set E ⊂ R n+1 with finite n-Hausdorff measure H n , if the n-dimensional Riesz transform

For 1 ≤ n &lt; d integers and \rho &gt;2 , we prove that an -dimensional Ahlfors–David regular measure \mu in \mathbb R^d is uniformly -rectifiable if only the -variation for Riesz transform with respect to a bounded operator L^2(\mu) . This result can be considered as partial solution well known open problem posed by G. David S. Semmes which relates boundedness of uniform rectifiability

We show that, given a set E Rn+1 with finite n-Hausdorff measure Hn, if the n-dimensional Riesz transform is bounded in L2(HnbE), then n-rectifiable. From this result we deduce that compact Hn(E) < 1 removable for Lipschitz harmonic functions and only it purely n-unrectifiable, thus proving analog of Vitushkin's conjecture higher dimensions.

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Ω</m:mi> <m:mo>⊊</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> {\Omega\subsetneq{\mathbb{R}}^{n+1}} be open and let μ some measure supported on <m:mo>∂</m:mo> <m:mo>⁡</m:mo> {\partial\Omega} such that <m:mi>μ</m:mi> <m:mo>⁢</m:mo> <m:mo>(</m:mo> <m:mi>B</m:mi> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>r</m:mi> <m:mo>)</m:mo> <m:mo>≤</m:mo> <m:mi>C</m:mi>...

We prove that, for ρ>2, the ρ-variation and oscillation smooth truncations of Cauchy transform on Lipschitz graphs are bounded in Lp 1<p<∞. The analogous result holds n-dimensional Riesz graphs, as well other singular integral operators with odd kernel. In particular, our results strengthen classical theorem L2 boundedness by Coifman, McIntosh, Meyer.

Let Ω⊂Rn+1, n≥1, be a corkscrew domain with Ahlfors–David regular boundary. In this article we prove that ∂Ω is uniformly n-rectifiable if every bounded harmonic function on Ω ε-approximable or satisfies suitable square-function Carleson measure estimate. particular, applies to the case when Ω=Rn+1∖E and E regular. Our results establish conjecture posed by Hofmann, Martell, Mayboroda, in which they proved converse statements. Here also obtain two additional criteria for uniform...

Let $Ω\subset\mathbb{R}^{n+1}$, $n\geq2$, be an open set with Ahlfors-David regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator $L$ in divergence form associated matrix $A$ real, merely bounded and possibly non-symmetric coefficients, which are also locally Lipschitz satisfy suitable Carleson type estimates. In this paper we show if $L^*$ is transpose of $A$, then $\partialΩ$ $n$-rectifiable only every solution $Lu=0$ $L^*v=0$ $Ω$...