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Zbl 0557.46020
Musielak, Julian
Orlicz spaces and modular spaces. (English)
[B] Lecture Notes in Mathematics. 1034. Berlin etc.: Springer-Verlag. 222 p. DM 28.00; {\$} 10.90 (1983).

The author has played a central role in the development of the theory of modular spaces for more than twenty years by now. These lecture notes are a survey on that theory. A modular $\rho$ on a vector space X is a map $\rho: Y\to [0,+\infty]$ verifying $\rho (x)=0\Leftrightarrow x=0$ and $\rho (\lambda x+(1-\lambda)y)\le \rho (x)+\rho (y)$ for $0\le \lambda \le 1$. The associated modular space is the vector space $X\sb{\rho}=\{x\in X:\rho (\lambda x)\to 0\quad as\quad \lambda \to 0\}.$ $X\sb{\rho}$ is endowed with a linear metrizable (normable when $\rho$ is convex) topology: for that topology, $x\sb k\to 0$ iff $\rho (\lambda x\sb k)\to 0$ for every $\lambda >0$. Replacing "every" by "some" we get a weaker type of convergence, the "modular convergence". \par After a few generalities (e.g. conjugate modular on the dual space qhwn $\rho$ is convex, modular on a tensor product...) come the main examples, namely the "generalized Orlicz spaces" $L\sp{\phi}(\Omega,\Sigma,\mu)$, modular spaces of ($\mu$-classes of) measurable functions on the measure space ($\Omega$,$\Sigma$,$\mu)$ associated to the modular $\rho (x)=\int\sb{\Omega}\phi (t,\vert x(t)\vert)d\mu (t),$ where $\phi$ (t,u) is a non-negative function defined on $\Omega \times {\bbfR}\sb+$, measurable with respect to t, increasing and continuous with respect to u, null only for $u=0$. When $\phi$ is convex in u, $L\sp{\phi}$ is a Banach space. The above mentioned modular convergence is useful when $\phi$ does not verify the so-called $''\Delta\sb 2$ condition". Several results are presented, which are generally well known for classical Orlicz spaces, and were generalized more or less recently to the spaces $L\sp{\phi}$ where $\phi$ depends on the integration variable: when $\phi$ is convex, characterization of the dual of $L\sp{\phi}$; condition of uniform convexity; interpolation theorem of Riesz-Thorin type for sublinear or linear operators $P: L\sp{\phi\sb i}\to L\sp{\psi\sb i}$, $i=0,1$; when $\Omega$ is a measurable subset of ${\bbfR}\sp n$ and $\mu$ is the Lebesgue measure, study of the translation operators $x(t)\to x(t- v)$, of the convolution operators with kernels $K\sb w$, and compacity criteria in $L\sp{\phi}$. Let us notice that the author's hypotheses seem to be sometimes superfluous. For instance, $L\sp{\phi}(\Omega,\Sigma,\mu)$ is complete, without assuming $\mu$ $\sigma$-finite: it suffices to observe that every function $x\in L\sp{\phi}$ is null outside some set of $\sigma$-finite measure. \par Other examples of modular spaces are considered, such as Orlicz-Sobolev spaces, or the spaces of functions of finite generalized variation. \par The last chapters are devoted to the investigation of modulars associated in a natural way to a given family of modulars. For example, to a sequence of modulars $\rho\sb n$, $n\ge 1$, on a vector space X, is associated the modular $\rho\sb 0=\sup\sb{n}\rho\sb n$, and also a modular $\rho$, such that $X\sb{\rho}=\cap\sb{n}X\sb{\rho\sb n}$ with the projective limit topology. This is applied to spaces of infinitely differentiable functions, with $\rho$ ${}\sb n(x)=\int \phi (D\sp nx),$ where $\phi$ is a convex Orlicz function: a characterization of the subspace $X\sb{\rho\sb 0}$ of $X\sb{\rho}$ is given. Another example is furnished by the Hardy-Orlicz spaces $H\sp{\phi}$ of analytic functions in the open disk, with the modular $\rho\sb o(x)=\sup\sb{r}\rho (r,x)$, where $\rho (r,x)=\int\sp{2\pi}\sb{0}\phi (\vert x(re\sp{it})\vert)dt/2\pi,$ $0<r<1$. Here $\phi$ is logarithmically convex, not necessarily convex. We may cite also a modular space $Y\sp{\rho}$ associated to the integral equation $$ x(t)=a\int\sb{\Omega}k(t,s,\vert x(s)\vert)d\mu (s)+x\sb 0(t) $$ where k(t,s,u), t,s in $\Omega$, $u\in {\bbfR}\sb+$, is a convex Orlicz function with respect to u. We have a family of modulars $\rho (t,x)=\int\sb{\Omega}k(t,s,\vert x(s)\vert)d\mu (s),$ $t\in \Omega$, and $\rho$ is given by $\rho (x)=\int\sb{\Omega}\rho (t,x)d\mu (t).$ The functions $x\sb 0$ and x lie in $X\sb{\rho}$. There is a theorem of existence and unicity of the solution x. Let us mention also a problem of approximation of functions by non-linear singular integrals, formulated in terms of families of modulars. \par Helpful historical comments are gathered at the end of the volume.
[Ph.Turpin]

MSC 2000:: *46E30 Spaces of measurable functions
46E35 Sobolev spaces and generalizations
46-02 Research monographs (functional analysis)
45G10 Nonsingular nonlinear integral equations

Keywords: modular spaces; modular convergence; conjugate modular on the dual space; modular on a tensor product; generalized Orlicz spaces; characterization of the dual; uniform convexity; interpolation theorem of Riesz-Thorin type for sublinear or; linear operators; translation operators; convolution operators; compacity criteria; Orlicz-Sobolev spaces; spaces of functions of finite generalized variation; projective limit; $\rho \sb n(x)=\int \phi (D\sp nx),$; convex Orlicz function; Hardy-Orlicz spaces; logarithmically convex; integral equation; approximation of functions by non-linear singular integrals; interpolation theorem of Riesz-Thorin type for sublinear or linear operators

Cited in: Zbl 1222.46002 Zbl 1135.46003 Zbl 1067.41014 Zbl 1067.46036 Zbl 0842.46005 Zbl 0716.46028 Zbl 0874.46022 Zbl 0703.46020 Zbl 0661.46023 Zbl 0656.46022 Zbl 0639.46036 Zbl 0637.47036 Zbl 0622.46019 Zbl 0612.46029 Zbl 0608.47068 Zbl 0602.46022 Zbl 0602.46021

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