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

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

Over the previous fifteen years, inspired by way of regularity difficulties in evolution equations, there was great growth within the research of Banach space-valued services and approaches.

The contents of this broad and robust toolbox were typically scattered round in examine papers and lecture notes. gathering this assorted physique of fabric right into a unified and obtainable presentation fills a spot within the present literature. The relevant 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 obtainable to graduate scholars and researchers with a heritage in useful research or comparable areas.

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

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

**Sample text**

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. Proof. By strong measurability we may assume X to be separable. 9. The assumption of the proposition implies that ˆ ˆ 1 1 φi ◦ f dµ = φi f dµ 0 µ(A) A µ(A) A for all i and all A ∈ A with 0 < µ(A) < ∞. But this implies that φi ◦ f almost everywhere for all i, and thus f ∈ C µ-almost everywhere. 0 The next result gives a useful criterion for testing when two integrable functions are equal almost everywhere.

45. Let f : S ´ → X be Pettis integrable and let C ⊆ X be 1 closed and convex. If µ(A) (P ) - A f dµ ∈ C for all A ∈ A with 0 < µ(A) < ∞, then f (s) ∈ C for almost all s ∈ S. 3 Duality of Bochner spaces One of the cornerstones of the theory of classical Lebesgue spaces is the fact that Lp (S) provides representations of all bounded linear functionals on Lp (S) when p ∈ [1, ∞), p1 + p1 = 1, and (S, A , µ) is σ-finite. 3 Duality of Bochner spaces 37 the Bochner spaces Lp (S; X), the extent to which this duality remains valid turns out to depend on the Banach space X, and leads to another example (cf.

Clearly we have x∗A φ . This concludes the proof for p = 1 in case µ is σ-finite, for then A is countable and φ is represented by the function g = A∈A 1A ⊗x∗A in L∞ (S; X ∗ ). Note that in the absence of σ-finiteness g would fail to be strongly µ-measurable. Suppose now that 1 < p < ∞. We must show that A∈A x∗A q < ∞, for then φ is represented by the function g = A∈A 1A ⊗ x∗A in Lq (S; X ∗ ). Fix N 1 and select atoms A1 , . . , AN ∈ A. We have N x∗An q N 1/q = sup c n=1 p N 1 n=1 |cn | x∗An . Fix ε > 0 and choose elements xAn ∈ X of norm one such that xAn , x∗An N x∗An − ε/N.