Analysis in Banach Spaces: Volume II: Probabilistic Methods and Operator Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 67)

This second volume of Analysis in Banach Spaces, Probabilistic Methods and Operator Theory, is the successor to Volume I, Martingales and Littlewood-Paley Theory. It presents a thorough study of the fundamental randomisation techniques and the operator-theoretic aspects of the theory. The first two chapters address the relevant classical background from the theory of Banach spaces, including notions like type, cotype, K-convexity and contraction principles. In turn, the next two chapters provide a detailed treatment of the theory of R-boundedness and Banach space valued square functions developed over the last 20 years. In the last chapter, this content is applied to develop the holomorphic functional calculus of sectorial and bi-sectorial operators in Banach spaces. Given its breadth of coverage, this book will be an invaluable reference to graduate students and researchers interested in functional analysis, harmonic analysis, spectral theory, stochastic analysis, and the operator-theoretic approach to deterministic and stochastic evolution equations. Content: Intro Preface Contents Symbols and notations Standing assumptions Random sums 6.1 Basic notions and estimates 6.1.a Symmetric random variables and randomisation 6.1.b Kahaneâ#x80 #x99 s contraction principle 6.1.c Norm comparison of different random sums 6.1.d Covariance domination for Gaussian sums 6.2 Comparison of different Lp-norms 6.2.a The discrete heat semigroup and hypercontractivity 6.2.b Kahaneâ#x80 #x93 Khintchine inequalities 6.2.c End-point bounds related to p=0 and q= 6.3 The random sequence spaces p(X) and p(X) 6.3.a Coincidence with square function spaces when X=Lq. 6.3.b Dual and bi-dual of Np(X) and Np(X)6.4 Convergence of random series 6.4.a ItÃá̂#x80 #x93 Nisio equivalence of different modes of convergence 6.4.b Boundedness implies convergence if and only if c0X 6.5 Comparison of random sums and trigonometric sums 6.6 Notes Type, cotype, and related properties 7.1 Type and cotype 7.1.a Definitions and basic properties 7.1.b Basic examples 7.1.c Type implies cotype 7.1.d Type and cotype for general random sequences 7.1.e Extremality of Gaussians in (co)type 2 spaces 7.2 Comparison theorems under finite cotype 7.2.a Summing operators. 7.2.b Pisierâ#x80 #x99 s factorisation theorem7.2.c Contraction principle with function coefficients 7.2.d Equivalence of cotype and Gaussian cotype 7.2.e Finite cotype in Banach lattices 7.3 Geometric characterisations 7.3.a Kwapienâ#x80 #x99 s characterisation of type and cotype 2 7.3.b Maureyâ#x80 #x93 Pisier characterisation of non-trivial (co)type 7.4 K-convexity 7.4.a Definition and basic properties 7.4.b K-convexity and type 7.4.c K-convexity and duality of the spaces ""pN(X) 7.4.d K-convexity and interpolation 7.4.e K-convexity with respect to general random variables. 7.5 Contraction principles for double random sums7.5.a Pisierâ#x80 #x99 s contraction property 7.5.b The triangular contraction property 7.5.c Duality and interpolation 7.5.d Gaussian version of Pisierâ#x80 #x99 s contraction property 7.5.e Double random sums in Banach lattices 7.6 Notes R-boundedness 8.1 Basic theory 8.1.a Definition and comparison with related notions 8.1.b Testing R-boundedness with distinct operators 8.1.c First examples: multiplication and averaging operators 8.1.d R-boundedness versus boundedness on ""p N(X) 8.1.e Stability of R-boundedness under set operations. 8.2 Sources of R-boundedness in real analysis8.2.a Pointwise domination by the maximal operator 8.2.b Inequalities with Muckenhoupt weights 8.2.c Characterisation by weighted inequalities in Lp 8.3 Fourier multipliers and R-boundedness 8.3.a Multipliers of bounded variation on the line 8.3.b The Marcinkiewicz multiplier theorem on the line 8.3.c Multipliers of bounded rectangular variation 8.3.d The product-space multiplier theorem 8.3.e Necessity of Pisierâ#x80 #x99 s contraction property 8.4 Sources of R-boundedness in operator theory 8.4.a Duality and interpolation. "The present volume develops the theory of integration in Banach spaces, martingales and UMD spaces, and culminates in a treatment of the Hilbert transform, Littlewood-Paley theory and the vector-valued Mihlin multiplier theorem. Over the past fifteen years, motivated by regularity problems in evolution equations, there has been tremendous progress in the analysis of Banach space-valued functions and processes. The contents of this extensive and powerful toolbox have been mostly scattered around in research papers and lecture notes. Collecting this diverse body of material into a unified and accessible presentation fills a gap in the existing literature. The principal audience that we have in mind consists of researchers who need and use Analysis in Banach Spaces as a tool for studying problems in partial differential equations, harmonic analysis, and stochastic analysis. Self-contained and offering complete proofs, this work is accessible to graduate students and researchers with a background in functional analysis or related areas."--Publisher's description for v. 1