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Orlicz spaces and modular spaces. (English) Zbl 0557.46020
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 $$\rho (\lambda x+(1-\lambda)y)\leq \rho (x)+\rho (y)$$ for $$0\leq \lambda \leq 1$$. The associated modular space is the vector space $$X_{\rho}=\{x\in X:\rho (\lambda x)\to 0\quad as\quad \lambda \to 0\}.$$ $$X_{\rho}$$ is endowed with a linear metrizable (normable when $$\rho$$ is convex) topology: for that topology, $$x_ k\to 0$$ iff $$\rho (\lambda x_ k)\to 0$$ for every $$\lambda >0$$. Replacing ”every” by ”some” we get a weaker type of convergence, the ”modular convergence”.
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” $$L^{\phi}(\Omega,\Sigma,\mu)$$, modular spaces of ($$\mu$$-classes of) measurable functions on the measure space ($$\Omega$$,$$\Sigma$$,$$\mu)$$ associated to the modular $$\rho (x)=\int_{\Omega}\phi (t,| x(t)|)d\mu (t),$$ where $$\phi$$ (t,u) is a non-negative function defined on $$\Omega \times {\mathbb{R}}_+$$, measurable with respect to t, increasing and continuous with respect to u, null only for $$u=0$$. When $$\phi$$ is convex in u, $$L^{\phi}$$ is a Banach space. The above mentioned modular convergence is useful when $$\phi$$ does not verify the so-called $$''\Delta_ 2$$ condition”. Several results are presented, which are generally well known for classical Orlicz spaces, and were generalized more or less recently to the spaces $$L^{\phi}$$ where $$\phi$$ depends on the integration variable: when $$\phi$$ is convex, characterization of the dual of $$L^{\phi}$$; condition of uniform convexity; interpolation theorem of Riesz-Thorin type for sublinear or linear operators $$P: L^{\phi_ i}\to L^{\psi_ i}$$, $$i=0,1$$; when $$\Omega$$ is a measurable subset of $${\mathbb{R}}^ n$$ and $$\mu$$ is the Lebesgue measure, study of the translation operators $$x(t)\to x(t- v)$$, of the convolution operators with kernels $$K_ w$$, and compacity criteria in $$L^{\phi}$$. Let us notice that the author’s hypotheses seem to be sometimes superfluous. For instance, $$L^{\phi}(\Omega,\Sigma,\mu)$$ is complete, without assuming $$\mu$$ $$\sigma$$-finite: it suffices to observe that every function $$x\in L^{\phi}$$ is null outside some set of $$\sigma$$-finite measure.
Other examples of modular spaces are considered, such as Orlicz-Sobolev spaces, or the spaces of functions of finite generalized variation.
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 $$\rho_ n$$, $$n\geq 1$$, on a vector space X, is associated the modular $$\rho_ 0=\sup_{n}\rho_ n$$, and also a modular $$\rho$$, such that $$X_{\rho}=\cap_{n}X_{\rho_ n}$$ with the projective limit topology. This is applied to spaces of infinitely differentiable functions, with $$\rho$$ $${}_ n(x)=\int \phi (D^ nx),$$ where $$\phi$$ is a convex Orlicz function: a characterization of the subspace $$X_{\rho_ 0}$$ of $$X_{\rho}$$ is given. Another example is furnished by the Hardy-Orlicz spaces $$H^{\phi}$$ of analytic functions in the open disk, with the modular $$\rho_ o(x)=\sup_{r}\rho (r,x)$$, where $$\rho (r,x)=\int^{2\pi}_{0}\phi (| x(re^{it})|)dt/2\pi,$$ $$0<r<1$$. Here $$\phi$$ is logarithmically convex, not necessarily convex. We may cite also a modular space $$Y^{\rho}$$ associated to the integral equation $x(t)=a\int_{\Omega}k(t,s,| x(s)|)d\mu (s)+x_ 0(t)$ where k(t,s,u), t,s in $$\Omega$$, $$u\in {\mathbb{R}}_+$$, is a convex Orlicz function with respect to u. We have a family of modulars $$\rho (t,x)=\int_{\Omega}k(t,s,| x(s)|)d\mu (s),$$ $$t\in \Omega$$, and $$\rho$$ is given by $$\rho (x)=\int_{\Omega}\rho (t,x)d\mu (t).$$ The functions $$x_ 0$$ and x lie in $$X_{\rho}$$. 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.
 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 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 45G10 Other nonlinear integral equations