# Lebesgue and sobolev spaces with variable exponents pdf

Geometric properties of banach space valued bochner. Since then many researchers have tried to overcome this problem. On boundedness of weighted hardy operator in and regularity condition on boundedness of weighted hardy operator in and regularity condition. The aclcharacterization is that a function belongs to the sobolev space w1,p if and only if it has a lebesgue pintegrable representative which is. Those are a generalization of scalar valued lebesgue and sobolev spaces with variable exponent. In this article we extend the sobolev spaces with variable exponents to include the fractional case, and we prove a compact embedding theorem of these spaces into variable exponent lebesgue spaces. As applications, we prove a new landaukomogorov type inequality for the secondorder derivative and an embedding theorem and discuss the equivalent norms in the space w 0 1, p. Michael ruzicka m lebesgue and sobolev spaces with variable exponents, springer, heidelberg dordrecht london new york2011. Pdf renormalized solutions for nonlinear parabolic. Article in press oof please cite this article in press as. Approximation of functions in variableexponent lebesgue and. The authors provide a comprehensive survey of the state of the art concerning lebesgue and sobolev spaces with variable exponents. Lars diening, petteri harjulehto, peter hasto, michael ruzicka.

This paper established many of the basic properties of lebesgue and sobolev spaces in rn. If the function lies in the sobolev space with variable exponent, it is shown that. Lebesgue points in lebesgue spaces, the topic of section 3, are quite simple to handle and require no indepth knowledge of variable exponent spaces. For some of the latest advances in the study of variable exponent spaces see 3, 4, 11, 12, 16.

Lebesgue and sobolev spaces with variable exponents, lecture notes in math. Fractional sobolev spaces with variable exponents, sobolev capacity, relative capacity, choquet capacity, outer capacity. Sharapudinov 2012 some questions in the theory of approximation in lebesgue spaces with variable. Weighted variable sobolev spaces and capacity weighted variable sobolev spaces and. Research article estimates of fractional integral operators. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Thus this selfcontained monograph collecting all the basic properties of variable exponent lebesgue and sobolev spaces is timely and provides a muchneeded accessible reference work utilizing consistent notation and terminology. At last, we define malliavin derivatives in and discuss some properties of malliavin derivatives in 1. As an application we prove the existence and uniqueness of a solution for a nonlocal problem. Open problems in variable exponent lebesgue and sobolev spaces. Pdf lebesgue and sobolev spaces with variable exponents. In the third part we give a selection of applications of these results to partial di. Samkoon sobolev theorem for the riesz type potentials in lebesgue spaces with variable exponent z. The inequality, which is an analogue of the first jackson theorem, is shown to hold for the finite fourierhaar series, provided that the variable exponent satisfies the condition.

Interpolation theorems for variable exponent lebesgue spaces. Lebesgue and sobolev spaces with variable exponents, lecture notes in mathematics, vol. The classical weak herz type spaces can be traced to the work of hu, lu and yang 7,8 on the study of the boundedness for some operators. For more details we refer to the book by musielak 22 and the papers by edmunds et al. Approximation problems in the variable exponent lebesgue. The classical weak herz type spaces mainly include the weak herz spaces and the. We establish the mapping properties of the fractional integral operators with homogeneous kernels on morrey spaces with variable exponents. We obtain the factorization theorem for hardy space via the variable exponent lebesgue spaces. Thus this selfcontained monograph collecting all the basic properties of variable exponent lebesgue and sobolev spaces is. Thus this selfcontained monograph collecting all the basic properties of variable exponent lebesgue and sobolev spaces is timely and provides a muchneeded accessible reference work utilizing consistent notation and. Fractional sobolev spaces with variable exponents and fractional pxlaplacians article pdf available in electronic journal of qualitative theory of differential equations 201776 october. An overview of variable exponent lebesgue and sobolev spaces, future trends in geometric function theory d.

