## Lebesgue and Sobolev spaces with variable exponents.(English)Zbl 1222.46002

Lecture Notes in Mathematics 2017. Berlin: Springer (ISBN 978-3-642-18362-1/pbk; 978-3-642-18363-8/ebook). x, 509 p. (2011).
The book is devoted to Lebesgue and Sobolev spaces with variable exponents. Let $$(A,\Sigma,\mu)$$ be a $$\sigma$$-finite complete measure space and $$p:A\to[1,\infty]$$. For simplicity assume that $$p$$ is $$\mu$$-almost everywhere finite. For a $$\mu$$-measurable function $$u:A\to{\mathbb K}$$, where $${\mathbb K}$$ is the field of real or complex numbers, consider $$\varrho_{L^{p(\cdot)}(A)}(u):=\int_A|u(x)|^{p(x)}dx$$. The Lebesgue space with variable exponent $$L^{p(\cdot)}(A,\mu)$$ is defined as the collection of all $$\mu$$-measurable functions $$u:A\to{\mathbb K}$$ such that $$\varrho_{L^{p(\cdot)}(A,\mu)}(u/\lambda)<\infty$$ for some $$\lambda$$ depending on $$u$$. This is a Banach space when equipped with the so-called Luxemburg norm $\|u\|_{L^{p(\cdot)}(A,\mu)}=\inf\left\{\lambda>0: \varrho_{L^{p(\cdot)}(A,\mu)}(u/\lambda)\leq 1\right\}.$ It is clear that $$L^{p(\cdot)}(A,\mu)$$ is nothing but the Lebesgue space whenever $$p\in[1,\infty)$$ is constant.
Let $$k\in\{0,1,2,\dots\}$$. If $$\Omega\subset{\mathbb R}^n$$ is an open set equipped with the Lebesgue measure, then the Sobolev space with variable exponent $$W^{k,p(\cdot)}(\Omega)$$ is defined as the set of all functions $$u\in L^{p(\cdot)}(\Omega)$$ such that all their weak derivatives $$\partial_\alpha u$$ with $$|\alpha|\leq k$$ exist and belong to $$L^{p(\cdot)}(\Omega)$$. This is a Banach space when equipped with the norm
$\|u\|_{W^{k,p(\cdot)}(\Omega)}=\inf\left\{\lambda>0:\sum_{0\leq|\alpha|\leq k}\varrho_{L^{p(\cdot)}(\Omega)}(\partial_\alpha u/\lambda)\leq 1\right\}.$
For the first time Lebesgue spaces with variable exponents were considered by W. Orlicz [“Über konjugierte Exponentenfolgen.” Stud. Math. 3, 200–211 (1931; Zbl 0003.25203)]. Further, [H. Nakano, Modulared semi-ordered linear spaces. Tokyo Math. Book Series, Vol. 1. Tokyo: Maruzen (1950; Zbl 0041.23401)] constructed the theory of modular spaces. Lebesgue spaces with variable exponents are mentioned explicitly in that book as an example of modular spaces. Therefore, sometimes Lebesgue spaces with variable exponents are called Nakano spaces. Nakano’s theory was further generalized by J. Musielak and W. Orlicz [“On modular spaces.” Stud. Math. 18, 49–65 (1959; Zbl 0086.08901)], where in particular Musielak-Orlicz spaces (variable Orlicz spaces) were introduced. Later developments in the theory of modular spaces and Musielak-Orlicz spaces were summarized in [J. Musielak, Orlicz spaces and modular spaces. Lecture Notes in Mathematics 1034. Berlin etc.: Springer-Verlag. (1983; Zbl 0557.46020)]. To the reviewer’s knowledge, Sobolev spaces of functions with weak derivatives in Musielak-Orlicz spaces were studied for the first time by H. Hudzik [“A generalization of Sobolev spaces. I.” Funct. Approximatio, Comment. Math. 2, 67–73 (1976; Zbl 0338.46032)], [“A generalization of Sobolev space. II.” Funct. Approximatio, Comment. Math. 3, 77–85 (1976; Zbl 0355.46011)], [“On generalized Orlicz-Sobolev space.” Funct. Approximatio, Comment. Math. 4, 37–51 (1976; Zbl 0355.46012)], [“On problem of density of $$C_0^\infty(\Omega)$$ in generalized Orlicz-Sobolev space $$W_M^k (\Omega)$$ for every open set $$\Omega\subset {\mathbb R}^n$$.” Commentat. Math. 20, 65–78 (1977; Zbl 0385.46016)]. The present book consists of the introduction and three parts. The introduction (Chapter 1) contains a short history of spaces with variable exponents, an overview of the main results of the book and preliminaries from functional analysis and measure theory.
