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 therapy of the Hilbert remodel, Littlewood-Paley idea and the vector-valued Mihlin multiplier theorem.

Over the earlier fifteen years, stimulated through regularity difficulties in evolution equations, there was super growth within the research of Banach space-valued services and methods.

The contents of this vast and robust toolbox were often scattered round in learn papers and lecture notes. amassing this various physique of fabric right into a unified and obtainable presentation fills a niche within the latest literature. The important viewers that we've got in brain contains researchers who desire 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 entire proofs, this paintings is on the market to graduate scholars and researchers with a history in sensible research or similar areas.

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

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

**Example text**

28. Let (S, A ) be a measurable space (respectively, (S, A , µ) a measure space) and let X and Y be Banach spaces. If f : S → X and g : S → L (X, Y ) are strongly (µ-)measurable, then gf : S → Y is strongly (µ-)measurable. Proof. By assumption there exists a sequence (fn )n 1 of (µ-)simple functions converging pointwise to f (µ-almost everywhere). The functions gfn are strongly (µ-)measurable and satisfy limn→∞ gfn → gf pointwise (µ-almost everywhere). 23) now implies the strong (µ-)measurability of gf .

Here, A∆A = (A \ A ) ∪ (A \ A) = (A ∪ A ) \ (A ∩ A ) is the symmetric difference of A and A . 28. , by all balls with rational radius centred at point with rational coordinates), but the Lebesgue σ-algebra of Rd (the completion of the Borel σ-algebra) is only λ-countably generated, where λ is the Lebesgue measure. For A ∈ A we denote A |A = {A ∩ B : B ∈ A } = {B ∈ A : B ⊆ A}. This is a σ-algebra in A. The restriction of µ to A |A is denoted by µ|A . 29 (Separability of Bochner spaces). Let (S, A , µ) be a measure space, let 1 p < ∞, and let X be a Banach space.

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 . Hence, for almost all t ∈ T , ˆ ˆ F dµ (t) = lim Fn dµ (t) n→∞ S S ˆ ˆ = lim fn (s, t) dµ(s) = f (s, t) dµ(s).