Mar 22, 2019 we obtain the factorization theorem for hardy space via the variable exponent lebesgue spaces. On sobolev theorem for riesz type potentials in the lebesgue. Vakulovc a centro cemat, ist, portugal b universidade do algarve, faro 8000, portugal c rostov state university, russia received 28 june 2006 available online 12 february 2007 submitted by p. Lebesgue and sobolev spaces with variable exponents lars. The problem of how to extend sobolev functions was recognized early in the development of the sobolev spaces. A local estimate for the maximal function in lebesgue. Pdf boundedness of singular integral operators on weak. A vectorvalued estimate of multilinear calderonzygmund operators in herzmorrey spaces with variable exponents shen, conghui and xu, jingshi, hokkaido mathematical journal, 2017. Preliminaries on variable exponent lebesgue and morrey spaces. Lebesgue and sobolev spaces with variable exponents lecture. As in the classical case, this space coincides with the lebesgue sobolev space for integer. Maximal functions, riesz potentials and sobolevs inequality in generalized lebesgue spaces mizuta, yoshihiro and shimomura, tetsu, 2006. A generalization of the stein weiss inequality 26, that is, sobolev embedding with power weight on unbounded domains for variable exponents was con sidered in 24, where this generalization was obtained with a certain additional restriction on the parameters involved. Thus this selfcontained monograph collecting all the basic properties of variable exponent lebesgue and sobolev spaces.

Research article estimates of fractional integral operators on variable exponent lebesgue spaces canqintang, 1 qingwu, 1 andjingshixu 2 departmentofmathematics,dalianmaritimeuniversity,dalian,liaoning, china department of mathematics, hainan normal university, haikou, china correspondence should be addressed to jingshi xu. The standard reference article for basic properties is already 20 years old. Geometric properties of banach space valued bochnerlebesgue. In a recent e ort to complete the picture of the variable exponent lebesgue and sobolev spaces, almeida and samko 4 and gurka, harjulehto and nekvinda 16 introduced variable exponent bessel potential spaces l. Lebesgue and sobolev spaces with variable exponents, lecture notes in mathematics 2017,springer, heidelberg 2011. Notes on commutator on the variable exponent lebesgue spaces. Potential operators in variable exponent lebesgue spaces. Function spaces with variable exponents henning kempka. Since most of the results contained in this book are no more than ten years.

We show that if the exponent is bounded then almost every point is a lebesgue point. After discussing some approximation results of, sobolev spaces on h with variable exponents are introduced. Lebesgue and sobolev spaces with variable exponents the field of variable exponent function spaces has witnessed an explosive growth in recent years. Cruzuribe, sfo trinity college summer school and workshop harmonic analysis and related topics lisbon, june 2125, 2010. At that time, i was writing a book on variable lebesgue spaces with david cruzuribe.

Note that variable exponent lebesgue spaces are not translation invariant and hence convolution, being the main tool in the classical case, cannot be taken for granted. Littlewoodpaley theory for variable exponent lebesgue spaces and gagliardonirenberg inequality for riesz potentials mizuta, yoshihiro, nakai, eiichi, sawano, yoshihiro, and shimomura, tetsu, journal of the mathematical society of japan, 20. Lebesgue points in variable exponent sobolev spaces on metric measure spaces, complex analysis and free boundary flows, transactions of the institute of mathematics of the national academy of sciences of ukraine, 2004, vol. As is known, last two decades there is an increasing interest to the study of variable exponent spaces and operators with variable parameters in such spaces, we refer for instance to the surveying papers 15, 23, 32, 51, on the progress in this. Recently variable exponent lebesgue and sobolev spaces have attracted a lot of attention, mainly due to their use in pdes and variational problems with nonstandard growth two celebrated applications being the modelling of electrorheological. Pdf open problems in variable exponent lebesgue and. During the following ten years there were many scattered e. Lebesgue and sobolev spaces with variable exponents springer. Interpolation inequalities for derivatives in variable. Approximation of functions in variableexponent lebesgue and sobolev spaces by finite fourierhaar series. Singular operators within the framework of the spaces with variable exponents were studied in 14. Pdf the boundedness of fractional maximal operators on. Thus this selfcontained monograph collecting all the basic properties of variable exponent lebesgue and sobolev spaces is timely and provides a muchneeded accessible reference work utilizing consistent. Nonlinear eigenvalue problems in sobolev spaces with variable exponent teodoraliliana dinu communicated by george isac 2000 mathematics subject classi.

