×

Geometry of Orlicz spaces. (English) Zbl 0873.46018

The monograph under review is a modern survey of the initiated as well as advanced results (including a considerable body of the author’s own theorems) which related to the classical area of the functional analysis – the theory of Orlicz Spaces (OS). Their geometric properties declared in the book’s title are rather a “casus belli” than the subject of the exclusive author’s interest: these are just the topological properties (i.e. determinated by OS’s element composition) but the geometrical ones (i.e. determined by the unit ball’s geometry of OS) that are investigated at every turn in the text. Actually this difference is very formal: anyway one can say that a OS \(L_{M_1}\) possesses a geometric property (P) if there exists an OS \(L_{M_2}\) whose unit ball satisfies (P) such that \(L_{M_1}= L_{M_2}\) in the sense of element’s composition (what means \(M_1\sim M_2\) in the sense of Orlicz functions).
After preliminary Chapter 1 the other chapters are the following:
Chapter 2. Convexity and smoothness: 2.1. Extreme points and rotundity, 2.2. \(\lambda\)-property, 2.3. Locally uniform rotundity, 2.4. Mid-point locally uniform rotundity and uniform rotundity in every direction, 2.5. Full convexity and weakly uniform rotundity, 2.6. Smoothness.
Chapter 3. Other geometrical properties: 3.1. Normal structure, 3.2. H-property, 3.3. Nonsquareness, 3.4. Some geometrical characterization of reflexivity, 3.5. Roughness, girth and Radon-Nikodým property, 3.6. Ball-packing constants.
Chapter 4. Some applications of geometry of Orlicz spaces: 4.1. Best approximation, 4.2. Predictors, 4.3. Some optimal control problems.
Chapter 5. Geometry of Musielak-Orlicz spaces: 5.1. Musielak-Orlicz spaces, 5.2. Extreme points, rotundity and uniform rotundity, 5.3. Complex rotundities.
Note, that Orlicz sequences and Orlicz functions spaces are studied in a parallel way. The thorough examination of all properties in the Contents is very impressiv. The analysis provides exhaustive results in many cases. The monograph is convenient for reading due to the Index and author’s tables of results. By reducing the latter, we give some examples of the results from Chapters 2 and 3. We need the following notations.
\(M\) denotes an Orlicz function, i.e. a function defined on the real line by the formula \(M(u)= \int^{|u|}_0 p(t)dt\), where (i) \(p\) is right-continuous and nondecreasing on \([0,\infty)\); (ii) \(p(t)>0\;\forall t>0\); (iii) \(p(0)=0\) and \(\lim_{t\to\infty}p(t)= \infty\). \(N\) denotes the complementary Orlicz function for \(M\), i.e. \(N(v)=\int^{|v|}_0 q(s)ds\;\forall v>0\), where the right-inverse for \(p\) function \(q\) satisfies (i)–(iii).
\(M\in SC\) (strictly convex) whenever \(M[(u+ v)/2]<[M(u)+ M(v)]/2\) if \(u\neq v\).
\(M\in UC\) (uniformly convex) whenever \(\forall\varepsilon\), \(u_0>0\;\exists\delta>0\) such that
\(M[(u+ v)/2]\leq (1-\delta)[M(u)+ M(v)]/2\) if \(|u-v|\geq\varepsilon\max\{|u|,|v|\}\geq\varepsilon u_0\).
\(M\in\Delta_2\) (or, equivalently, \(N\in\nabla_2\)) whenever \(\exists K>2\) and \(\exists u_0\geq 0\) such that \(M(2u)\leq KM(u)\), \(u\geq u_0\).
The triple \((G,\Sigma,\mu)\) denotes the Lebesgue measure space with \(0<\mu(G)<\infty\); all the functions \(u,v\dots\) are supposed to be measurable on \((G,\Sigma,\mu)\).
The Orlicz space \(L_M= \left\{u:|u|^0:=\sup\left\{\int_G u(t)v(t)d\mu: \rho_N(v)\leq 1\right\}<\infty\right\}\) is defined by Orlicz modular \(\rho_M(u):= \int_G M(u(t))d\mu\). It is considered usually with two equivalent Banach norms: Orlicz norm \(|\;|^0\) and Luxemburg norm \(|u|:=\inf\{\lambda>0: \rho_M(u\backslash\lambda)\leq 1\}\) \((u\in L_M)\).
In what follows the notations \(L^0_M= (L_M,|\;|^0)\) and \(L_M= (L_M,|\;|)\) are used. The unit ball and sphere of a Banach space \(X\) are denoted through \(B(X)\) and \(S(X)\), respectively. \(X^*\) denotes as usually the conjugate space for \(X\).
\(X\) is called a rotund space (abbr. [R]) if for each three elements \(x,y,z\in B(X)\) the equality \(2x=y+z\) implies \(|x|=1\). \(X\) is said to be (weakly) uniformly rotund [(w)UR], whenever \(|x_n+ y_n|_X\to 2\) implies \(x_n- y_n\) (weakly) converges to 0 for every two sequences \(x_n,y_n\in B(X)\). \[ \boxed{\begin{matrix} \text{Geometric} & L^0_M & L_M\\ \text{Properties} & M\text{-Properties} & M\text{-Properties}\\ \text{R} &\text{SC} & \Delta_2,\text{ SC}\\ \text{UR} & \Delta_2,\text{ UC} &\Delta_2,\text{ UC}\\ \text{wUR} &\Delta_2,\text{ UC} &\Delta_2, \nabla_2,\text{ SC}\end{matrix}} \] A point \(x\in S(X)\) is called a uniformly non-\(\ell_n\) point ([UN-\(\ell^1_n\)]) if \(\exists\delta>0\) such that for any \(x_2,\dots,x_n\in S(X)\), \(\min\{|x+\varepsilon_2x_2+\cdots+ \varepsilon_nx_n|:\varepsilon_i= \pm1,\;i=2,\dots,n\}\leq n-\delta\). If each point in \(S(X)\) is an UN-\(\ell^1_n\) point, then \(X\) is called a locally uniformly non-\(\ell^1_n\) space [LUN-\(\ell^1_n\)].
For each \(x\in S(X)\) denote \(\varepsilon(x)= \sup\{\varepsilon>0\): there exist \(f_n,g_n\in S(X^*)\) with \(f_n(x),g_n(x)\to 1\) and \(\limsup|f_n- g_n|\geq\varepsilon\}\). If \(\varepsilon(x)>0\) for every \(x\in S(X)\) then \(X\) is called pointwise rough. If \(\varepsilon(X):= \inf\{\varepsilon(x): x\in S(X)\}>0\) then \(X\) is called rough. \[ \boxed{\begin{matrix} \text{Geometric} & L^0_M & L_M\\ \text{Properties} & M\text{-Properties} &M\text{-Properties}\\ \text{LUN-}\ell^1_n & \nabla_2 &\Delta_2\\ (n\geq 2)\\ \text{Pointwise} & \not\in\nabla_2 &\not\in\nabla_2\\ \text{rough}\\ \text{Rough} &\not\in\nabla_2 &\Delta_2\backslash\nabla_2\end{matrix}}. \]

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B20 Geometry and structure of normed linear spaces
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
PDFBibTeX XMLCite