Perturbations and sums of operators

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.

Function spaces

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

Extended calculi and powers of operators

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...

Dyadic operators and the T (1) theorem

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...

Nonlinear parabolic evolution equations in critical spaces

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.

Singular integral operators

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.

The Fourier transform and multipliers

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

Pointwise Properties of Martingales with Values in Banach Function Spaces

In this paper we consider local martingales with values in a UMD Banach function space. We prove that such martingales have a version which is a martingale field. Moreover, a new Burkholder–Davis–Gundy type inequality is obtained.

The UMD property for musielak–orlicz spaces

In this paper we show that Musielak–Orlicz spaces are UMD spaces under the so-called Δ₂ condition on the generalized Young function and its complemented function. We also prove that if the measure space is divisible, then a Musielak–Orlicz space has the UMD property if and only if it is reflexive. As a consequence we show that...