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 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.

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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).

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