Request pdf grand lebesgue spaces with respect to measurable functions let 1 lebesgue measure, the norm of the grand lebesgue spaces l p. On the approximation of periodic functions within the. Pdf we consider a generalized version of the small lebesgue spaces, introduced in 5 as the associate spaces of the grand lebesgue. We construct examples to show that our results are close to. A note on the boundedness of the hilbert transform in weighted grand lebesgue spaces. Compact sets in bide side grand lebesgue spaces, with. R n a measurable set of finite lebesgue measure, the norm of the grand lebesgue spaces l p. Inequalities with conjugate exponents in grand lebesgue spaces. We consider the banach function spaces, called fully measurable grand lebesgue spaces, associated with the function norm where denotes the norm of the lebesgue space of exponent, and and are measurable functions over a measure space, and almost everywhere. Hardy type operators on grand lebesgue spaces for non. As a direct consequence of our results any lorentzzygmund space, is an interpolation space in the sense of peetre between either two grand lebesgue spaces or between two small spaces. Characterization of interpolation between grand, small or classical lebesgue spaces author amiran gogatishvili joint work with alberto fiorenza, maria rosaria formica, tengiz kopaliani and jean michel rakotoson 1international conference new perspectives in the theory of function spaces and their applications npfsa2017 september 1723, 2017.
The results are proved also for the generalized small and grand spaces topics. Equivalent norms on grand and small lebesgue spaces grand lebesgue space small lebesgue space interpolation theorem compactness of sobolev imbedding theorem a. Operations on measurable functions sums, products, composition realvalued measurable functions. Fully measurable grand lebesgue spaces sciencedirect. We give equivalent, explicit expressions for the norms of the small and grand lebesgue spaces, which depend only on the nondecreasing rearrangement. Integral operators in bilateral grand lebesgue spaces. Density, duality and preduality in grand variable exponent. In 9, the authors studied the boundedness of the maximal operator between lp wi spaces. We prove that every such space can be expressed equivalently replacing and with functions defined everywhere on the interval.
Generalized singular integral on carleson curves in. Pdf a note on the boundedness of the hilbert transform. This book, the result of the authors long and fruitful collaboration, focuses on integral operators in new, nonstandard function spaces and presents a systematic study of the boundedness and compactness properties of basic, harmonic analysis integral operators in the following function spaces, among others. Real interpolation of small lebesgue spaces in a critical case. The boundedness of the maximal operator on the grand lebesgue spaces was studied in 9. Iterated grand and small lebesgue spaces, collectanea. Duality and reflexivity in grand lebesgue spaces core. Comprehensive study and some aspects of applications of one these spaces were delivered in the recently published books 1, 6, 23. Generalization of the notion of grand lebesgue space.
These facts make the study of weighted grand lebesgue spaces important. Grand l p spaces have been considered in various fields. Pdf identification of fully measurable grand lebesgue spaces. We introduce a socalled restricted, in particular, discrete version of banach grand lebesgue spaces gls. In this article we introduce and investigate a new class of rearrangement invariant r. Embeddings between grand, small, and variable lebesgue spaces oscar m. Finally, the fundamental function of the grand lebesgue space is estimated, and. An introductory course in lebesgue spaces rene erlin. Npfsa 2017 some aspects of fully measurable grand lebesgue spaces pankaj jain south asian university new delhi, india international conference on new perspective in. Pdf in this paper we estimate the norm of operator acting from one bilateral grand lebesgue space bgls into other bilateral grand lebesgue space.
However, the same is not true for grand lebesgue spaces see 9. The reader is referred to a recent survey paper for. Recently the in the weighted boundedness criteria for cauchy singular integral operator in weighted grand lebesgue spaces with respect to measurable function were established. Characterization of interpolation between grand, small or. Request pdf fully measurable grand lebesgue spaces using variable exponents, we build a new class of rearrangementinvariant banach function spaces, independent of the variable lebesgue spaces. Request pdf grand lebesgue spaces with respect to measurable functions let 1 small and variable lebesgue spaces d. We establish an interpolation formula for small lebesgue spaces in a critical case. Inclusions and the approximate identities of the generalized grand lebesgue spaces a. These spaces unify two nonstandard function spaces. Grand lebesgue spaces with respect to measurable functions. Pdf integral operators in bilateral grand lebesgue spaces.
