000			03823nam a22004815i 4500
001			978-3-319-00596-6
003			DE-He213
005			20140220082506.0
007			cr nn 008mamaa
008			131216s2014 gw | s |||| 0|eng d
020			_a9783319005966 _9978-3-319-00596-6
024	7		_a10.1007/978-3-319-00596-6 _2doi
050		4	_aQA331-355
072		7	_aPBKD _2bicssc
072		7	_aMAT034000 _2bisacsh
082	0	4	_a515.9 _223
100	1		_aTolsa, Xavier. _eauthor.
245	1	0	_aAnalytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory _h[electronic resource] / _cby Xavier Tolsa.
264		1	_aCham : _bSpringer International Publishing : _bImprint: Birkhäuser, _c2014.
300			_aXIII, 396 p. 8 illus. _bonline resource.
336			_atext _btxt _2rdacontent
337			_acomputer _bc _2rdamedia
338			_aonline resource _bcr _2rdacarrier
347			_atext file _bPDF _2rda
490	1		_aProgress in Mathematics, _x0743-1643 ; _v307
505	0		_aIntroduction -- Basic notation -- Chapter 1. Analytic capacity -- Chapter 2. Basic Calderón-Zygmund theory with non doubling measures -- Chapter 3. The Cauchy transform and Menger curvature -- Chapter 4. The capacity γ+ -- Chapter 5. A Tb theorem of Nazarov, Treil and Volberg -- Chapter 6. The comparability between γ and γ +, and the semiadditivity of analytic capacity -- Chapter 7. Curvature and rectifiability -- Chapter 8. Principal values for the Cauchy transform and rectifiability -- Chapter 9. RBMO(μ) and H1 atb(μ) -- Bibliography -- Index.
520			_aThis book studies some of the groundbreaking advances that have been made regarding analytic capacity and its relationship to rectifiability in the decade 1995–2005. The Cauchy transform plays a fundamental role in this area and is accordingly one of the main subjects covered. Another important topic, which may be of independent interest for many analysts, is the so-called non-homogeneous Calderón-Zygmund theory, the development of which has been largely motivated by the problems arising in connection with analytic capacity. The Painlevé problem, which was first posed around 1900, consists in finding a description of the removable singularities for bounded analytic functions in metric and geometric terms. Analytic capacity is a key tool in the study of this problem. In the 1960s Vitushkin conjectured that the removable sets which have finite length coincide with those which are purely unrectifiable. Moreover, because of the applications to the theory of uniform rational approximation, he posed the question as to whether analytic capacity is semiadditive. This work presents full proofs of Vitushkin’s conjecture and of the semiadditivity of analytic capacity, both of which remained open problems until very recently. Other related questions are also discussed, such as the relationship between rectifiability and the existence of principal values for the Cauchy transforms and other singular integrals. The book is largely self-contained and should be accessible for graduate students in analysis, as well as a valuable resource for researchers.
650		0	_aMathematics.
650		0	_aFunctions of complex variables.
650		0	_aPotential theory (Mathematics).
650		0	_aMathematical optimization.
650	1	4	_aMathematics.
650	2	4	_aFunctions of a Complex Variable.
650	2	4	_aPotential Theory.
650	2	4	_aCalculus of Variations and Optimal Control; Optimization.
710	2		_aSpringerLink (Online service)
773	0		_tSpringer eBooks
776	0	8	_iPrinted edition: _z9783319005959
830		0	_aProgress in Mathematics, _x0743-1643 ; _v307
856	4	0	_uhttp://dx.doi.org/10.1007/978-3-319-00596-6
912			_aZDB-2-SMA
999			_c92498 _d92498