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2016 | Buch

Definition of Symmetric Spaces Kapitel lesen Erstes Kapitel lesen

Foundations of Symmetric Spaces of Measurable Functions

Lorentz, Marcinkiewicz and Orlicz Spaces

verfasst von: Ben-Zion A. Rubshtein, Genady Ya. Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova

Verlag: Springer International Publishing

Buchreihe : Developments in Mathematics

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

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SUCHEN

Über dieses Buch

Key definitions and results in symmetric spaces, particularly Lp, Lorentz, Marcinkiewicz and Orlicz spaces are emphasized in this textbook. A comprehensive overview of the Lorentz, Marcinkiewicz and Orlicz spaces is presented based on concepts and results of symmetric spaces. Scientists and researchers will find the application of linear operators, ergodic theory, harmonic analysis and mathematical physics noteworthy and useful.

This book is intended for graduate students and researchers in mathematics and may be used as a general reference for the theory of functions, measure theory, and functional analysis. This self-contained text is presented in four parts totaling seventeen chapters to correspond with a one-semester lecture course. Each of the four parts begins with an overview and is subsequently divided into chapters, each of which concludes with exercises and notes. A chapter called “Complements” is included at the end of the text as supplementary material to assist students with independent work.

Inhaltsverzeichnis

Frontmatter

Symmetric Spaces: The Spaces Lp,L1 ∩ L∞, L1 + L∞

Frontmatter
Chapter 1. Definition of Symmetric Spaces
Abstract
In this chapter we begin the study of symmetric spaces. The definition of the symmetric spaces is based on two important notions: the distribution function η  | f |  and the decreasing rearrangement f ∗ of a function f on \(\mathbb{R}^{+}\). Most of the chapter is devoted to studying properties of η  | f |  and f ∗.
Ben-Zion A. Rubshtein, Genady Ya. Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova
Chapter 2. Spaces L p ,  1 ≤ p ≤ ∞
Abstract
In this chapter we study the class of L p spaces, 1 ≤ p ≤ ∞, which is one of the most important classes of symmetric spaces. We begin with the Hölder and Minkowski inequalities and prove that L p is a symmetric space for all 1 ≤ p ≤ ∞. In the case 1 ≤ p < ∞, we show that L p is separable and describe its dual.
Ben-Zion A. Rubshtein, Genady Ya. Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova
Chapter 3. The Space L 1 ∩L ∞
Abstract
The space L 1 ∩L ∞ , which we study in this chapter, consists of all bounded integrable functions equipped with the norm \(\|\cdot \|_{\mathbf{L}_{1}\cap \mathbf{L}_{\infty }} =\max (\|\cdot \|_{\mathbf{L}_{1}},\,\|\cdot \|_{\mathbf{L}_{\infty }})\). We show that \((\mathbf{L}_{1} \cap \mathbf{L}_{\infty },\,\|\cdot \|_{\mathbf{L}_{1}\cap \mathbf{L}_{\infty }})\) is a symmetric space and describe the closure L ∞ 0 of L 1 ∩L ∞ in L ∞ . Given two equimeasurable functions f and g, we treat an approximation of g in the L 1 ∩ L ∞ -norm by shifted functions f ∘θ, where θ is a measure-preserving transformation. Step functions and integrable simple functions are applied for this purpose.
Ben-Zion A. Rubshtein, Genady Ya. Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova
Chapter 4. The Space L 1 +L ∞
Abstract
In this chapter, we study the sum L 1 +L ∞ of the spaces L 1 and L ∞ . We show that L 1 +L ∞ equipped with a natural norm \(\|\cdot \|_{\mathbf{L}_{1}+\mathbf{L}_{\infty }}\) is a symmetric space. The norm \(\|\cdot \|_{\mathbf{L}_{1}+\mathbf{L}_{\infty }}\) can be written in the form ∫ 0 1 f ∗ dm using the maximal property of decreasing rearrangements f ∗. We also describe embeddings of L 1 and L ∞ into L 1 +L ∞ and the closure R 0 of L 1 in L 1 +L ∞ .
Ben-Zion A. Rubshtein, Genady Ya. Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova

Symmetric Spaces: The Embedding Theorem. Properties (A), (B), (C)

