Orliczgarling sequence spaces of difference operator and their. Korn inequalities in orlicz spaces tcd mathematics. Thanks for contributing an answer to mathematics stack exchange. Lebesgue, lorentz, zygmund, and orlicz spaces receive detailed treatment, as do the classical interpolation theorems and their applications in harmonic analysis.
Extreme cases of weak type interpolation with applications. Asking for help, clarification, or responding to other answers. Alberto fiorenza, miroslav krbec, indices of orlicz spaces and some applications. Interpolation of markov maps on quantum orlicz spaces l e labuschagne1 and w a majewski2 1. Free interpolation in hardy spaces is characterized by the wellknown carleson condition.
It should be pronounced clearly that orlicz spaces from the functional analysis point of. Pdf extrapolation and interpolation in generalized. Ams transactions of the american mathematical society. Xjm, where x 0,1, 1 s p orlicz space l is an interpolation space between lp and lq but the constant cin the estimation of the norm can be large. It is shown that under suitable conditions on the type of these spaces and the boundedness of the classical. As an application, we prove some martingale inequalities with weak orlicz space norm. Interpolation and harmonic majorants in big hardyorlicz. Department of mathematics, palestine polytechnic university, visitor. This book presents a systematic treatment of generalized orlicz spaces also known as musielakorlicz spaces with minimal assumptions on the generating. Interpolation of bilinear operators between quasibanach spaces.
Orlicz spaces and generalized orlicz spaces petteri. On interpolation in hardy orlicz spaces mahmud masri. Andreas hartmann, free interpolation in hardy orlicz spaces natan kruglyak, lech maligranda, lars persson, a carlson type inequality with blocks and interpolation alberto fiorenza, miroslav krbec, indices of orlicz spaces and some applications. Interpolation of linear operators on orlicz and bmo. Ming fan, complex interpolation functors with a family of quasipower function parameters. We used properties of random processes in orlicz space described in articles bykozachenko and pashko 1988, tegza 2002,antoniniet al 2002. For the smirnov and the nevanlinna classes, interpolating sequences have been characterized in a recent paper in terms of the existence of harmonic.
The nfunctions involved satisfy certain consistency conditions. The main field of application are the degenerate elliptic differential operators. The result extends to hardy orlicz spaces contained in the scale of classical hardy spaces hp, p0. These results generalize our results for linear operators obtained in 7. Marcinkiewicz interpolation theorem in orlicz spaces 41 c.
The result extends to hardy orlicz spaces contained in the scale of classical hardy spaces hp, p 0. History media compliance careers affiliate program. Natan kruglyak, lech maligranda, lars persson, a carlson type inequality with blocks and interpolation. Fund fwf project f5508n26, which is part of the special research program quasi. A generalization of the marcinkiewicz interpolation theorem to orlicz spaces contains the conditions 1. Maligranda of lulei dedicated to professor lars erik persson on the occasion of his 50th birthday received march 10, 1994 abstract. Interpolation theorems involving operators with change of measures and weights were proved in lo, 91 and later in 6. We investigate composition operators on hardyorlicz spaces when the orlicz function.
Instytut matematyczny polskiej akademi naukwarszawa, 1985. I found holder inequalities for orlicz norms as well as some inequalities about convolutions, but i could not find the above interpolation inequalities. Citeseerx document details isaac councill, lee giles, pradeep teregowda. The study of orlicz sequence spaces was initiated with a certain specific purpose in banach space theory. Marcinkiewicz theorems for orlicz and lorentz gamma spaces 17 theorem a. Interpolation of weighted orlicz spaces researchgate. Strong and weak lebesgue spaces, lorentz spaces, and orlicz spaces.
First, necessary and sufficient conditions for their continuity are presented in several forms. Interpolation inequalities for derivatives in orlicz spaces. An extension of the marcinkiewicz interpolation theorem is proved, yielding a necessary and sufficient condition for every quasilinear operator, satisfying given endpoint estimates of weak type, to be bounded from an orlicz space into another. In the particular case of the variable lebesgue spaces, this was proved very recently in 16. The result extends to hardyorlicz spaces contained in the scale of classical hardy spaces h p, p 0.
Stein and taifjleson gave a characterization for f e lpirn to be in the spaces. For the smirnov and the nevanlinna classes, interpolating. Buy orlicz spaces and interpolation seminars in mathematics on free shipping on qualified orders. Lipschitz orlicz spaces and the laplace equation by a. For the smirnov and the nevanlinna classes, interpolating sequences have been characterized in a recent paper in terms of the. Interpolation of operators from a suitable couple of sobolev spaces provided that both the function and its complementary function satisfy the condition or. For the smirnov and the nevanlinna classes, interpolating sequences have been characterized in a recent paper in terms.
