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Preface; Contents; Symbols and notations; Standing assumptions; 1 Bochner spaces; 1.1 Measurability; 1.1.a Functions on a measurable space (S; A); 1.1.b Functions on a measure space (S; A ;); 1.1.c Operator-valued functions; 1.2 Integration; 1.2.a The Bochner integral; 1.2.c The Pettis integral; 1.3 Duality of Bochner spaces; 1.3.a Elementary duality results; 1.3.b Duality and the Radon-Nikodým property; 1.3.c More about the Radon-Nikodým property; 1.4 Notes; 2 Operators on Bochner spaces; 2.1 The Lp-extension problem ; 2.1.a Boundedness of T IX for positive operators T.
2.1.b Boundedness of T IH for Hilbert spaces H2.1.c Counterexamples; 2.2 Interpolation of Bochner spaces; 2.2.a The Riesz-Thorin interpolation theorem; 2.2.b The Marcinkiewicz interpolation theorem; 2.2.c Complex interpolation of the spaces Lp(S; X); 2.2.d Real interpolation of the spaces Lp(S; X); 2.3 The Hardy-Littlewood maximal operator; 2.3.a Lebesgue points and differentiation; 2.3.b Convolutions and approximation; 2.4 The Fourier transform; 2.4.a The inversion formula and Plancherel's theorem; 2.4.b Fourier type; 2.4.c The Schwartz class S(Rd; X).
2.4.d The space of tempered distributions S0(RdX); 2.5 Sobolev spaces and differentiability; 2.5.a Weak derivatives; 2.5.b The Sobolev spaces Wk; p(D; X); 2.5.c Almost everywhere differentiability; 2.5.d The fractional Sobolev spaces Ws; p(Rd; X); 2.6 Conditional expectations; 2.6.a Uniqueness; 2.6.b Existence; 2.6.c Conditional limit theorems; 2.6.d Inequalities and identities; 2.7 Notes; 3 Martingales; 3.1 Definitions and basic properties; 3.1.a Difference sequences; 3.1.b Paley-Walsh martingales; 3.1.c Stopped martingales; 3.2 Martingale inequalities; 3.2.a Doob's maximal inequalities.
3.2.b Rademacher variables and contraction principles3.2.c John-Nirenberg and Kahane-Khintchine inequalities; 3.2.d Applications to inequalities on; 3.3 Martingale convergence; 3.3.a Forward convergence; 3.3.b Backward convergence; 3.3.c The Itô-Nisio theorem for martingales; 3.3.d Martingale convergence and the RNP; 3.4 Martingale decompositions; 3.4.a Gundy decomposition; 3.4.b Davis decomposition; 3.5 Martingale transforms; 3.5.a Basic properties; 3.5.b Extrapolation of Lp-inequalities; 3.5.c End-point estimates in L1 ; 3.5.d Martingale type and cotype.
3.6 Approximate models for martingales3.6.a Universality of Paley-Walsh martingales; 3.6.b The Rademacher maximal function; 3.6.c Approximate models for martingale transforms; 3.7 Notes; 4 UMD spaces; 4.1 Motivation; 4.1.a Square functions for martingale difference sequences; 4.1.b Unconditionality; 4.2 The UMD property; 4.2.a Definition and basic properties; 4.2.b Unconditionality of the Haar decomposition; 4.2.c Examples and constructions; 4.2.d Stein's inequality for conditional expectations; 4.2.e Boundedness of martingale transforms; 4.3 Banach space properties implied by UMD.
2.1.b Boundedness of T IH for Hilbert spaces H2.1.c Counterexamples; 2.2 Interpolation of Bochner spaces; 2.2.a The Riesz-Thorin interpolation theorem; 2.2.b The Marcinkiewicz interpolation theorem; 2.2.c Complex interpolation of the spaces Lp(S; X); 2.2.d Real interpolation of the spaces Lp(S; X); 2.3 The Hardy-Littlewood maximal operator; 2.3.a Lebesgue points and differentiation; 2.3.b Convolutions and approximation; 2.4 The Fourier transform; 2.4.a The inversion formula and Plancherel's theorem; 2.4.b Fourier type; 2.4.c The Schwartz class S(Rd; X).
2.4.d The space of tempered distributions S0(RdX); 2.5 Sobolev spaces and differentiability; 2.5.a Weak derivatives; 2.5.b The Sobolev spaces Wk; p(D; X); 2.5.c Almost everywhere differentiability; 2.5.d The fractional Sobolev spaces Ws; p(Rd; X); 2.6 Conditional expectations; 2.6.a Uniqueness; 2.6.b Existence; 2.6.c Conditional limit theorems; 2.6.d Inequalities and identities; 2.7 Notes; 3 Martingales; 3.1 Definitions and basic properties; 3.1.a Difference sequences; 3.1.b Paley-Walsh martingales; 3.1.c Stopped martingales; 3.2 Martingale inequalities; 3.2.a Doob's maximal inequalities.
3.2.b Rademacher variables and contraction principles3.2.c John-Nirenberg and Kahane-Khintchine inequalities; 3.2.d Applications to inequalities on; 3.3 Martingale convergence; 3.3.a Forward convergence; 3.3.b Backward convergence; 3.3.c The Itô-Nisio theorem for martingales; 3.3.d Martingale convergence and the RNP; 3.4 Martingale decompositions; 3.4.a Gundy decomposition; 3.4.b Davis decomposition; 3.5 Martingale transforms; 3.5.a Basic properties; 3.5.b Extrapolation of Lp-inequalities; 3.5.c End-point estimates in L1 ; 3.5.d Martingale type and cotype.
3.6 Approximate models for martingales3.6.a Universality of Paley-Walsh martingales; 3.6.b The Rademacher maximal function; 3.6.c Approximate models for martingale transforms; 3.7 Notes; 4 UMD spaces; 4.1 Motivation; 4.1.a Square functions for martingale difference sequences; 4.1.b Unconditionality; 4.2 The UMD property; 4.2.a Definition and basic properties; 4.2.b Unconditionality of the Haar decomposition; 4.2.c Examples and constructions; 4.2.d Stein's inequality for conditional expectations; 4.2.e Boundedness of martingale transforms; 4.3 Banach space properties implied by UMD.