zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Painlevé’s problem and the semiadditivity of analytic capacity. (English) Zbl 1060.30031
A compact set $E$ in the complex plane $\bbfC$ is said to be removable (for bounded holomorphic functions) if for every open set $\Omega$ containing $E$, every bounded function holomorphic in $\Omega\setminus E$ has a holomorphic extension in $\Omega$. {\it L. V. Ahlfors} in 1947 [Duke Math. J. 14, 1--11 (1947; Zbl 0030.03001)] introduced the notion of analytic capacity of $E$; this is defined by $$ \gamma(E)=\sup\{\vert f^\prime(\infty)\vert :f:C\setminus E\to C\text{ holomorphic, } \vert f\vert \leq 1 \}, $$ where $f^\prime(\infty):=\lim_{z\to\infty} z(f(z)-f(\infty)).$ Ahlfors showed that $E$ is removable if and only if $\gamma(E)=0$ and posed the problem of the geometric characterization of the sets $E$ with $\gamma(E)=0$. This is a fundamental problem in Complex Analysis; it is called Painlevé’s problem. One of the main results in the present paper is that analytic capacity is semi-additive: $$ \gamma(E\cup F)\leq const.(\gamma(E)+\gamma(F)). $$ This fact was conjectured by {\it A. G. Vitushkin} [Russ. Math. Surv. 22, 139--200 (1967; Zbl 0164.37701)]. The semi-additivity of $\gamma$ has the following consequences on Painlevé’s problem: If $E$ has $\sigma$-finite length, then $\gamma(E)=0$ if and only if $H^1(E\cap\Gamma)=0$ for all rectifiable curves $\Gamma$ ($H^1$ is the one-dimensional Hausdorff measure). For compact sets of finite length this had been proved by {\it G. David} [Rev. Mat. Iberoam. 14, 369--479 (1998; Zbl 0913.30012)]; David’s theorem was previously known as Vitushkin’s conjecture. For sets $E$ which are not $\sigma$-finite the above characterization of removable sets does not hold; this was proved by {\it P. Mattila} [Ann. Math. 123, 303--309 (1986; Zbl 0589.28006)]. Tolsa obtained the semi-additivity of $\gamma$ by relating the analytic capacity with another quantity, the analytic capacity $\gamma_+(E)$ defined in terms of Cauchy transforms of measures supported on $E$: $$ \gamma_+(E)=\sup\{\mu(E): \mu\text{ Radon measure on $E$ with }\Vert \tfrac{1}{z}* \mu\Vert _\infty\leq 1\}. $$ Tolsa proved that $$ \gamma (E)\leq const.\ \gamma_+(E). $$ This inequality, together with its converse (which is relatively easy) and the semi-additivity of $\gamma_+$ (proved by the author in [Indiana Math. J. 51, 317--343 (2002; Zbl 1041.31002)]) give the semi-additivity of $\gamma$. In [Indiana Math. J. 51, 317--343 (2002; Zbl 1041.31002)] Tolsa has also shown that $\gamma_+$ has a rather precise description in terms of curvature of measures. The curvature $c(\mu)$ of a measure $\mu$ is defined as follows: $$ c^2(\mu)=\iiint \frac{1}{R(x,y,z)^2}\,d\mu(x) \,d\mu(y) \,d\mu(z), $$ where $R(x,y,z)$ is the radius of the circle passing through $x,y,z$. This quantity, introduced by {\it M. S. Melnikov} [Sb. Math. 186, No. 6, 827--846 (1995; Zbl 0840.30008)] is one of the main tools in the study of removable sets. A consequence of the equivalence of $\gamma$ and $\gamma_+$ is a characterization of removable sets previously conjectured by Melnikov: $E$ is non-removable if and only if it supports a Radon measure with linear growth and finite curvature. Further information and references on the connection of Painlevé’s problem with Cauchy transforms, T(b)-theorems, the continuous analytic capacity, and the theory of rational approximation, as well as more details on the main steps of the proofs of the important results in the paper under review can be found in two recent survey papers by the author: 1. {Painlevé’s problem, analytic capacity and curvature of measures}, Proceedings of the Fourth European Congress, 2004. 2. {Painlevé’s problem and analytic capacity}, Lecture notes of a minicourse given at El Escorial, 2004. Also, in the paper [“Bilipschitz maps, analytic capacity, and the Cauchy integral”, to appear in Ann. Math.] the author continues the research of the present paper.