We prove variants of the classical gagliardonirenberg inequalities in both additive and multiplicative forms, with orlicz norms replacing the usual v norms. Primarily aimed at researchers and phd students interested in orlicz spaces or generalized orlicz spaces, this book can be used as a basis for advanced graduate courses in analysis. Construction of a model of gaussian stationary random process. In this paper, we investigate the behavior of solutions to 2d kleingordon equation in the framework of orlicz norm. Interpolation of markov maps on quantum orlicz spaces. Approximation of modified lagrange interpolation in orlicz spaces. On interpolation of the fourier maximal operator in orlicz spaces. Korn inequalities in orlicz spaces martin fuchs abstract. Examines the sample path behavior of stochastic processes. In this work two versions of weak orlicz spaces that appear in the literature, ma and ma. Extreme cases of weak type interpolation with applications to orliczsobolev embeddings article january 2003 with 14 reads how we measure reads. Mathematics faculty of engineering centre for mathematical sciences publishing date. An optimal interpolation theorem of marcinkiewicz type in.
Mathematica indicate direct continuation of banachs scientific program. Interpolation inequalities for derivatives in orlicz spaces agnieszka kalamajska and katarzyna pietruskapaluba abstract. Convergence for a family of neural network operators in orlicz spaces. Abstract quantum dynamical maps are defined and studied for quantum statistical physics based on orlicz spaces. It should be pronounced clearly that orlicz spaces from the functional analysis. Interpolation in orlicz spaces involving weights core. As outlined for example in the monographs of m alek, ne cas, rokyta, r u zi cka 18, of duvaut and lions 7 and of zeidler 26, the well. And applying the interpolation theorem, we obtain some embedding relationships among weak orlicz martingale spaces.
In this paper, the authors give the marcinkiewiczzygmund inequality based on the zeros of the first kind chebyshev polynomials in orlicz norm. Gaussian stationary random process in some orlicz spaces. The spaces are named for wladyslaw orlicz and zygmunt william birnbaum, who. Approximation by means of nonlinear kantorovich sampling type. As a consequence, we obtain boundedness results for several classical operators as well as a sobolev inequality in this setting. There are only very few papers which are concerned with the interpolation theory of these spaces.
Orlicz spaces and interpolation seminars in mathematics. On best interpolation in orlicz spaces springerlink. Interpolation of operators, volume 129 pure and applied. Composition operators and multiplication operators between two orlicz function spaces are investigated. The spaces are named for wladyslaw orlicz and zygmunt william birnbaum, who first defined them in 1931 besides the l p spaces, a variety of function spaces arising naturally in analysis. It is proved to be true for weak orlicz martingale spaces by weak atomic decomposition. The analysis we conducted in this article, which is based on profiles decompositions, emphasizes the distinguished role played by the 1oscillating component of the sequence of the cauchy data. A good place to start are some surveys written by andrea cianchi. We shall generalize this for orlicz spaces by means of the inequality of lemma 2.
The extension of the best approximation operator is. Studia mathematica indicate direct continuation of banachs scientific program. The freedom gained in considering more general kind of spaces can be used for selecting the interpolants in an adequate class of functions. Sharpley, r interpolation of operators, academic press, boston, 1988. In the mathematical analysis, and especially in real and harmonic analysis, a birnbaum orlicz space is a type of function space which generalizes the l p spaces. Strong and weak lebesgue spaces, lorentz spaces, and. In this paper, we use the interpolation method introduced by gustavssonpeetre stud. As application, the degree of approximation by two kinds of modified lagrange interpolatory polynomials in orlicz spaces is studied. These notes are devoted to the analysis on a capacity space, with capacities as substitutes of measures, of the orlicz function spaces. The spaces are named for wladyslaw orlicz and zygmunt william birnbaum, who first defined them in 1931. The existence of harmonic majorants in such orlicz spaces will also be discussed in the general situation. In this article, we show that orliczlorentz spaces. It is proved to be true for weak orlicz martingale spaces by weak atomic decomposition of weak hardy martingale spaces. Orlicz spaces and generalized orlicz spaces springerlink.
Pdf convergence for a family of neural network operators in. Andreas hartmann, free interpolation in hardy orlicz spaces. Some fixed point results for generalized weak contraction mappings in modular spaces. Wladyslaw orlicz 1903 1990 mactutor history of mathematics. Finally, central results are provided for sobolev spaces, including poincare and sobolevpoincare inequalities in norm and modular forms. Marcinkiewicz type interpolation theorems for weak orlicz martingale spaces. Extrapolation and interpolation in generalized orlicz spaces authors. Free interpolation in hardy spaces is caracterized by the wellknown carleson condition. Convex functions on dual orlicz spaces 3 cesaro means are order bounded in l as well. Extrapolation and interpolation in generalized orlicz spaces 3 our.
