Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis's Analysis in Banach Spaces : Volume I: Martingales and PDF

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

ISBN-10: 3319485199

ISBN-13: 9783319485195

ISBN-10: 3319485202

ISBN-13: 9783319485201

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

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

The contents of this vast and robust toolbox were generally scattered round in learn papers and lecture notes. accumulating this various physique of fabric right into a unified and obtainable presentation fills a spot within the latest literature. The important viewers that we have 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 delivering entire proofs, this paintings is on the market to graduate scholars and researchers with a history in useful research or similar areas.

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

Example text

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. If c pN 1, then ψc := n=1 |cn |1An defines an element of Lp (S; X) of norm 1 and N N |cn | n=1 x∗An |cn | xAn , x∗An = ε + ψc , φ ε+ ε+ φ . n=1 Since ε > 0 was arbitrary, combination of the above inequalities gives N x∗An q φ q. n=1 Since N proof. 1 and A1 , . . 1 it has been established that if 1 p ∞ and p1 + 1q = 1, then every function g ∈ Lq (S; X ∗ ) determines a functional φg ∈ (Lp (S; X))∗ by the formula ˆ f, φg = f (s), g(s) dµ(s).

17). Our definition is prompted by the observation that no meaningful definition of the Bochner integral of a simple function f can be given if f is supported on sets of infinite measure. In the scalar-valued setting, this problem is usually circumvented by defining the Lebesgue integral first for non-negative simple functions, in which case the integral is allowed to take the value ∞. In the present vector-valued setting we are forced to working with µ-simple functions only. In view of this, our definition of L∞ (S; X) is the most natural one.

This vector does not belong to c0 and as a consequence f fails to be Pettis integrable. This example also explains the need of considering arbitrary sets A ∈ A in the definition of the Pettis integral and not just the set S. 39 and define g : S → c0 by g(t) := f (t) for t ´∈ (0, 1) and g(t) := −f (t − 1) for t ∈ (1, 2). Then the weak integral τ (c0 , l1 )- S g(t) dt equals 0 and therefore belongs to c0 , whereas both ´1 ´2 τ (c0 , l1 )- 0 g(t) dt and τ (c0 , l1 )- 1 g(t) dt belong to l∞ \ c0 . 39 is essentially the only example of a strongly measurable function which is weakly integrable but not Pettis integrable.

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Analysis in Banach Spaces : Volume I: Martingales and Littlewood-Paley Theory by Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis

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