The weighted grand lebesgue space of 2 periodic functions is defined by the norm, 2 1 0 1 sup p 2 p p l p t f f x w x dx. The classical grand lebesgue spaces lp were introduced in the paper by t. Approximation in weighted generalized grand lebesgue spaces. Iterated grand and small lebesgue spaces iterated grand and small lebesgue spaces anatriello, giuseppina 201106 00. We study the inverse problem of approximation theory in generalized grand lebesgue spaces. Boundedness for hardylittlewood maximal operator and hilbert transform in weighted grand l. Grand and small lebesgue spaces and their analogs core. The generalized grand lebesgue spaces considered in 7 are evidently included in the family of the function norm. In the weighted case, grand lebesgue spaces on the bounded sets of a euclidean space were introduced. In this paper, we show that the interpolation spaces between grand, small or classical lebesgue are so called lorentzzygmund spaces or more generally spaces. The book also touches on basic harmonic analysis in the aforementioned spaces. The lecture notes were prepared in latex by ethan brown, a former student in the class. Another approach regarding variable smoothness and integrability to study herz type hardy spaces was used in.
Sbordone 19 when they studied the problem of the integrability of the jacobian jf,x of the order preserving mapping f f 1,f n. The variable exponent lebesgue space represents the special. Pdf this thesis aims at the utilization of elementary methods to prove that the classical lebesgue spaces lp are the intermediate spaces for grand. Hyperbolic equation, parabolic equation, coercive operator, oscillation, singular coefficient. Fiorenzaa direct approach to the duality of grand and small lebesgue spaces. In this article we find some sufficient conditions for the set in the bilateral grand lebesgue space to be compact set. Moreover, it also deals with weak, lorentz and the more recent variable exponent and grand lebesgue spaces with considerable detail to the proofs. Some aspects of fully measurable grand lebesgue spaces. In this paper we estimate the norm of operator acting from one bilateral grand lebesgue space bgls into other bilateral grand lebesgue space. Fiorenza in, and they have found applications, for example, in boundary value problems 2, 3, besov embeddings 4, 5, and dimensionfree sobolev embeddings. Pdf we consider the banach function spaces, called fully measurable grand lebesgue spaces, associated with the function norm. He used professor viaclovskys handwritten notes in producing them. Grand spaces proved to be useful in application to partial differential equations.
Fully measurable grand lebesgue spaces request pdf. Pdf identification of fully measurable grand lebesgue. Lp space, a special banach space of functions or rather, equivalence classes standard probability space, a nonpathologic probability space. The generalized grand lebesgue spaces appeared in 12, where the existence and uniqueness of the nonhomogeneous nharmonic equations were studied. Boundedness for hardylittlewood maximal operator and.
Unique in its sole dedication, this book explores lebesgue spaces, distribution functions and nonincreasing rearrangement. Embeddings between grand, small, and variable lebesgue. Browse other questions tagged realanalysis functionalanalysis measuretheory lebesgueintegral lebesguemeasure or ask your own question. Inequalities with conjugate exponents in grand lebesgue spaces rene erlin castillo and humberto rafeiroy abstract in this paper we want to show the validity of the generalized doubleparametric hilbert inequality with conjugate exponents in the framework of grand lebesgue spaces as a particular case of a more general. Banach function spaces, socalled composed grand lebesgue spaces cgls, in particular, integral grand. In this paper we consider the spaces defined by a norm with an analogous expression, where. We give conditions on the exponent function pthat imply the existence of embeddings between the grand, small and variable lebesgue spaces. This disambiguation page lists mathematics articles associated with the same title. We construct examples to show that our results are close to optimal. Equivalent norms on grand and small lebesgue spaces. A note on the boundedness of sublinear operators on grand.
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