In the mathematical analysis, and especially in real and harmonic analysis, a birnbaumorlicz space is a type of function space which generalizes the l p spaces. Our interest here is in the direction of the interpolation of operators between orlicz spaces and also between their possible. For given data, an interpolant is sought, so that a certain convex functional defined by a youngs function in the corresponding orlicz space is minimized. The interpolation constant c, in general, is less than 4 and, in many cases, we can give much better estimates for c. The purpose of these notes is to give an exposition of basic results in the theory of orlicz spaces and some results about operators between them. Orlicz spaces and generalized orlicz spaces petteri harjulehto. Keywords sobolev inequalities orlicz spaces rearrangement invariant spaces interpolation. On interpolation of the fourier maximal operator in orlicz. The result extends to hardy orlicz spaces contained in the scale of classical hardy spaces h p, p 0. Boyd 4 extended in 1967 this theorem to rearrangementinvariant spaces and his constant in the estimation of the norm increases to. We finish the paper with an example of a separated blaschke sequence that is interpolating for certain hardy orlicz spaces without being interpolating for slightly smaller ones. The text includes a wide range of techniques and applications, and will serve as an amenable introduction and useful reference to the modern theory of interpolation of operators.
Applications of orlicz spaces crc press book presents previously unpublished material on the fundumental pronciples and properties of orlicz sequence and function spaces. Marcinkiewicz type interpolation theorems for weak orlicz. High frequency approximation of solutions to kleingordon. Double sequence spaces by means of orlicz functions. It is a great honor to be asked to write this article for the proceedings of the conference. This book presents a systematic treatment of generalized orlicz spaces also known as musielak orlicz spaces with minimal assumptions on the generating. We use gradient estimates for solutions of elliptic equations to obtain korns inequality for elds with zero trace from orliczsobolev classes. Fractional sobolev spaces and interpolation in unbounded lipschitz domains. Is there something like an interpolation theorem for. Capacitary orlicz spaces, calderon products and interpolation abstract. The goal is to study some aspects of the classical theory of orlicz spaces for these spaces including the classical theory of. School of statistical, computer and mathematical sciences. Since then, the theory of modulars and modular spaces is widely applied in the study of interpolation theory 22, 26 and various orlicz spaces 40. For example, cianchi, andreaifrnzama on some aspects of the theory of orlicz sobolev spaces.
Is there something like an interpolation theorem for fredholm. While preparing an upcoming post on interpolation inequalities and interpolation spaces, i noticed that many of the interesting examples came in the form of variations on the notion of an l p space such as the weak l p spaces and lorentz spaces. Our theory applies to l p spaces, interpolation spaces, exponential spaces and many others. Extension of the best approximation operator in orlicz spaces. It introduces and develops a technique centered on the use of equivalent. In the field of mathematical analysis, an interpolation space is a space which lies in between two other banach spaces. We present have the salient features from the theory of orlicz function spaces, l. In this paper we deal with the interpolation from lebesgue spaces l p and l q, into an orlicz space l.
Pdf marcinkiewicz interpolation theorems for orlicz and. Generalized orlicz spaces, also known as musielak orlicz spaces and nakano spaces, are a class of banach spaces that include a number of spaces of interest in harmonic anal ysis and pdes as. In this section, we focus on employing the technique of atomic decomposition to establish two marcinkiewicz type interpolation theorems for weak orlicz martingale spaces. It is the purpose of this paper to generalize the interpolation theorems of torchinsky l l to weighted orlicz spaces. L maligranda, orlicz spaces and interpolation, campinas 1989, m m rao and z d ren, theory. For a detailed treatment of interpolation theory, we refer to 2. The main applications are in sobolev spaces, where spaces of functions that have a noninteger number of derivatives are interpolated from the spaces of functions with integer number of derivatives. On interpolation in hardy orlicz spaces mahmud masri department of mathematics, faculty of science, annajah national university, palestine sabbatical. Interpolation of bilinear operators between quasibanach spaces loukas grafakos. Maligranda, orlicz spaces and interpolation, campinas 1989. Moreover, we study the problem of the convergence in the setting of orlicz spaces. We investigate composition operators on hardy orlicz spaces when the orlicz function.
Interpolation of weighted orlicz spaces sciencedirect. Composition and multiplication operators between orlicz. Spaces of measurable functions l pspaces, orlicz spaces, kothe function spaces, lorentz spaces, rearrangement invariant spaces, ideal spaces, etc. Pdf on the interpolation constant for orlicz spaces. For given data, an interpolant is sought, so that a certain convex functional defined by a youngs function in the corresponding orlicz space is minimi the freedom gained in considering more general kind of spaces can be used for selecting the interpolants in an adequate class of functions. An interpolation theorem for weak orlicz spaces generalized by nfunctions satisfying m.
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