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.

**Read Online or Download Analysis in Banach Spaces : Volume I: Martingales and Littlewood-Paley Theory PDF**

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**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 .