30C85Capacity and harmonic measure in the complex plane
31A15Potentials and capacity, harmonic measure, extremal length (two-dimensional)
42B20Singular and oscillatory integrals, several variables
42B35Function spaces arising in harmonic analysis
Full Text: DOI arXiv
[1] Ahlfors, L. V., Bounded analytic functions.Duke Math. J., 14 (1947), 1--11. · Zbl 0030.03001 · doi:10.1215/S0012-7094-47-01401-4
[2] Christ, M., AT (b) theorem with remarks on analytic capacity and the Cauchy integral.Colloq. Math. 60/61 (1990), 601--628. · Zbl 0758.42009
[3] David, G.,Wavelets and Singular Integrals on Curves and Surfaces. Lecture Notes in Math., 1465. Springer-Verlag, Berlin, 1991.
[4] -- Unrectifiable 1-sets have vanishing analytic capacity.Rev. Mat. Iberoamericana, 14 (1998), 369--479. · Zbl 0913.30012
[5] -- Analytic capacity, Calderón-Zygmund operators, and rectificability.Publ. Mat., 43 (1999), 3--25.
[6] David, G. &Mattila, P., Removable sets for Lipschitz harmonic functions in the plane.Rev. Mat. Iberoamericana, 16 (2000), 137--215. · Zbl 0976.30016
[7] Davie, A. M., Analytic capacity and approximation problems.Trans. Amer. Math. Soc., 171 (1972), 409--444. · Zbl 0263.30032 · doi:10.1090/S0002-9947-1972-0350009-9
[8] Davie, A. M. &Øksendal, B., Analytic capacity and differentiability properties of finely harmonic functions.Acta Math., 149 (1982), 127--152. · Zbl 0527.31001 · doi:10.1007/BF02392352
[9] Garnett, J.,Analytic Capacity and Measure. Lecture Notes in Math., 297. Springer-Verlag, Berlin-New York, 1972. · Zbl 0253.30014
[10] Jones, P. W., Rectifiable sets and the traveling salesman problem.Invent. Math., 102, (1990), 1--15. · Zbl 0731.30018 · doi:10.1007/BF01233418
[11] Jones, P. W. &Murai, T., Positive analytic capacity but zero Buffon needle probability.Pacific J. Math., 133 (1988), 99--114. · Zbl 0653.30016
[12] Léger, J. C., Menger curvature and rectifiability.Ann. of Math. (2), 149 (1999), 831--869. · Zbl 0966.28003 · doi:10.2307/121074
[13] Mattila, P., Smooth maps, null-sets for integralgeometric measure and analytic capacity.Ann. of Math. (2), 123 (1986), 303--309. · Zbl 0589.28006 · doi:10.2307/1971273
[14] --Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability. Cambridge Stud. Adv. Math., 44, Cambridge Univ. Press, Cambridge, 1995.
[15] Mattila, P. Rectifiability, analytic capacity, and singular integrals, inProceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998).Doc. Math., 1998, Extra Vol. II, 657--664 (electronic).
[16] Mattila, P., Melnikov, M. S. &Verdera, J., The Cauchy integral, analytic capacity, and uniform rectifiability.Ann. of Math. (2), 144 (1996), 127--136. · Zbl 0897.42007 · doi:10.2307/2118585
[17] Mattila, P. &Paramonov, P. V., On geometric properties of harmonic Lip1-capacity.Pacific J. Math., 171 (1995), 469--491. · Zbl 0852.31004
[18] -- On density properties of the Riesz capacities and the analytic capacity {$\gamma$}+.Tr. Mat. Inst. Steklova, 235 (2001), 143--156 (Russian); English translation inProc. Steklov Inst. Math., 235 (2001), 136--149.
