FUNCTION SPACES WITH VARIABLE EXPONENTS – AN INTRODUCTION –
FUNCTION SPACES WITH VARIABLE EXPONENTS – AN INTRODUCTION –
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DOI：
10.32219/isms.77.2_187
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
M. Izuki;E. Nakai;Y. Sawano
中科院分区：
文献类型：
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作者：
M. Izuki;E. Nakai;Y. Sawano
本文旨在对变指数函数空间作一个初步的介绍，并对相关的函数空间作一综述。在给出变指数广义Lebesgue空间Lp（·）（Rn）的基本性质和初等性质之后，我们给出了Diening（2004），Cruz-Uribe，Fiorenza and Neugebauer（2003，2004），Nekvinda（2004）和Lerner（2005）定理的重新证明.它们可能比原始的更简单。此外，我们还讨论了与Lp（·）（Rn）有关的问题。例如，我们给出了Lerner关于模不等式的定理的另一种证明和变指数Sobolev空间中密度的详细证明。进一步，我们将描述Lp（·）（Rn）上的分数次积分算子和Calderón-Zygmund算子的最新结果.最后，我们回顾了最近的结果，（无需证明）在几个变指数函数空间上，例如，变增长条件的广义Morrey和Campanato空间，哈代空间Hp（·）（Rn），Besov空间B p（·），q（·）（R）和Triebel-Lizorkin空间Fs（·）p（·），q（·）（R）等.前言近年来，在调和分析，偏微分方程，在位势论和应用数学中，许多作者研究了变指数函数空间。特别是，具有可变指数的函数空间在电子流体力学[177]和图像恢复[17，65，108]的应用领域中是必要的。Kovacik和Rakosnik [101]给出了变指数广义Lebesgue空间在变增长系数非线性偏微分方程Dirichlet边值问题中的应用。另一个应用于微分方程的简单例子可以在[49，p.438，Example]中找到，其中Fan和Zhao隐含地表明可变Lebesgue空间可以用来控制微分方程的非线性项。变指数Lebesgue空间的理论可以追溯到Orlicz的论文[163]（1931）和中野的书[158，159]（1950，1951）。特别地，所谓的Musielak-Orlicz空间的定义在[158，89节]中清楚地写了出来，而Orlicz似乎主要对函数空间的完备性感兴趣。后来，Sharapudinov [208]（1979）和Kováčik和Rákosık [101]（1991）阐明了变指数Lebesgue空间和变指数Sobolev空间的基本性质。这一重要成果引起了人们对变指数函数空间的热烈讨论。一个值得注意的事实是，范和赵独立调查勒贝格空间与可变指数和索伯列夫空间与可变指数。证明Hardy-Littlewood极大算子M在变指数广义Lebesgue空间Lp（·）（Rn）上的有界性是这一领域的重要问题之一。一旦这一点确立，我们的经验使我们感到，这种有界性可以2010年数学学科分类。46 E30、42 B25、46 E35、42 B20、26 A33、42 B30、42 B35、46 E15、46 A80、46 B10。
This paper is oriented to an elementary introduction to function spaces with variable exponents and a survey of related function spaces. After providing basic and elementary properties of generalized Lebesgue spaces Lp(·)(Rn) with variable exponents, we give rearranged proofs of the theorems by Diening (2004), Cruz-Uribe, Fiorenza and Neugebauer (2003, 2004), Nekvinda (2004) and Lerner (2005). They are maybe simpler than the originals. Moreover, we deal with topics related to Lp(·)(Rn). For example, we present an alternative proof for Lerner’s theorem on the modular inequality and a detailed proof of the density in Sobolev spaces with variable exponents. Furthermore, we will describe the recent results of fractional integral operators and Calderón-Zygmund operators on Lp(·)(Rn). Finally, we survey recent results (without proofs) on several function spaces with variable exponents, for example, generalized Morrey and Campanato spaces with variable growth condition, Hardy spaces Hp(·)(Rn), Besov spaces B p(·),q(·)(R ) and Triebel-Lizorkin spaces F s(·) p(·),q(·)(R ), etc. Preface Recently, in harmonic analysis, partial differential equations, potential theory and applied mathematics, many authors investigate function spaces with variable exponents. In particular, function spaces with variable exponents are necessary in the field of electronic fluid mechanics [177] and the applications to the image restoration [17, 65, 108]. Kovacik and Rakosnik [101] gave an application of generalized Lebesgue spaces with variable exponents to Dirichlet boundary value problems for nonlinear partial differential equations with coefficients of a variable growth. Another simple example of the application to differential equations can be found in [49, p. 438, Example], where Fan and Zhao implicitly showed that the variable Lebesgue spaces can be used to control the non-linear term of differential equations. The theory of Lebesgue spaces with variable exponents dates back to Orlicz’s paper [163] (1931) and Nakano’s books [158, 159] (1950, 1951). In particular, the definition of so-called Musielak-Orlicz spaces is clearly written in [158, Section 89], while it seems that Orlicz is mainly interested in completeness of function spaces. Later, Sharapudinov [208] (1979) and Kováčik and Rákosńık [101] (1991) clarified fundamental properties of Lebesgue spaces with variable exponents and Sobolev spaces with variable exponents. This important achievement nowadays leads to the hot discussion of function spaces with variable exponents. A noteworthy fact is that Fan and Zhao independently investigated Lebesgue spaces with variable exponents and Sobolev spaces with variable exponents. One of the important problems in this field is to prove the boundedness of the HardyLittlewood maximal operator M on generalized Lebesgue spaces Lp(·)(Rn) with variable exponents. Once this is established, our experience makes us feel that this boundedness can 2010 Mathematics Subject Classification. 46E30, 42B25, 46E35, 42B20, 26A33, 42B30, 42B35, 46E15, 46A80, 46B10.