Applications of Orlicz spaces. (English) Zbl 0997.46027

Rao, M. M.; Ren, Z. D.

Pure and Applied Mathematics, Marcel Dekker. 250. New York, NY: Marcel Dekker. xi, 464 p. (2002).

This is a comparison volume to the authors’ “Theory of Orlicz spaces” (1991; Zbl 0724.46032). The book under review comprises ten chapters, and the captions of the chapters are:
Chapter I. Introduction and backward material.
Chapter II. Nonsquare and von Neumann-Jordan constants.
Chapter III. Normal structure and WCS coefficients.
Chapter IV. Jung constants of Orlicz spaces.
Chapter V. Packing in Orlicz spaces.
Chapter VI: Fourier analysis in Orlicz spaces.
Chapter VII. Applications to prediction analysis.
Chapter VIII. Applications to stochastic analysis.
Chapter IX. Nonlinear PDEs and Orlicz spaces.
Chapter X. Miscellaneous applications.
The authors discuss embedding theorems for the Orlicz-Sobolev spaces in Chapter IX. The Beurling-Orlicz algebras, Riesz angles of Orcliz spaces, embedding theorems for sequence spaces, differentiable properties of Orlics spaces and \(L^{\Phi^{-1}}\)-spaces are presented in the best chapter.
It is imperative to quote some results in order to have a glimpse of the theory given in this book.
Result 1. If the \(N\)-function \(\Phi\in \Delta_2\cap\nabla_2\), then
(i) \(J(L^{(\Phi)}(\mathbb{R}^+))< 2\), and
(ii) \(J(L^\Phi(\mathbb{R}^+))<2\).
Result 2. Let \(\Phi\) be an \(N\)-function. Then
(i) \(\Phi\not\in \Delta_2(0)\Rightarrow \text{WCS}(\ell^\Phi)= \text{WCS}(m^\Phi)= 1\),
(ii) \(\Phi\in \Delta_2(0)\Rightarrow \text{WCS}(\ell^\Phi)> 1\).
Result 3: Let \(X\) be an infinite-dimensional Banach space. Let \(P(X)\) denote a packing constant and let \(K(X)\) be the Kottman constant of \(X\). Then \(P(X)= {K(X)\over 2+ K(X)}\).
Result 4. A subspace \(N\subset L^\Phi(\mu)\) is the range of a constrative projection \(\Leftrightarrow\) it is a Banach lattice.
Result 5. A \((L^{(\Phi)}[0,1])< 2\Leftrightarrow \Phi\in \nabla_2(\infty)\).
Result 6. Let \(\Phi\) be an \(N\)-function. Then \(\ell^\Phi\) is reflexive \(\Leftrightarrow \Phi\in \Delta_2(0)\cap \nabla_2(0)\).
The printing and get-up are attractive. References, Notation and Index are appended.

Reviewer: K.Chandrasekhara Rao (Karaikudi)

MSC:

46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

normal structure; packing; nonlinear PDEs; Orlicz spaces; von Neumann-Jordan constants; WCS coefficients; Jung constants; Fourier analysis; prediction analysis; stochastic analysis; embedding theorems; Orlicz-Sobolev spaces; Beurling-Orlicz algebras; Riesz angles; embedding theorems for sequence spaces; differentiable properties; Kottman constant; Banach lattice