Frontmatter
Chapter 5. Embeddings L1 ∩ L∞ ⊆ X ⊆ L1 + L∞ ⊆ L0
Abstract
In this chapter, we prove the main embedding theorem for symmetric spaces. The theorem asserts that for every symmetric space X, there are continuous embeddings L 1 ∩L ∞   ⊆ X  ⊆ L 1 +L ∞ and inequalities \(2\| \cdot \|_{\mathbf{L}_{1}\cap \mathbf{L}_{\infty }} \geq (\varphi _{\mathbf{X}})^{-1}(1) \times \|\cdot \|_{\mathbf{X}} \geq\) \(\geq \|\cdot \|_{\mathbf{L}_{1}+\mathbf{L}_{\infty }}.\) The space L 0 of all measurable functions and the embedding L 1 +L ∞  ⊂ L 0 are also considered.
Ben-Zion A. Rubshtein, Genady Ya. Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova
Chapter 6. Embeddings. Minimality and Separability. Property (A)
Abstract
In this chapter we study minimal and separable symmetric spaces. The minimal part X 0 of a symmetric space X is the closure of L 1 ∩L ∞ in X, and X is minimal if X 0 = X. We show that every separable symmetric space is minimal, and the converse is true under the additional condition ϕ X (0+) = 0. We consider also an important property (A), which is equivalent to separability.
Ben-Zion A. Rubshtein, Genady Ya. Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova
Chapter 7. Associate Spaces
Abstract
In this chapter, we study associate spaces X 1 of symmetric spaces X. The space X 1 is defined by the duality \(\langle f,g\rangle =\int gf\,dm\), f ∈ X, g ∈ X 1, and the norm \(\|\cdot \|_{\mathbf{X}^{1}}\) is induced by the canonical embedding of X 1 into the dual space X ∗ of X. We show that \((\mathbf{X}^{1},\|\cdot \|_{\mathbf{X}^{1}})\) is a symmetric space and that the canonical embedding of X 1 into X ∗ is surjective if and only if the space X is separable, i.e., X has property (A).
Ben-Zion A. Rubshtein, Genady Ya. Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova
Chapter 8. Maximality. Properties (B) and (C)
Abstract
In this chapter, we study the second associate space X 11 of a symmetric space X. To describe properties of the natural embedding X ⊆ X 11, we two consider important properties (B) and (C) of the space X. We show that X has property (C) if and only if the natural embedding X → X 11 is isometric. If in addition to (C), X has property (B), then X = X 11, i.e., the symmetric space X is maximal, and the natural embedding X → X 11 is an isometric isomorphism between X and X 11.
Ben-Zion A. Rubshtein, Genady Ya. Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova

Lorentz and Marcinkiewicz Spaces

Frontmatter
Chapter 9. Lorentz Spaces
Abstract
In this chapter, we study Lorentz spaces \(\boldsymbol{\varLambda }_{W}\). It is shown that every Lorentz space \(\boldsymbol{\varLambda }_{W}\) with concave weight function W is a maximal symmetric space. We also describe conditions of minimality and separability of Lorentz spaces.
Ben-Zion A. Rubshtein, Genady Ya. Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova
Chapter 10. Quasiconcave Functions
Abstract
In this chapter we study quasiconcave functions in conjunction with fundamental functions of symmetric spaces. We show that every concave function is also quasiconcave, and conversely, for every quasiconcave function V, there exists its least concave majorant \(\widetilde{V }\), satisfying inequality \(\frac{1} {2}\widetilde{V } \leq V \leq \widetilde{ V }\). We also show that fundamental functions of symmetric spaces are quasiconcave.
Ben-Zion A. Rubshtein, Genady Ya. Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova
Chapter 11. Marcinkiewicz Spaces
Abstract
In this chapter we study Marcinkiewicz spaces M V constructed by a quasiconcave weight V. The space M V is equipped with the norm \(\|f\|_{\mathbf{M}_{V }} =\| V _{{\ast}}f^{{\ast}{\ast}}\|_{\mathbf{L}_{\infty }}\), where \(V _{{\ast}}(x) = \frac{x} {V (x)}\), and \(f^{{\ast}{\ast}}(x) = \frac{1} {x}\int _{0}^{x}f^{{\ast}}\,dm\) is the maximal Hardy–Littlewood function of f. We show that \((\mathbf{M}_{V },\|\cdot \|_{\mathbf{M}_{V }})\) is a maximal symmetric space. The associate space \(\boldsymbol{\varLambda }_{W}^{1}\) of every Lorentz space \(\boldsymbol{\varLambda }_{W}\) with a concave weight W is a Marcinkiewicz space and \(\|\cdot \|_{\mathbf{M}_{V }} =\| \cdot \|_{\boldsymbol{\varLambda }_{W}^{1}}\). Conversely, the associate space M V 1 coincides with a Lorentz space \(\boldsymbol{\varLambda }_{W}\) with a concave weight W that is equivalent to V.
Ben-Zion A. Rubshtein, Genady Ya. Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova
Chapter 12. Embedding ⊆ X ⊆ MV*
Abstract
In this chapter, we prove the embedding theorem for classes of symmetric spaces having the same fundamental functions. The embedding theorem asserts that for every symmetric space X with a given fundamental function V = φ X , there are continuous embeddings \(\boldsymbol{\varLambda }_{\widetilde{V }}^{0} \subseteq \mathbf{X} \subseteq \mathbf{M}_{V _{{\ast}}}\). This means that the minimal part \(\boldsymbol{\varLambda }_{\widetilde{V }}^{0}\) of the Lorentz space \(\boldsymbol{\varLambda }_{\widetilde{V }}\) is the smallest symmetric space whose (concave) fundamental function \(\widetilde{V }\) is equivalent to V, while the Marcinkiewicz space \(\mathbf{M}_{V _{{\ast}}}\) is the largest symmetric space X with \(\varphi _{\mathbf{X}} =\varphi _{\mathbf{M}_{V_{{\ast}}}} = V\). The renorming theorem and other consequences of the embedding theorem are considered.
Ben-Zion A. Rubshtein, Genady Ya. Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova

