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2016 | Buch

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Analysis in Banach Spaces

Volume I: Martingales and Littlewood-Paley Theory

verfasst von: Prof. Tuomas Hytönen, Jan van Neerven, Prof. Mark Veraar, Prof. Dr. Lutz Weis

Verlag: Springer International Publishing

Buchreihe : Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The present volume develops the theory of integration in Banach spaces, martingales and UMD spaces, and culminates in a treatment of the Hilbert transform, Littlewood-Paley theory and the vector-valued Mihlin multiplier theorem.

Over the past fifteen years, motivated by regularity problems in evolution equations, there has been tremendous progress in the analysis of Banach space-valued functions and processes.

The contents of this extensive and powerful toolbox have been mostly scattered around in research papers and lecture notes. Collecting this diverse body of material into a unified and accessible presentation fills a gap in the existing literature. The principal audience that we have in mind consists of researchers who need and use Analysis in Banach Spaces as a tool for studying problems in partial differential equations, harmonic analysis, and stochastic analysis. Self-contained and offering complete proofs, this work is accessible to graduate students and researchers with a background in functional analysis or related areas.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Bochner spaces

Abstract

This chapter sets up the general framework in which we work throughout these volumes. After introducing the relevant notions of measurability for functions taking values in a Banach space, we proceed to define the Bochner integral and the Bochner spaces L ^p(S;X), which are the vector-valued counterparts of the Lebesgue integral and the classical L ^p-spaces, respectively. We also briefly discuss the weaker Pettis integral. The chapter concludes with a detailed investigation of duality of the Bochner spaces and the related Radon–Nikodým property.

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

Chapter 2. Operators on Bochner spaces

Abstract

Here we bring in the colourful cast of operators acting on the spaces that we study. The first two sections have a general character: we discuss the fundamental problem of extending an operator from an L ^p-space to the corresponding Bochner space, and study different interpolation techniques for operators on these spaces. The next three sections are concerned with distinguished particular operators of classical analysis (the Hardy–Littlewood maximal operator, the Fourier transform, and partial derivatives) on Bochner spaces L ^p(ℝ ^d;X) with a Euclidean domain. In the final section, returning to abstract measure spaces, we develop the theory of conditional expectations, a necessary prerequisite for the discussion of martingales in the subsequent chapter.

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

Chapter 3. Martingales

Abstract

This chapter is a lengthy introduction to the general theory of discrete-time martingales, biased towards the needs of the subsequent chapters. Throughout, we discuss martingales with a σ-finite measure space domain and a Banach space range, but the development is self-contained and does not assume any results from the conventional set-up of a probability space domain and/or a real-valued range. Each section is devoted to a different aspect of the martingale theory, including norm inequalities, convergence results, decompositions and transforms, and norm approximation of general martingales by simpler ones. This systematic development of the general martingale theory will substantially streamline the discussion of UMD spaces in the subsequent chapter.

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

Chapter 4. UMD spaces

Abstract

After motivation from classical inequalities for real-valued martingales on the one hand, and the general theory of unconditionality in Banach spaces on the other hand, we introduce the central notion of UMD (unconditional martingale differences) Banach spaces. This chapter is primarily concerned with probabilistic implications and equivalencies of the UMD property, leaving the (harmonic) analytic side of the theory for the next chapter. We also show that UMD spaces enjoy reflexivity, among various other nontrivial Banach spaces properties, and we prove Burkholder's characterisation of the UMD property by suitable concave functions, which leads to sharp constants in the fundamental martingale inequalities in the scalar-valued case.

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

Chapter 5. Hilbert transform and Littlewood–Paley theory

Abstract

In this final chapter, we connect the probabilistic notion of UMD spaces with various norm inequalities of harmonic analysis. We begin with the equivalence of the UMD property of X with the boundedness of the classical Hilbert transform on L ^p(ℝ;X) and proceed to more general vector-valued Fourier multiplier operators and Littlewood–Paley inequalities, including versions in several variables and in the periodic setting. Applications of the general theory are illustrated by two independent sections dealing with operators on Schatten classes, and interpolation of Sobolev spaces.

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

Backmatter

Titel: Analysis in Banach Spaces
verfasst von: Prof. Tuomas Hytönen
Jan van Neerven
Prof. Mark Veraar
Prof. Dr. Lutz Weis
Copyright-Jahr: 2016
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-48520-1
Print ISBN: 978-3-319-48519-5
DOI: https://doi.org/10.1007/978-3-319-48520-1

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