A function \(\varphi:[0,\infty)\to[0,\infty]\) is said to be a (weak) \(\Phi\)-functionif \(\varphi(0)=0\), \(\varphi\) is continuous at \(0\), \(\lim_{t\to\infty}\varphi(t)=\infty\), and \(\varphi(t)/t\) is almost increasing. Consider a \(\sigma\)-finite complete measure space \((A,\Sigma,\mu)\). A function \(\varphi:A\times[0,\infty)\to[0,\infty]\) is said to be a generalized (weak) \(\Phi\)-function if \(x\mapsto\varphi(x,|f(x)|)\) is \(\mu\)-measurable for every \(\mu\)-measurable function \(f\) and \(\varphi(x,\cdot)\) is a weak \(\Phi\)-function for \(\mu\)-almost every \(x\in A\). For a \(\mu\)-measurable function \(f\), let\(\varrho_\varphi(f):=\int_A\varphi(x,|f(x)|)\,d\mu(x)\). The set \(L^\varphi(A,\mu)\)of all \(\mu\)-measurable functions \(f\) such that \(\varrho_\varphi(\lambda f)<\infty\)for some \(\lambda>0\) is said to be a generalized Orlicz space (also known as a Musielak-Orlicz space). This is a quasi-normed space with respect to the quasi-norm \(\|f\|_{L^\varphi(A,\mu)}:=\inf\{\lambda>0:\varrho_\varphi(f/\lambda)\le 1\}\).
The aim of the book under review is to study properties of generalized Orlicz spacesand the main operators of harmonic analysis in them.
Chapter 1, entitled “Introduction”, is devoted to a brief history and motivations.It also contains a list of properties and tools of standard Lebesgue spaces \(L^p\), which are not available in \(L^\varphi(A,\mu)\) with \(\varphi(x,t)=t^{p(x)}\).The latter spaces are usually called variable exponent Lebesgue spaces (or Nakano spaces).
Chapter 2, entitled “\(\Phi\)-functions”, contains a detailed study of \(\Phi\)-functions, their generalized inverses defined by \(\varphi^{-1}(\tau):=\inf\{t\ge 0:\varphi(t)\ge\tau\}\), and theirconjugate functions defined by \(\varphi^*(u)=\sup_{t\ge 0}\{tu-\varphi(t)\}\).Further, generalized versions (depending on two variables) of \(\Phi\)-functionsare considered.
In Chapter 3, called “Generalized Orlicz spaces”, these spaces are discussedfrom the Functional Analysis point of view. In particular, convergence in these spaces, their completeness, separability, uniform convexity, and reflexivity are treated in detail. This chapter is concluded with a sufficient condition on a generalized \(\Phi\)-function \(\varphi\) guaranteeing that the set \(C_0^\infty(\Omega)\) of all smooth compactly supported functions on an open set \(\Omega\subset{\mathbb R}^n\) is dense in the generalized Orlicz space \(L^\varphi(\Omega)\).
Chapter 4, entitled “Maximal and averaging operators”, deals with the boundednessof the Hardy-Littlewood maximal operator and averaging operators on generalizedOrlicz spaces. This chapter is the heart of the book.Let \(\Omega\subset\mathbb{R}^n\) be an open set. For a function\(f\in L_{\mathrm{loc}}^1(\Omega)\), the Hardy-Littlewood maximal operator is defined by\(Mf(x):=\sup_{B\ni x}\frac{1}{|B|}\int_{B\cap \Omega}|f(y)|\, dy\), where the supremumis taken over all open balls \(B\) containing \(x\). Further, for a family\({\mathcal B}\) of open bounded sets \(U\subset\mathbb{R}^n\), the averaging operator is definedby \(T_{\mathcal{B}}f(x):=\sum_{U\in\mathcal{B}}\frac{\chi_{U\cap\Omega}(x)}{|U|}\int_{U\cap\Omega}|f(y)|\, dy\).Sufficient conditions for the boundedness and weak-type estimates for \(M\) areproved. Further, sufficient conditions for the uniformboundedness of \(T_{\mathcal{B}}\) are established. These results are applied to the studyof mollifiers on generalized Orlicz spaces.
In Chapter 5, entitled “Extrapolation and interpolation”, the Rubio de Franciaextrapolation techniques are adapted to the setting of generalized Orlicz spaces.This allows the authors to obtain the boundedness of vector-valued maximal operators,Calderón-Zygmund singular integral operators, Riesz potentials, and fractionalmaximal operators in the setting of generalized Orlicz spaces. Further, the complex interpolation of generalized Orlicz spaces is also discussed.
Chapter 6, entitled “Sobolev spaces”, is devoted to the generalized Sobolev spaces built upon generalized Orlicz spaces. Three main themes of this chapter are Poincaréinequalities, Sobolev-type embedding theorems, and the density of regularfunctions in generalized Sobolev spaces.
In Chapter 7, entitled “Special cases”, various particular cases of generalizedOrlicz spaces are discussed briefly. Among them are the case of variable exponent Lebesgue spaces with \(\varphi(x,t)=t^{p(x)}\), the so-called “double phase cases” with \(\varphi(x,t)=t^p+a(x)t^q\)and \(\varphi(x,t)=t^p+a(x)t^p\log (e+t)\), and the caseof classical Orlicz spaces with \(\varphi(x,t)\) independent of the first variable.
The list of references contains 131 items.

Reviewer: Alexei Yu. Karlovich (Lisboa)