Painlevé's problem and the semiadditivity of analytic capacity

By Xavier Tolsa. 2002

325 citations recorded by the source corpus.

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

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5e892dcf675f8f875572b3746f285d98255a5053

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Citations in the corpus

A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa (2009, methodology): “The John Nirenberg inequalityEverything is now prepared for the main result, whi h is a simultaneous generalisation of theJohn Nirenberg inequalities in Tolsa's RBMO(µ) spa e on Rn, and in the lassi al BMO(µ) spa eon abstra t homogeneous spa es (thanks to Proposition 4.7).”
Systems of dyadic cubes in a doubling metric space (2010, background): “These random dyadic systems have been instrumental in the development of the non-doubling theory of singular integrals [23, 24] and its applications to analytic capacity [27, 28] as well as sharp one-weight [9, 13] and two-weight [16, 25] inequalities for classical singular integrals.”
Menger curvature and Lipschitz parametrizations in metric spaces (2005, background): “In [13] Tolsa proved that a compact set F ⊂ C is not removable for bounded analytic functions if and only if F supports a positive Radon measure μ such that μ(B) ≤ d(B) for every ball B in C and c(z1, z2, z3) 2 dμz1 dμz2 dμz3 <∞.”
Holomorphic Approximation: The Legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan (2018, background): “kin-type results is due to Paramonov [134]. Major results on the behavior of the (continuous) capacity and estimates of Cauchy integrals over curves were obtained by M. Mel’nikov [120, 122], X. Tolsa [162, 163], and Mel’nikovand Tolsa [121]. 3 Approximation on unbounded sets in Riemann surfaces It s”
A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa (2009, methodology): “Some highlights of this theory, each building on the previous one, are the introduction of the regularised BMO space by Tolsa [20], the proof of a non-homogeneous Tb theorem by Nazarov, Treil and Volberg [18], and the solution of the Painlevé problem, again by Tolsa [21].”
Bergman-type singular integral operators and the characterization of Carleson measures for Besov-Sobolev spaces on the complex ball (2009, methodology): “This theory has found applications in Tolsa’s solution of the Vitushkin Conjecture, [17–19].”
Besov‐Morrey spaces and Triebel‐Lizorkin‐Morrey spaces for nondoubling measures (2009, background): “In [8, 9] Volberg, Nazarov, and Treil investigated the analytic capacity and the function space with the measure non-doubling and in [29, 30] Tolsa proved the subadditivity and the bi-Lipschitz invariance of the analytic capacity.”
Uniform rectifiability and harmonic measure, II: Poisson kernels in $L^{p}$ imply uniform rectifiability (2012, methodology): “g the scale invariant higher integrability condition (1.20), then ∂Ω is Uniformly Rectiﬁable (cf. Theorem 1.22). We observe that this result is in the spirit of the solution of the Painleve´ problem ([To], but see also [Ch], [MMV], [Da] and [Vo]), in which analytic information is used to establish r”
Boundedness of Calderón–Zygmund Operators on Non-homogeneous Metric Measure Spaces (2010, background): “However, recently, many results on the classical Calderón-Zygmund theory have been proved still valid if the measure doubling condition is replaced by a less demanding condition; see, for example, [13, 16, 17, 14, 18, 12, 2] and the references therein.”
GENERALIZED FRACTIONAL INTEGRALS AND THEIR COMMUTATORS OVER NON-HOMOGENEOUS METRIC MEASURE SPACES (2013, background): “1 Introduction During the past ten to fifteen years, considerable attention has been paid to the study of the classical theory of harmonic analysis on Euclidean spaces with non-doubling measures only satisfying the polynomial growth condition (see, for example, [11, 10, 35, 36, 37, 38, 39, 40, 5, 27”
BOUNDEDNESS OF MULTILINEAR COMMUTATORS OF CALDERÓN-ZYGMUND OPERATORS ON ORLICZ SPACES OVER NON-HOMOGENEOUS SPACES (2012, background): “However, in recent years, many results in the classical theory of Calderón-Zygmund operators on R have been proved still valid with the Lebesgue measure m replaced by a Radon measure μ satisfying the polynomial growth condition (see, for example, [22, 23, 24, 25, 26, 27, 17, 3, 7, 8]).”
The planar Cantor sets of zero analytic capacity and the local T(b)-theorem (2002, background): “The proof in [ T3 ] also involves and induction argument and an appropriate T(b)-Theorem as in the present paper.”
The Hardy space H1 on non-homogeneous metric spaces (2010, methodology): “A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa.”