[19] Melnikov, M. S., Estimate of the Cauchy integral over an analytic curve.Mat. Sb. (N.S.), 71 (113), (1966), 503--514 (Russian); English translation inAmer. Math. Soc. Transl. Ser. 2, 80 (1969), 243--256.
[20] -- Analytic capacity: a discrete approach and the curvature of measure.Mat. Sb., 186:6 (1995), 57--76 (Russian); English translation inSb. Math., 186 (1995), 827--846.
[21] Melnikov, M. S. & Verdera, J.. A geometric proof of theL 2 boundedness of the Cauchy integral on Lipschitz graphs.Internat. Math. Res. Notices 1995, 325--331. · Zbl 0923.42006
[22] Mateu, J., Tolsa, X. &Verdera, J., The planar Cantor sets of zero analytic capacity and the localT(b)-theorem.J. Amer. Math. Soc., 16 (2003), 19--28. · Zbl 1016.30020 · doi:10.1090/S0894-0347-02-00401-0
[23] Murai, T.,A Real Variable Method for the Cauchy Transform, and Analytic Capacity. Lecture Notes in Math. 1307. Springer-Verlag, Berlin, 1988. · Zbl 0645.30016
[24] Nazarov, F., Treil, S. &Volberg, A., TheTb-theorem on non-homogeneous spaces that proves a conjecture of Vitushkin. Preprint, Centre de Recerca Matemàtica, Barcelona, 2002.
[25] -- Accretive systemTb-theorems on nonhomogeneous spaces.Duke Math. J., 113 (2002), 259--312. · Zbl 1055.47027 · doi:10.1215/S0012-7094-02-11323-4
[26] Nazarov, F., Treil, S. & Volberg, A. TheTb-theorem on non-homogeneous spaces. To appear inActa Math. (2003). · Zbl 1065.42014
[27] Pajot, H.,Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral. Lecture Notes in Math., 1799. Springer-Verlag, Berlin, 2002. · Zbl 1043.28002
[28] Suita, N., On subadditivity of analytic capacity for two continua.Kodai Math. J., 7 (1984), 73--75. · Zbl 0548.30018 · doi:10.2996/kmj/1138036857
[29] Tolsa, X., Curvature of measures, Cauchy singular integral and analytic capacity. Ph.D. Thesis, Universitat Autònoma de Barcelona, 1998. · Zbl 0912.42009
[30] --L 2-boundedness of the Cauchy integral operator for continuous measures.Duke Math. J., 98 (1999), 269--304. · Zbl 0945.30032 · doi:10.1215/S0012-7094-99-09808-3
[31] -- Principal values for the Cauchy integral and rectifiability.Proc. Amer. Math. Soc., 128 (2000), 2111--2119. · Zbl 0944.30022 · doi:10.1090/S0002-9939-00-05264-3
[32] -- On the analytic capacity {$\gamma$}+.Indiana Univ. Math. J., 51 (2002), 317--343. · Zbl 1041.31002 · doi:10.1512/iumj.2002.51.2202
[33] Verdera, J., Removability, capacity and approximation, inComplex Potential Theory (Montreal, PQ 1993), pp. 419--473. NATO Adv. Sci. Int. Ser. C Math. Phys. Sci., 439. Kluwer, Dordrecht, 1994. · Zbl 0809.30001
[34] -- On theT(1)-theorem for the Cauchy integral.Ark. Mat., 38, (2000), 183--199. · Zbl 1039.42011 · doi:10.1007/BF02384497
[35] Verdera, J., Melnikov, M. S. &Paramonov, P. V.,C 1-approximation and the extension of subharmonic functions.Mat. Sb., 192:4 (2001), 37--58 (Russian); English translation inSb. Math., 192 (2001), 515--535.
[36] Vitushkin, A. G., The analytic capacity of sets in problems of approximation theory.Uspekhi Mat. Nauk, 22:6 (1967), 141--199. (Russian); English translation inRussian Math. Surveys, 22 (1967), 139--200. · Zbl 0157.39402
[37] Vitushkin, A. G. &Melnikov, M. S., Analytic capacity and rational approximation, inLinear and Complex Analysis Problem Book pp. 495--497. Lecture Notes in Math., 1403. Springer-Verlag, Berlin, 1984.