The first part is about Lebesgue spaces with variable exponents. Chapter 2 describes properties of semi-modular and modular spaces, as well as properties of Musielak-Orlicz spaces such as separability, uniform convexity, duality and reflexivity. This chapter has an overlap with Musielak’s book cited above.
In Chapter 3, completeness, reflexivity, separability, and uniform convexity of Lebesgue spaces with variable exponents are derived from more general results of Chapter 2. The norm dual formula is also established. It is not assumed that $$p$$ is bounded, $$p$$ can even take the value infinity on a set of positive measure. Let $$p_-:=\text{ess\,inf}_{x\in A}p(x)$$ and $$p_+:=\text{ess\,sup}_{x\in A}p(x)$$. For instance, it is shown that $$L^{p(\cdot)}(A,\mu)$$ is reflexive whenever $$1<p_-$$ and $$p_+<\infty$$.
Chapter 4 is central in the first part. It deals with the boundedness of the Hardy-Littlewood maximal operator $$M$$ on $$L^{p(\cdot)}(\Omega)$$ where $$\Omega\subset{\mathbb R}^n$$ is an open set equipped with the Lebesgue measure. In contrast to previous investigations, the authors are able to treat the case of unbounded exponents. One says that a function $$\alpha:\Omega\to{\mathbb R}$$ is locally log-Hölder continuous if there exists a constant $$c_1>0$$ such that $$|\alpha(x)-\alpha(y)|\leq c_1/\log(e+1/|x-y|)$$ for all $$x,y\in\Omega$$ and that $$\alpha$$ satisfies the log-Hölder decay condition if there exists an $$\alpha_\infty\in{\mathbb R}$$ and a constant $$c_2>0$$ such that $$|\alpha(x)-\alpha_\infty|\leq c_2/\log(e+|x|)$$ for all $$x\in\Omega$$. If both conditions are satisfied then one says that $$\alpha$$ is globally log-Hölder continuous. The main result of Chapter 4 says that if $$1/p$$ is globally log-Hölder continuous on $${\mathbb R}^n$$ and $$p_->1$$ then $$M$$ is bounded on $$L^{p(\cdot)}({\mathbb R}^n)$$. It is also shown that $$p_->1$$ is necessary for the bounededness of $$M$$ on $$L^{p(\cdot)}({\mathbb R}^n)$$.
Chapter 5 starts with A. K. Lerner’s proof [“Some remarks on the Hardy-Littlewood maximal function on variable $$L^p$$ spaces.” Math. Z. 251, No. 3, 509–521 (2005; Zbl 1092.42009)] of the fact that the global log-Hölder continuity of $$1/p$$ is not necessary for the boundedness of $$M$$. In particular, it is shown that if $$\alpha$$ is small enough, then $$M$$ is bounded on $$L^{p(\cdot)}({\mathbb R}^n)$$ with $$p(x)=2-\alpha(1+\sin(\log\log(e+|x|+1/|x|)))$$. Clearly, this function is discontinuous at zero and at infinity. Chapter 5 contains a more abstract treatment of the boundedness of the maximal operator in terms of the so-called class $${\mathcal A}$$. This class consists of those exponents for which a suitable collection of averaging operators is bounded. The main result of Chapter 5 says that if $$1<p_-$$ and $$p_+<\infty$$ then the following statements are equivalent: (a) $$M$$ is bounded on $$L^{p(\cdot)}({\mathbb R}^n)$$; (b) $$p\in{\mathcal A}$$; (c) $$p':=p/(p-1)\in{\mathcal A}$$; (d) $$M$$ is bounded on $$L^{sp(\cdot)}({\mathbb R}^n)$$ for some $$s\in(1/p_-,1)$$; (e) $$M$$ is bounded on $$L^{p'(\cdot)}({\mathbb R}^n)$$.