This paper is devoted to the problem of extendability in the fractional sobolev spaces with variable exponent and its relation with the trace operator. In section 4 we study lebesgue points in sobolev spaces. Weighted sobolev theorem in lebesgue spaces with variable exponent n. Nonlinear eigenvalue problems in sobolev spaces with. Fractional integral operators with homogeneous kernels. Pdf fractional sobolev spaces with variable exponents. Malliavin derivatives in spaces with variable exponents. On sobolev theorem for riesz type potentials in the lebesgue spaces with variable exponent by v. Fractional sobolev spaces with variable exponents and.

Approximation of functions in variableexponent lebesgue and sobolev spaces by finite fourierhaar series i i sharapudinovapproximation of functions and their conjugates in variable lebesgue spaces s. Our main result is the following compact embedding theorem into variable exponent. Finally the properties of the banach valued bochnersobolev spaces with variable exponent are also given. At last, we define malliavin derivatives in and discuss some properties of malliavin derivatives in. Many results for variable exponent spaces were obtained, we can refer,,,, and the references therein. Intuitively, a sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. The field of variable exponent function spaces has witnessed an explosive growth in recent years. Function spaces with variable exponents international. Some relations between the variable exponent mazya spaces and the variable exponent sobolev spaces are also achieved. For potential operators there was proved a sobolevspanne type lp. Sobolev spaces are named after the russian mathematician sergei sobolev.

One of such spaces is the lebesgue and sobolev spaces with variable exponent. Variable lebesgue spaces foundation and harmonic analysis. On sobolev theorem for riesz type potentials in the. Twoweight estimates potential operators in variable exponent lebesgue spaces. At the end, we give an application of the previous results for the wellposedness of a class of quasilinear equations with variable exponent. Fractional integral operators with homogeneous kernels on morrey spaces with variable exponents. The book is also having a rich bibliography of 399 entries, a long list of symbols and an index. Weighted sobolev theorem in lebesgue spaces with variable. In appendix a we prove some results on complex interpolation. On sobolev theorem for riesz type potentials in the lebesgue spaces with variable exponent by. On grand and small lebesgue and sobolev spaces and some. Classical sobolev spaces can be characterized in many di.

The lebesgue spaces lprn with variable exponent and the corresponding variable sobolev spaces w k,p r n are of interest for their applications to modelling problems in physics, and to the study of variational integrals and partial differential equations with non. A noteworthy fact is that fan and zhao independently investigated lebesgue spaces with variable exponents and sobolev spaces with variable. Renormalized solutions for nonlinear parabolic systems in the lebesgue sobolev spaces with variable exponents article pdf available in journal of mathematical physics, analysis, geometry 141. As an application, it is proved that if the commutator of coifman, rochberg and weiss b, t is bounded on the variable exponent lebesgue spaces, then. Interpolation in variable exponent spaces introduction.

In this context, the interpolation inequalities of these spaces are of particular interest. Lebesgue and sobolev spaces with variable exponents, lecture notes in mathematics 2017,springer. As in the classical case, this space coincides with the lebesguesobolev space for integer. An exposition of the morrey spaces can be found in the books 19 and 36. On january 27, 2011, miroslav krbec, who should have been the second author of this paper, sent me a message that contained an idea for a proof of a local boundednesstype result for the maximal operator in variable exponent lebesgue spaces.

Approximation of functions in variableexponent lebesgue. Pdf open problems in variable exponent lebesgue and sobolev. Ruzicka 2011 lebesgue and sobolev spaces with variable exponents lecture notes in math. Here, is the modulus of continuity in defined in terms of steklov functions. Lebesgue and sobolev spaces with variable exponents. At the turn of the millennium several factors contributed to start a period of systematic intense study of variable exponent spaces. Approximation problems in the variable exponent lebesgue spaces. Johnnirenberg inequalities with variable exponents on probability spaces hao, zhiwei, jiao, yong, and wu, lian, tokyo journal of mathematics, 2015. Traces and fractional sobolev extension domains with. As an application, it is proved that if the commutator of coifman, rochberg and weiss b, t is bounded on the variable exponent lebesgue spaces, then b is a bounded mean oscillation bmo function. Estimates of fractional integral operators on variable exponent lebesgue spaces. A local estimate for the maximal function in lebesgue spaces. As an application we prove the existence and uniqueness of a solution for a nonlocal problem involving the fractional pxlaplacian. We show the interpolation inequalities for derivatives in variable exponent lebesguesobolev spaces by applying the boundedness of the hardylittlewood maximal operator on l p.

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