##
**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.

Helpful historical comments are gathered at the end of the volume.

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.

Helpful historical comments are gathered at the end of the volume.

Reviewer: Ph.Turpin

### 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 |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

45G10 | Other nonlinear integral equations |