Chapter 6 consists of a fairly straightforward application of methods from Chapter 4 to other operators such as the Riesz potential, the Fefferman-Stein sharp maximal operator, the Calderón-Zygmund singular integral operator. One sample result is as follows. Let $$p\in{\mathcal A}$$ be such that $$1<p_-$$ and $$p_+<\infty$$. Then the Calderón-Zygmund singular integral operator is bounded on $$L^{p(\cdot)}({\mathbb R}^n)$$.
Chapter 7 collects some techniques which allow one to transfer some results from one setting to another, more general, setting. In Section 7.1, a Riesz-Thorin type interpolation theorem for Lebesgue spaces with variable exponents is proved. Notice that for the more general setting of Musielak-Orlicz spaces such a result is contained in Section 14 of Musielak’s book. Section 7.2 deals with the Rubio de Francia extrapolation in the setting of Lebesgue spaces with variable exponent. This subsection has an overlap with the recent monograph by [D. Cruz-Uribe, J. M. Martell and C. Pérez, Weights, extrapolation and the theory of Rubio de Francia. Basel: Birkhäuser (2011; Zbl 1234.46003)]. Another interesting transfer technique considered in Section 7.4 allows one to generalize many statements for balls to (possible unbounded) John domains.
The second part of the book is about Sobolev spaces with variable exponents. In Chapter 8, the completeness, reflexivity, separability, and uniform convexity of Sobolev spaces with variable exponents are proved under minimal assumptions on $$p$$. More sophisticated results like Sobolev embeddings and Poincaré inequalities are proved by recourse to results on maximal and other operators.
Chapter 9 deals with the density of smooth functions in Sobolev spaces with variable exponents. Several sufficient conditions for the density are presented and several counterexamples when the density does not hold are given.
In Chapter 10 two kinds of capacities are introduced: the Sobolev capacity and the relative capacity. For a constant exponent these definitions agree with classical ones. Capacities are used to understand the pointwise behavior of Sobolev functions. These capacities are compared with each other and with the variable dimension Hausdorff measure.
Chapter 11 is devoted to the study of fine properties of Sobolev functions which are defined only up to a set of measure zero. The authors pick a good representative from every equivalence class of Sobolev functions and show that this representative, called quasicontinuous, has many good properties. The main tools in this chapter are the capacities introduced in Chapter 10. Each Sobolev function has a quasicontinuous representative under a natural assumption on the variable exponent. Removable sets are studied in terms of capacities. It is proved that if $$p$$ is globally log-Hölder continuous, then each point is a Lebesgue point except for a set of Sobolev $$p(\cdot)$$-capacity zero. It is shown by example that for more general exponents this is not the case.
Chapter 12 deals with other spaces of Sobolev type, that is, spaces of functions of some (possible fractional) smoothness. In particular, homogeneous Sobolev, Bessel potential, Besov, and Triebel-Lizorkin spaces with variable exponents are studied.
In the third part of the book, applications of the results of the first two parts to partial differential equations are developed. In Chapter 13, partial differential equations with non-standard growth are considered. The Laplace equation can be generalized to the variable exponent setting as $$\text{div}(p(x)|\nabla u(x)|^{p(x)-2}\nabla u)=0$$. In this case the Sobolev space $$W^{1,p(\cdot)}(\Omega)$$ is the natural space in which to look for a solution. The approach of this chapter is based on use the of capacities and fine properties of functions from Chapters 10 and 11.
In Chapter 14, the authors use the theory of Calderón-Zygmund operators to prove regularity results for the Poisson problem and the Stokes problem, to show the solvability of the divergence equation and to prove Korn’s inequality. The last section of the chapter is devoted to the existence theory of so-called electrorheological fluids. This section nicely illustrates how all the previously developed theory is used. The reader can find more information on electrorheological fluids in the monograph by [M. Ružička, Electrorheological fluids: modeling and mathematical theory. Lecture Notes in Mathematics 1748. Berlin: Springer (2000; Zbl 0962.76001)].
The majority of the results presented in the monograph were obtained by the authors and their collaborators. The list of references contains 399 items and, certainly, it is far from being complete. For instance, the pioneering papers by H. Hudzik on properties of Musielak-Orlicz-Sobolev spaces (see above) are not mentioned at all. Nevertheless, the books is a useful source of unified information on Lebesgue and Sobolev spaces with variable exponents.

### 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 42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory 35Q35 PDEs in connection with fluid mechanics 76A05 Non-Newtonian fluids 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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