Document Zbl 0557.46020

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Berlin etc.: Springer-Verlag. 222 p. DM 28.00; $ 10.90 (1983). </div> <div class="abstract">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”. <br class="zbmathjax-paragraph">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. <br class="zbmathjax-paragraph">Other examples of modular spaces are considered, such as Orlicz-Sobolev spaces, or the spaces of functions of finite generalized variation. <br class="zbmathjax-paragraph">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. <br class="zbmathjax-paragraph">Helpful historical comments are gathered at the end of the volume.<div class="reviewer"> Reviewer: <a href="/authors/?q=Ph+Turpin">Ph.Turpin</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A3888333">Cited in <strong>20</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A3888333">Cited in <strong>1269</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A46E30" title="MSC2020">46E30</a> </td> <td class="space"> Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A46E35" title="MSC2020">46E35</a> </td> <td class="space"> Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A46-02" title="MSC2020">46-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to functional analysis </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A45G10" title="MSC2020">45G10</a> </td> <td class="space"> Other nonlinear integral equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Amodular+spaces">modular spaces</a>; <a href="/?q=ut%3Amodular+convergence">modular convergence</a>; <a href="/?q=ut%3Aconjugate+modular+on+the+dual+space">conjugate modular on the dual space</a>; <a href="/?q=ut%3Amodular+on+a+tensor+product">modular on a tensor product</a>; <a href="/?q=ut%3Ageneralized+Orlicz+spaces">generalized Orlicz spaces</a>; <a href="/?q=ut%3Acharacterization+of+the+dual">characterization of the dual</a>; <a href="/?q=ut%3Auniform+convexity">uniform convexity</a>; <a href="/?q=ut%3Ainterpolation+theorem+of+Riesz-Thorin+type+for+sublinear+or">interpolation theorem of Riesz-Thorin type for sublinear or</a>; <a href="/?q=ut%3Alinear+operators">linear operators</a>; <a href="/?q=ut%3Atranslation+operators">translation operators</a>; <a href="/?q=ut%3Aconvolution+operators">convolution operators</a>; <a href="/?q=ut%3Acompacity+criteria">compacity criteria</a>; <a href="/?q=ut%3AOrlicz-Sobolev+spaces">Orlicz-Sobolev spaces</a>; <a href="/?q=ut%3Aspaces+of+functions+of+finite+generalized+variation">spaces of functions of finite generalized variation</a>; <a href="/?q=ut%3Aprojective+limit">projective limit</a>; <a href="/?q=ut%3A%5C%28%5Crho+_+n%28x%29%3D%5Cint+%5Cphi+%28D%5E+nx%29%2C%5C%29">$\rho _ n(x)=\int \phi (D^ nx),$</a>; <a href="/?q=ut%3Aconvex+Orlicz+function">convex Orlicz function</a>; <a href="/?q=ut%3AHardy-Orlicz+spaces">Hardy-Orlicz spaces</a>; <a href="/?q=ut%3Alogarithmically+convex">logarithmically convex</a>; <a href="/?q=ut%3Aintegral+equation">integral equation</a>; <a href="/?q=ut%3Aapproximation+of+functions+by+non-linear+singular+integrals">approximation of functions by non-linear singular integrals</a>; <a href="/?q=ut%3Ainterpolation+theorem+of+Riesz-Thorin+type+for+sublinear+or+linear+operators">interpolation theorem of Riesz-Thorin type for sublinear or linear operators</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0557.46020" data-ciurl="" data-biburl="/bibtex/03888333.bib" data-amsurl="/amsrefs/03888333.bib" data-xmlurl="/xml/03888333.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/03888333.pdf" title="Zbl 0557.46020 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/BFb0072210" aria-label="DOI for “Orlicz spaces and modular spaces”" title="10.1007/BFb0072210">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> </article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; 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