Orlicz Spaces

Frontmatter
Chapter 13. Definition and Examples of Orlicz Spaces
Abstract
In this chapter, we begin our study of Orlicz functions Φ and corresponding Orlicz spaces L Φ . We prove that \((\mathbf{L}_{\varPhi },\|\cdot \|_{\mathbf{L}_{\varPhi }})\) is a maximal symmetric space, find the fundamental functions \(\varphi _{\mathbf{L}_{ \varPhi }}\) of L Φ , and consider several simple examples.
Ben-Zion A. Rubshtein, Genady Ya. Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova
Chapter 14. Separable Orlicz Spaces
Abstract
In this chapter, we study conditions of separability for Orlicz spaces L Φ . We consider Young classes Y Φ , the subspaces H Φ , and their embeddings H Φ  ⊆ Y Φ  ⊆ L Φ . We show that the equality H Φ  = Y Φ  = L Φ is equivalent to separability of L Φ . This and other equivalents of separability studied earlier in Chaps. 6 and 7 can be expressed in term of an Orlicz function Φ. The (Δ 2) condition is described to this end in detail.
Ben-Zion A. Rubshtein, Genady Ya. Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova
Chapter 15. Duality for Orlicz Spaces
Abstract
In this chapter, we consider associate spaces L Φ 1 of Orlicz spaces L Φ . Using the Legendre transform of an Orlicz function Φ, we define the conjugate Orlicz function Ψ just as the Legendre transform of Φ. We prove that the spaces L Φ 1 and L Ψ coincide as sets and \(\|\cdot \|_{\mathbf{L}_{\varPsi }} \leq \|\cdot \|_{\mathbf{L}_{\varPhi }^{1}} \leq 2\| \cdot \|_{\mathbf{L}_{\varPsi }}\). The duality between L Φ and L Ψ is studied in detail.
Ben-Zion A. Rubshtein, Genady Ya. Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova
Chapter 16. Comparison of Orlicz Spaces
Abstract
In this chapter, we study embedding of Orlicz spaces in terms of corresponding Orlicz functions. We characterize the class of Orlicz functions corresponding to the same Orlicz space. The Zygmund classes are considered as examples of Orlicz spaces.
Ben-Zion A. Rubshtein, Genady Ya. Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova
Chapter 17. Intersections and Sums of Orlicz Spaces
Abstract
In this chapter, we study the intersections and the sums of Orlicz spaces. In particular, for every pair of mutually conjugate Orlicz functions Φ and Ψ, the Orlicz spaces (L Φ ∩L ∞ ,  L Ψ +L 1) and (L Φ +L ∞ ,  L Ψ ∩L 1) are described. In particular, the spaces L p +L q and L p ∩L q , 1 ≤ p, q ≤ ∞, are considered in greater detail.
Ben-Zion A. Rubshtein, Genady Ya. Grabarnik, Mustafa A. Muratov, Yulia S. Pashkova
Backmatter
Metadaten
Titel
Foundations of Symmetric Spaces of Measurable Functions
verfasst von
Ben-Zion A. Rubshtein
Genady Ya. Grabarnik
Mustafa A. Muratov
Yulia S. Pashkova
Copyright-Jahr
2016
Electronic ISBN
978-3-319-42758-4
Print ISBN
978-3-319-42756-0
DOI
https://doi.org/10.1007/978-3-319-42758-4