##
**Variable Lebesgue spaces. Foundations and harmonic analysis.**
*(English)*
Zbl 1268.46002

Applied and Numerical Harmonic Analysis. New York, NY: Birkhäuser/Springer (ISBN 978-3-0348-0547-6/hbk; 978-3-0348-0548-3/ebook). ix, 312 p. (2013).

The aim of the book under review is to provide a readable introduction to variable Lebesgue spaces and to the fundamentals of harmonic analysis on them and, further, to give an overview of the latest developments in this rapidly growing field. The authors not only summarize existing work, but also include many new previously unpublished facts and give new proofs of several known results. The monograph is a very useful and nice complement to another recent book on the topic written by [L. Diening et al., Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics 2017. Berlin: Springer (2011; Zbl 1222.46002)]. For \(\Omega\subset\mathbb R^n\) and a Lebesgue measurable function \(p:\Omega\to[1,\infty]\), put \(\Omega_\infty=\{x\in\Omega:p(x)=\infty\}\). Further, for a function \(f\) on \(\Omega\), consider the modular defined by \(\rho_{p(\cdot),\Omega}(f)=\int_{\Omega\setminus\Omega_\infty}|f(x)|^{p(x)}dx+\|f\|_{L^\infty(\Omega_\infty)}\). The variable Lebesgue space \(L^{p(\cdot)}(\Omega)\) is the set of all Lebesgue measurable functions \(f\) such that \(\rho_{p(\cdot),\Omega}(f/\lambda)<\infty\) for some \(\lambda>0\). This is a Banach space with respect to the norm \(\|f\|_{L^{p(\cdot)}(\Omega)}=\inf\{\lambda>0:\rho_{p(\cdot),\Omega}(f/\lambda)\leq 1\}\).

The book consists of six chapters and an appendix. The first chapter is introductory. It contains an overview and a short history of variable Lebesgue spaces. The second chapter is devoted to the basic functional analysis of variable Lebesgue spaces. Hölder’s and Minkowski’s inequalities are proved. Convergence in norm, in modular, in measure and relations between them are studied. The density of several canonical sets in \(L^{p(\cdot)}\) is discussed. It is shown that the dual space \((L^{p(\cdot)}(\Omega))^*\) is isomorphic to the variable Lebesgue space \(L^{p'(\cdot)}(\Omega)\), where \(1/p(x)+1/p'(x)=1\) for \(x\in\Omega\), if and only if \(p\) is bounded. Finally, a generalization of the Lebesgue differentiation theorem is given. Every result in this chapter is proved directly, following wherever possible the classical theory of standard Lebesgue spaces. Special attention is paid to the differences between the cases of bounded and unbounded exponents.

Chapter 3 is devoted to the boundedness of the Hardy-Littlewood maximal function \(M\) on variable Lebesgue spaces with nice exponents. One says that a function \(r:\Omega\to{\mathbb R}\) belongs to \(LH_0(\Omega)\) if there is a constant \(C_0>0\) such that \(|r(x)-r(y)|\leq C_0/(-\log|x-y|)\) for all \(x,y\in\Omega\) such that \(|x-y|<1/2\). Further, it is said that \(r\in LH_\infty\) if there exist constants \(C_\infty\) and \(r_\infty\) such that \(|r(x)-r_\infty|\leq C_\infty/\log(e+|x|)\) for all \(x\in\Omega\). Finally, \(LH(\Omega)=LH_0(\Omega)\cap LH_\infty(\Omega)\). The key result of Chapter 3 (Theorem 3.16) says that, if \(1/p\in LH(\Omega)\), then \(\|t\chi_{\{x: Mf(x)>t\}}\|_{L^{p(\cdot)}\Omega}\leq C\|f\|_{L^{p(\cdot)}(\Omega)}\). If, in addition, \(p\) is bounded away from one, then \(M\) is bounded on \(L^{p(\cdot)}(\Omega)\).

Chapter 4 discusses the boundedness of the Hardy-Littlewood maximal function \(M\) on variable Lebesgue spaces with exponents beyond the class \(LH(\Omega)\). It is shown that neither \(LH_\infty(\Omega)\) nor \(LH_0(\Omega)\) are necessary for the boundedness of \(M\). The condition \(LH_\infty(\Omega)\) at infinity may be substituted by a weaker condition \(N_\infty(\Omega)\) introduced by Nekvinda. This condition is also far from being necessary. The local condition \(LH_0(\Omega)\) can be substituted by the following weaker condition. One says that \(p\in K_0(\Omega)\) if there exists a constant \(C_K\) such that \(\|\chi_Q\|_{L^{p(\cdot)}(\Omega)}\|\chi_Q\|_{L^{p'(\cdot)}(\Omega)}\|\leq C_K|Q|\) for any cube \(Q\). It is shown (Theorem 4.52) that, if \(p\) is bounded away from one and infinity and \(p\in K_0({\mathbb R}^n)\cap N_\infty({\mathbb R}^n)\), then \(M\) is bounded on \(L^{p(\cdot)}({\mathbb R}^n)\). This chapter also contains a discussion (without proof) of a necessary and sufficient condition, due to Diening, for the maximal operator to be bounded.

In Chapter 5, other important operators from harmonic analysis are studied on variable Lebesgue spaces by means of an extension of the Rubio de Francia extrapolation theory to the setting of variable Lebesgue spaces. It is shown that, if an operator satisfies weighted norm inequalities on the classical Lebesgue spaces, then it satisfies norm inequalities on variable Lebesgue spaces, assuming some regularity of the exponent. Two classes of operators are studied in detail. These are Calderón-Zygmund singular integrals and Riesz potentials. This chapter has a natural overlap with the recent monograph [D. V. Cruz-Uribe, J. M. Martell and C. Pérez, Weights, extrapolation and the theory of Rubio de Francia. Operator Theory: Advances and Applications 215. Basel: Birkhäuser (2011; Zbl 1234.46003)].

