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2019 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Orlicz Spaces and Generalized Orlicz Spaces

verfasst von: Dr. Petteri Harjulehto, Peter Hästö

Verlag: Springer International Publishing

Buchreihe : Lecture Notes in Mathematics

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

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Über dieses Buch

This book presents a systematic treatment of generalized Orlicz spaces (also known as Musielak–Orlicz spaces) with minimal assumptions on the generating Φ-function. It introduces and develops a technique centered on the use of equivalent Φ-functions. Results from classical functional analysis are presented in detail and new material is included on harmonic analysis. Extrapolation is used to prove, for example, the boundedness of Calderón–Zygmund operators. Finally, central results are provided for Sobolev spaces, including Poincaré and Sobolev–Poincaré inequalities in norm and modular forms.

Primarily aimed at researchers and PhD students interested in Orlicz spaces or generalized Orlicz spaces, this book can be used as a basis for advanced graduate courses in analysis.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
We hope that our tools and exposition will aid in generalizing further results from the variable exponent setting to the generalized Orlicz setting.
Petteri Harjulehto, Peter Hästö
Chapter 2. Φ-Functions
Abstract
As mentioned in the introduction, we approach Φ-functions and Orlicz spaces by slightly more robust properties than the commonly used convexity.
Petteri Harjulehto, Peter Hästö
Chapter 3. Generalized Orlicz Spaces
Abstract
In the previous chapter, we investigated properties of Φ-functions. In this chapter, we use them to derive results for function spaces defined by means of Φ-functions.
Petteri Harjulehto, Peter Hästö
Chapter 4. Maximal and Averaging Operators
Abstract
For the rest of the book, we always consider subsets of \({\mathbb {R}^n}\) and the Lebesgue measure. By Ω we always denote an open set in \({\mathbb {R}^n}\).
Petteri Harjulehto, Peter Hästö
Chapter 5. Extrapolation and Interpolation
Abstract
In this chapter, we develop two techniques which allow us to transfer results of harmonic analysis from one setting to another: extrapolation and interpolation.
Petteri Harjulehto, Peter Hästö
Chapter 6. Sobolev Spaces
Abstract
In this chapter we study Sobolev spaces with generalized Orlicz integrability. We point out the novelties in this new setting and assume that the readers are familiar with classical Sobolev spaces.
Petteri Harjulehto, Peter Hästö
Chapter 7. Special Cases
Abstract
In this chapter, we consider our conditions and results in some special cases, namely variable exponent spaces and their variants, for double phase and degenerate double phase growth, as well as for Orlicz growth without x-dependence.
Petteri Harjulehto, Peter Hästö
Backmatter
Metadaten
Titel
Orlicz Spaces and Generalized Orlicz Spaces
verfasst von
Dr. Petteri Harjulehto
Peter Hästö
Copyright-Jahr
2019
Electronic ISBN
978-3-030-15100-3
Print ISBN
978-3-030-15099-0
DOI
https://doi.org/10.1007/978-3-030-15100-3