Skip to main content
main-content

Über dieses Buch

This book targets graduate students and researchers who want to learn about Lebesgue spaces and solutions to hyperbolic equations. It is divided into two parts.

Part 1 provides an introduction to the theory of variable Lebesgue spaces: Banach function spaces like the classical Lebesgue spaces but with the constant exponent replaced by an exponent function. These spaces arise naturally from the study of partial differential equations and variational integrals with non-standard growth conditions. They have applications to electrorheological fluids in physics and to image reconstruction. After an introduction that sketches history and motivation, the authors develop the function space properties of variable Lebesgue spaces; proofs are modeled on the classical theory. Subsequently, the Hardy-Littlewood maximal operator is discussed. In the last chapter, other operators from harmonic analysis are considered, such as convolution operators and singular integrals. The text is mostly self-contained, with only some more technical proofs and background material omitted.

Part 2 gives an overview of the asymptotic properties of solutions to hyperbolic equations and systems with time-dependent coefficients. First, an overview of known results is given for general scalar hyperbolic equations of higher order with constant coefficients. Then strongly hyperbolic systems with time-dependent coefficients are considered. A feature of the described approach is that oscillations in coefficients are allowed. Propagators for the Cauchy problems are constructed as oscillatory integrals by working in appropriate time-frequency symbol classes. A number of examples is considered and the sharpness of results is discussed. An exemplary treatment of dissipative terms shows how effective lower order terms can change asymptotic properties and thus complements the exposition.

Inhaltsverzeichnis

Frontmatter

Introduction to the Variable Lebesgue Spaces

Frontmatter

Chapter 1. Introduction and Motivation

Abstract
We begin with an intuitive introduction to the variable Lebesgue spaces, briefly sketch their history, and give some of the contemporary motivations for studying these spaces.
David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth

Chapter 2. Properties of Variable Lebesgue Spaces

Abstract
In this chapter we develop the function space properties of variable Lebesgue spaces. We begin with the basic properties and notation for exponent functions. We then define the modular and the norm, and prove that L p(.) is a Banach space. We prove a version of Hölder’s inequality, define the associate norm, and then characterize the dual space when p + < ∞. We conclude with a version of the Lebesgue differentiation theorem.
David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth

Chapter 3. The Hardy–Littlewood Maximal Operator

Abstract
In this chapter we turn to the study of harmonic analysis on the variable Lebesgue spaces. Our goal is to establish sufficient conditions for the Hardy–Littlewood maximal operator to be bounded on L p(.); in the next chapter we will show how this can be used to prove norm inequalities on L p(.) for the other classical operators of harmonic analysis. We begin with a brief review of the maximal operator on the classical Lebesgue spaces and introduce our principal tool, the Calderón–Zygmund decomposition.
David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth

Chapter 4. Extrapolation in Variable Lebesgue Spaces

Abstract
In this chapter we develop a general theory for proving norm inequalities for the other classical operators in harmonic analysis. Our main result is a powerful generalization of the Rubio de Francia extrapolation theorem. This approach, first developed in [22] and then treated as part of a more general framework in [27], lets us use the theory of weighted norm inequalities to prove the corresponding estimates in variable Lebesgue spaces. This greatly reduces the work required, since it lets us use the well-developed theory of weights.
David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth

Backmatter

Asymptotic Behaviour of Solutions to Hyperbolic Equations and Systems

Frontmatter

Chapter 1. Introduction

Abstract
These notes provide an introduction to and a survey on recent results about the long-time behaviour of solutions to systems of hyperbolic partial differential equations with time-dependent coefficients. Particular emphasis is given to questions about the sharpness of estimates.
David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth

Chapter 2. Equations with constant coefficients

Abstract
Before we pursue the analysis of equations and systems with time-dependent coefficients, it is instructive to understand what happens in the case of equations with constant coefficients. One of the very helpful observations available in this case is that after a Fourier transform in the spatial variable x we obtain an ordinary differential equation with constant coefficients which can be solved almost explicitly once we know its characteristics. This works well for frequencies where the characteristics are simple. If they become multiple, the representation breaks down and other methods are required. In the presentation of this part we follow [50], to which we refer for the detailed arguments and complete proofs of the material in this chapter.
David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth

Chapter 3. Some interesting model cases

Abstract
We now look at equations with time-dependent coefficients. In this chapter we will review two scale invariant model cases, which can be treated by means of special functions. They both highlight a structural change in the behaviour of solutions when lower order terms become effective. This change is a true variable coefficient phenomenon, it can not arise for equations with constant coefficients as treated in Chapter 2.
David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth

Chapter 4. Time-dependent hyperbolic systems

Abstract
In this chapter we will provide a diagonalisation based approach to obtain the high-frequency asymptotic properties of the representation of solutions for more general uniformly strictly hyperbolic systems. The exposition is based on ideas from the authors’ paper [52].
David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth

Chapter 5. Effective lower order perturbations

Abstract
If lower order terms are too large to be controlled, it becomes important to investigate the behaviour of solutions for bounded frequencies. We will restrict ourselves to situations where an asymptotic construction for ξ → 0 becomes important and provide some essential estimates for this.
David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth

Chapter 6. Examples and counter-examples

Abstract
Both in Chapters 4 and 5 we made symbol like assumptions on coefficients, e.g., we considered hyperbolic systems
$${\rm{D}}_tU = {\sum \limits_{k=1}^{n}}A_k(t){{\rm D}_{x_k}}U$$
(6.0.1)
with coefficient matrices \(A_k(t)\in\mathcal{T}\left\{{0}\right\}\), meaning that derivatives of the coefficients are controlled by
$$\|{\rm D}_t^lA_k(t)\|\leq \, C_l \, \left(\frac {1} {1+t}\right)^l.$$
(6.0.2)
David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth

Chapter 7. Related topics

Abstract
Most of the results presented here were based on diagonalisation procedures in order to deduce asymptotic information on the representations of solutions. This is natural and has a long history in the study of hyperbolic equations and coupled systems. For diagonalisation schemes in broader sense and their application we also refer to [21]. Some more applications are discussed there too.
David Cruz-Uribe, Alberto Fiorenza, Michael Ruzhansky, Jens Wirth

Backmatter

Weitere Informationen

Premium Partner

    Bildnachweise