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
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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
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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
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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
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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
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s characterisation of type and cotype 2
7.3.b Maureyâ#x80
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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
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s contraction property
7.5.b The triangular contraction property
7.5.c Duality and interpolation
7.5.d Gaussian version of Pisierâ#x80
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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
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s contraction property
8.4 Sources of R-boundedness in operator theory
8.4.a Duality and interpolation.