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 rework, Littlewood-Paley concept and the vector-valued Mihlin multiplier theorem.

Over the prior fifteen years, stimulated via regularity difficulties in evolution equations, there was super development within the research of Banach space-valued features and approaches.

The contents of this large and robust toolbox were often scattered round in study papers and lecture notes. accumulating this different physique of fabric right into a unified and available presentation fills a niche within the present literature. The relevant viewers that we've 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 out there to graduate scholars and researchers with a historical past 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 resources for Analysis in Banach Spaces : Volume I: Martingales and Littlewood-Paley Theory**

**Example text**

29 below will show that such a subspace is always contained in a subspace of the form Lp (T, B , ν|B ; Y ) for some sub-σ-algebra B ⊆ B, with ν|B σ-finite, and a separable closed subspace Y of X. Admitting this result for the moment, it follows that there is no loss of generality in assuming that also ν is σ-finite. Step 2 – We first assume that ν is a finite measure. Then we may view F as a Bochner integrable function with values in L1 (T ; X). 24. By this proposition, f satisfies (i). 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) < ∞.

Suppose that f : S → X is a function with the property that f, x∗ belongs to L1 (S) for all x∗ ∈ Y . Then Tf x∗ := f, x∗ , x∗ ∈ Y defines a bounded linear operator Tf : Y → L1 (S). 2 Integration 23 Proof. It is clear that Tf is well defined and linear. We claim that it is also closed, from which the boundedness follows by the closed graph theorem. Suppose that limn→∞ x∗n = x∗ in Y and limn→∞ Tf x∗n = g in Lp (S). By passing to a subsequence (if p ∈ [1, ∞); this is not needed for p = ∞) we may assume that limn→∞ Tf x∗n = g almost everywhere on S.

29 below will show that such a subspace is always contained in a subspace of the form Lp (T, B , ν|B ; Y ) for some sub-σ-algebra B ⊆ B, with ν|B σ-finite, and a separable closed subspace Y of X. Admitting this result for the moment, it follows that there is no loss of generality in assuming that also ν is σ-finite. Step 2 – We first assume that ν is a finite measure. Then we may view F as a Bochner integrable function with values in L1 (T ; X). 24. By this proposition, f satisfies (i). 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) < ∞.