Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

In Section 21 the basic theory of sectorial operators is developed, including examples and the concept of sectorial approximation In Section 22 we introduce some notation for certain spaces of holomorphic functions on sectors A functional calculus for sectorial operators is constructed in Section 23 along the lines of the abstract framework of Chapter 1 Fundamental properties like the composition rule are proved In Section 25 we give natural extensions of the functional calculus to larger function spaces in the case where the given operator is bounded and/or invertible In this way a panorama of functional calculi is developed In Section 26 some mixed topics are discussed, eg, adjoints and restrictions of sectorial operators and some fundamental boundedness and some first approximation results Section 27 contains a spectral mapping theorem

The Functional Calculus for Sectorial Operators

Indispensable for students, invaluable for researchers, this comprehensive treatment of contemporary quasi-Monte Carlo methods, digital nets and sequences, and discrepancy theory starts from scratch with detailed explanations of the basic concepts and then advances to current methods used in research. As deterministic versions of the Monte Carlo method, quasi-Monte Carlo rules have increased in popularity, with many fruitful applications in mathematical practice. These rules require nodes with good uniform distribution properties, and digital nets and sequences in the sense of Niederreiter are known to be excellent candidates. Besides the classical theory, the book contains chapters on reproducing kernel Hilbert spaces and weighted integration, duality theory for digital nets, polynomial lattice rules, the newest constructions by Niederreiter and Xing and many more. The authors present an accessible introduction to the subject based mainly on material taught in undergraduate courses with numerous examples, exercises and illustrations.

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Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration

Multilinear Calderón–Zygmund Theory

http://grupo.us.es/anaresba/trabajos/carlosperez/1preprint.pdf

New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory

We prove the Kato conjecture for elliptic operators on Jfin. More precisely, we establish that the domain of the square root of a uniformly complex elliptic operator L =-div (AV) with bounded measurable coefficients in IEtn iS the

The solution of the Kato square root problem for second order elliptic operators on Rn

with constants Cfi;p1;p2 depending only on fi;p1;p2 and p := p1p2 p1+p2 hold. The flrst result of this type is proved in [4], and the purpose of the current paper is to extend the range of exponents p1 and p2 for which (2) is known. In particular, the case p1 =2 ,p2 = 1 is solved to the a‐rmative. This was

/pdf/on-calderon-s-conjecture-lh16tlutd6.pdf

On Calderon s conjecture

$L^p$ estimates on the bilinear Hilbert transform for $2 < p < \infty$

A note on the almost sure central limit theorem

We give a simplified proof that the Carleson operator is of weak type (2, 2). This estimate is the main ingredient in the proof of Carleson’s theorem on almost everywhere convergence of Fourier series of functions in L2([0, 1]).

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2000/0007/0004/MRL-2000-0007-0004-a001.pdf

Michael T. Lacey

Papers

The solution of the Kato square root problem for second order elliptic operators on Rn

On Calderon s conjecture

$L^p$ estimates on the bilinear Hilbert transform for $2 < p < \infty$

A note on the almost sure central limit theorem

A proof of boundedness of the Carleson operator