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.@en

This second volume of Analysis in Banach Spaces, Probabilistic Methods and Operator Theory, is the successor to Volume I, Martingales and Littlewood-Paley Theory. It presents a thorough study of the fundamental randomisation techniques and the operator-theoretic aspects of the theory. The first two chapters address the relevant classical background from the theory of Banach spaces, including notions like type, cotype, K-convexity and contraction principles. In turn, the next two chapters provide a detailed treatment of the theory of R-boundedness and Banach space valued square functions developed over the last 20 years. In the last chapter, this content is applied to develop the holomorphic functional calculus of sectorial and bi-sectorial operators in Banach spaces. Given its breadth of coverage, this book will be an invaluable reference to graduate students and researchers interested in functional analysis, harmonic analysis, spectral theory, stochastic analysis, and the operator-theoretic approach to deterministic and stochastic evolution equations. @en

This third volume of Analysis in Banach Spaces offers a systematic treatment of Banach space-valued singular integrals, Fourier transforms, and function spaces. It further develops and ramifies the theory of functional calculus from Volume II and describes applications of these new notions and tools to the problem of maximal regularity of evolution equations. The exposition provides a unified treatment of a large body of results, much of which has previously only been available in the form of research papers. Some of the more classical topics are presented in a novel way using modern techniques amenable to a vector-valued treatment. Thanks to its accessible style with complete and detailed proofs, this book will be an invaluable reference for researchers interested in functional analysis, harmonic analysis, and the operator-theoretic approach to deterministic and stochastic evolution equations.@en

In this chapter, we complement the discussion of three major themes of Fourier analysis that we have studied in the previous Volumes.@en

This chapter presents an in-depth study of several classes of vector-valued function spaces defined by smoothness conditions.@en

In this chapter we address a couple of topics in the theory of H∞-calculus centering around the question what can be said about an operator of the form A+B when A and B have certain “good” properties such as being (R-)sectorial or admitting a bounded H∞-calculus.@en

In this chapter we address two strongly interwoven topics: How to verify the boundedness of the H∞-calculus of an operator and how to represent and estimate its fractional powers. For concrete operators such as the Laplace operator or elliptic partial differential operators, the fractional domain spaces can often be identifed with certain function spaces considered in Chapter 14 and the imaginary powers of the operator are related to singular integral and pseudo-differential operators treated in Chapters 11 and 13.@en

Before addressing this question for the Calderón{Zygmund type operators of the kind studied in Chapter 11, we investigate a number of related objects in a simpler dyadic model. Besides serving as an introduction to some of the key techniques, it turns out that these dyadic operators can be, and will be, also used as building blocks of the proper singular integral operators towards the end of the chapter.@en

The mapping properties of T will of course heavily depend on the assumptions made on the kernel K that we will discuss in more detail in this chapter.@en

As we have seen in the preceding sections, in the context of inhomogeneous linear evolution equations, maximal regularity enables one to set up an isomorphism between the space of data (initial value and inhomogeneity) and the solution space.@en

We present a general method to extend results on Hilbert space operators to the Banach space setting by representing certain sets of Banach space operators Γ on a Hilbert space. Our assumption on Γ is expressed in terms of α-boundedness for a Euclidean structure α on the underlying Banach space X. This notion is originally motivated by R- or γ-boundedness of sets of operators, but for example any operator ideal from the Euclidean space l2n to X defines such a structure. Therefore, our method is quite flexible. Conversely we show that Γ has to be α-bounded for some Euclidean structure α to be representable on a Hilbert space. By choosing the Euclidean structure α accordingly, we get a unified and more general approach to the Kwapień-Maurey factorization theorem and the factorization theory of Maurey, Nikišin and Rubio de Francia. This leads to an improved version of the Banach function space-valued extension theorem of Rubio de Francia and a quantitative proof of the boundedness of the lattice Hardy-Littlewood maximal operator. Furthermore, we use these Euclidean structures to build vectorvalued function spaces. These enjoy the nice property that any bounded operator on L2 extends to a bounded operator on these vector-valued function spaces, which is in stark contrast to the extension problem for Bochner spaces. With these spaces we define an interpolation method, which has formulations modelled after both the real and the complex interpolation method. Using our representation theorem, we prove a transference principle for sectorial operators on a Banach space, enabling us to extend Hilbert space results for sectorial operators to the Banach space setting. We for example extend and refine the known theory based on R-boundedness for the joint and operator-valued H∞-calculus. Moreover, we extend the classical characterization of the boundedness of the H∞- calculus on Hilbert spaces in terms of BIP, square functions and dilations to the Banach space setting. Furthermore we establish, via the H∞-calculus, a version of Littlewood-Paley theory and associated spaces of fractional smoothness for a rather large class of sectorial operators. Our abstract setup allows us to reduce assumptions on the geometry of X, such as (co)type and UMD. We conclude with some sophisticated counterexamples for sectorial operators, with as a highlight the construction of a sectorial operator of angle 0 on a closed subspace of Lp for 1 < p < ∞ with a bounded H∞-calculus with optimal angle ωH∞(A) > 0.@en

The R  -boundedness of certain families of vector-valued stochastic convolution operators with scalar-valued square integrable kernels is the key ingredient in the recent proof of stochastic maximal L p   -regularity, 2<p<∞  , for certain classes of sectorial operators acting on spaces X=L q (μ)  , 2≤q<∞  . This paper presents a systematic study of R  -boundedness of such families. Our main result generalises the afore-mentioned R  -boundedness result to a larger class of Banach lattices X  and relates it to the ℓ 1   -boundedness of an associated class of deterministic convolution operators. We also establish an intimate relationship between the ℓ 1   -boundedness of these operators and the boundedness of the X  -valued maximal function. This analysis leads, quite surprisingly, to an example showing that R  -boundedness of stochastic convolution operators fails in certain UMD Banach lattices with type 2  .@en

It is well-known that in Banach spaces with finite cotype, the R-bounded and γ-bounded families of operators coincide. If in addition X is a Banach lattice, then these notions can be expressed as square function estimates. It is also clear that R-boundedness implies γ-boundedness. In this note we show that all other possible inclusions fail. Furthermore, we will prove that R-boundedness is stable under taking adjoints if and only if the underlying space is K-convex.@en