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

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.

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.

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

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.

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.

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 l²_n 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 L^p for 1 < p < ∞ with a bounded H^∞-calculus with optimal angle ω_H^∞(A) > 0.

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.