Finally, in Chapter 6, the basic theory of variable Sobolev spaces is developed. Their basic function space properties are proved. The density of smooth functions in \(W^{1,p(\cdot)}\) is discussed. The Meyers-Serrin theorem, the Poincaré inequality, and the Sobolev embedding theorem are proved in the setting of variable exponent spaces. The appendix contains a list of 24 open problems.

Each chapter is concluded with “Notes and Further Results”. These sections occupy about one fifth of the book and contain very extensive and interesting historical comments, remarks, and links to other results. The list of references contains 364 sources.

The book consists of six chapters and an appendix. The first chapter is introductory. It contains an overview and a short history of variable Lebesgue spaces. The second chapter is devoted to the basic functional analysis of variable Lebesgue spaces. Hölder’s and Minkowski’s inequalities are proved. Convergence in norm, in modular, in measure and relations between them are studied. The density of several canonical sets in \(L^{p(\cdot)}\) is discussed. It is shown that the dual space \((L^{p(\cdot)}(\Omega))^*\) is isomorphic to the variable Lebesgue space \(L^{p'(\cdot)}(\Omega)\), where \(1/p(x)+1/p'(x)=1\) for \(x\in\Omega\), if and only if \(p\) is bounded. Finally, a generalization of the Lebesgue differentiation theorem is given. Every result in this chapter is proved directly, following wherever possible the classical theory of standard Lebesgue spaces. Special attention is paid to the differences between the cases of bounded and unbounded exponents.

Chapter 3 is devoted to the boundedness of the Hardy-Littlewood maximal function \(M\) on variable Lebesgue spaces with nice exponents. One says that a function \(r:\Omega\to{\mathbb R}\) belongs to \(LH_0(\Omega)\) if there is a constant \(C_0>0\) such that \(|r(x)-r(y)|\leq C_0/(-\log|x-y|)\) for all \(x,y\in\Omega\) such that \(|x-y|<1/2\). Further, it is said that \(r\in LH_\infty\) if there exist constants \(C_\infty\) and \(r_\infty\) such that \(|r(x)-r_\infty|\leq C_\infty/\log(e+|x|)\) for all \(x\in\Omega\). Finally, \(LH(\Omega)=LH_0(\Omega)\cap LH_\infty(\Omega)\). The key result of Chapter 3 (Theorem 3.16) says that, if \(1/p\in LH(\Omega)\), then \(\|t\chi_{\{x: Mf(x)>t\}}\|_{L^{p(\cdot)}\Omega}\leq C\|f\|_{L^{p(\cdot)}(\Omega)}\). If, in addition, \(p\) is bounded away from one, then \(M\) is bounded on \(L^{p(\cdot)}(\Omega)\).

Chapter 4 discusses the boundedness of the Hardy-Littlewood maximal function \(M\) on variable Lebesgue spaces with exponents beyond the class \(LH(\Omega)\). It is shown that neither \(LH_\infty(\Omega)\) nor \(LH_0(\Omega)\) are necessary for the boundedness of \(M\). The condition \(LH_\infty(\Omega)\) at infinity may be substituted by a weaker condition \(N_\infty(\Omega)\) introduced by Nekvinda. This condition is also far from being necessary. The local condition \(LH_0(\Omega)\) can be substituted by the following weaker condition. One says that \(p\in K_0(\Omega)\) if there exists a constant \(C_K\) such that \(\|\chi_Q\|_{L^{p(\cdot)}(\Omega)}\|\chi_Q\|_{L^{p'(\cdot)}(\Omega)}\|\leq C_K|Q|\) for any cube \(Q\). It is shown (Theorem 4.52) that, if \(p\) is bounded away from one and infinity and \(p\in K_0({\mathbb R}^n)\cap N_\infty({\mathbb R}^n)\), then \(M\) is bounded on \(L^{p(\cdot)}({\mathbb R}^n)\). This chapter also contains a discussion (without proof) of a necessary and sufficient condition, due to Diening, for the maximal operator to be bounded.

In Chapter 5, other important operators from harmonic analysis are studied on variable Lebesgue spaces by means of an extension of the Rubio de Francia extrapolation theory to the setting of variable Lebesgue spaces. It is shown that, if an operator satisfies weighted norm inequalities on the classical Lebesgue spaces, then it satisfies norm inequalities on variable Lebesgue spaces, assuming some regularity of the exponent. Two classes of operators are studied in detail. These are Calderón-Zygmund singular integrals and Riesz potentials. This chapter has a natural overlap with the recent monograph [D. V. Cruz-Uribe, J. M. Martell and C. Pérez, Weights, extrapolation and the theory of Rubio de Francia. Operator Theory: Advances and Applications 215. Basel: Birkhäuser (2011; Zbl 1234.46003)].

Finally, in Chapter 6, the basic theory of variable Sobolev spaces is developed. Their basic function space properties are proved. The density of smooth functions in \(W^{1,p(\cdot)}\) is discussed. The Meyers-Serrin theorem, the Poincaré inequality, and the Sobolev embedding theorem are proved in the setting of variable exponent spaces. The appendix contains a list of 24 open problems.

Each chapter is concluded with “Notes and Further Results”. These sections occupy about one fifth of the book and contain very extensive and interesting historical comments, remarks, and links to other results. The list of references contains 364 sources.

Reviewer: Alexei Yu. Karlovich (Lisboa)

### MSC:

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

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 |

42B25 | Maximal functions, Littlewood-Paley theory |

42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |

46A80 | Modular spaces |