Analysis in Banach Spaces : Volume I: Martingales and by Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis

By Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis

The current quantity develops the speculation of integration in Banach areas, martingales and UMD areas, and culminates in a remedy of the Hilbert rework, Littlewood-Paley conception and the vector-valued Mihlin multiplier theorem.

Over the prior fifteen years, encouraged by way of regularity difficulties in evolution equations, there was large growth within the research of Banach space-valued features and strategies.

The contents of this large and strong toolbox were often scattered round in learn papers and lecture notes. gathering this different physique of fabric right into a unified and obtainable presentation fills a spot within the current literature. The significant viewers that we have got in brain involves researchers who want and use research in Banach areas as a device for learning difficulties in partial differential equations, harmonic research, and stochastic research. Self-contained and providing whole proofs, this paintings is available to graduate scholars and researchers with a historical past in sensible research or comparable areas.

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Extra resources for Analysis in Banach Spaces : Volume I: Martingales and Littlewood-Paley Theory

Example text

K. The closure of this set is denoted by conv V . 12. Suppose that µ(S) = 1. If f : S → X is a Bochner integrable function, then ˆ f dµ ∈ conv{f (s) : s ∈ S}. S Proof. By strong measurability we may take X to be separable. 9. Then ˆ ˆ ˆ φi f dµ = φi (f ) dµ 0 dµ = 0 S for all i, and therefore ´ S S S f dµ ∈ C. In the converse direction we have the following result. 13. Let f´ : S → X be Bochner integrable and let C ⊆ X be 1 closed and convex. If µ(A) f dµ ∈ C for all A ∈ A with 0 < µ(A) < ∞, A then f (s) ∈ C for almost all s ∈ S.

For part (2) we may assume that µ(S) < ∞. Fix 0 < ε < 1. We may furthermore assume that X is separable and that f ∞ = 1. Let (xn )n 1 be an ε-net in X. Put An = {s ∈ S : f (s) − xn ε/4} and define B1 = A1 , Bn+1 := n n An+1 \ j=1 Bj . The functions gn := j=1 1Bn ⊗ xj satisfy gn ∞ 1 + ε/4 and sup f (s) − gn (s) ε/4. s∈ n j=1 Bn Therefore the functions fn := (1 + ε/4)−1 gn satisfy fn f (s) − fn (s) sup s∈ n j=1 fn − gn ∞ ∞ 1 and + ε/4 Bn 1− 1 (1 − ε/4) + ε/4 = ε/2. 1 + ε/4 In view of n 1 Bn = n 1 An = S we have µ( n 1 Bn ) ↑n µ(S).

The identity in (ii) is evidently true if F is a linear combination of functions of the form 1A ⊗(1B ⊗x) with µ(A) < ∞ and ν(B) < ∞. The´ general case follows ´ by approximation: if Fn → F in L1 (S; L1 (T ; X)), then S F dµ = limn→∞ S Fn dµ in L1 (T ; X), and by passing to a subsequence we may assume that ˆ ˆ F dµ (t) = lim Fn dµ (t) n→∞ S S for almost all t ∈ T . Also, Fn → F in L1 (S; L1 (T ; X)) implies fn → f in L1 (S × T ; X) and hence, by the Fubini theorem and passing to a further subsequence, fn (·, t) → f (·, t) in L1 (S; X) for almost all t ∈ T .

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