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 thought and the vector-valued Mihlin multiplier theorem.
Over the prior fifteen years, stimulated via regularity difficulties in evolution equations, there was large growth within the research of Banach space-valued features and procedures.
The contents of this broad and strong toolbox were regularly scattered round in study papers and lecture notes. amassing this different physique of fabric right into a unified and available presentation fills a niche within the present literature. The primary 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 delivering entire proofs, this paintings is available to graduate scholars and researchers with a history in useful research or comparable areas.
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Additional info for Analysis in Banach Spaces : Volume I: Martingales and Littlewood-Paley Theory
Suppose g is as stated in the proposition. Since g(s, ·) = F (s) in Lp (T ; X) for almost all s ∈ S, for almost all s ∈ S we have g(s, t) = f (s, t) for almost all t ∈ T . Hence, by Fubini’s theorem, g(s, t) = f (s, t) for (µ × ν)-almost all (s, t) ∈ S × T . 26. If ν is σ-finite, the result also holds for p = ∞. 29. We continue with a criterion for separability of Bochner spaces. We shall need the following terminology. 27. A measure space (S, A , µ) is called: (a) countably generated, if there exists a sequence (Sn )n 1 in A which generates A .
If f : S → L (X, Y ) and g : S → L (Y, Z) are strongly (µ-)measurable, then g ◦ f : S → L (X, Z) is strongly (µ-)measurable. Proof. Let x ∈ X be fixed. Since s → f (s)x is strongly (µ-)measurable, the preceding proposition shows that s → gf (s)x is strongly (µ-)measurable for all x ∈ X, so that g ◦ f is strongly (µ-)measurable. 2 Integration In this section we discuss the vector-valued extension of the Lebesgue integral, the so-called Bochner integral. At various places in this book we will also need its ‘weak’ companion, the Pettis integral.
Occasionally we will write, when F is a sub-σ-algebra of A , Lp (S, F ; X) for the Lp -space with respect to the measure space (S, F , µ|F ). It coincides with the closed linear subspace of Lp (S; X) consisting of all equivalence classes of functions with a representative that is strongly µ-measurable with respect to F . We write Lp (S, F ) := Lp (S, F ; K). 17. For a strongly µ-measurable function f : S → X the following assertions are equivalent: (1) f ∈ L∞ (S; X); (2) f, x∗ ∈ L∞ (S) for all x∗ ∈ X ∗ .
Analysis in Banach Spaces : Volume I: Martingales and Littlewood-Paley Theory by Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis