On the convolution operator in Morrey spaces Nursultanov, Erlan D.; Suragan, Durvudkhan
Journal of mathematical analysis and applications,
11/2022, Letnik:
515, Številka:
1
Journal Article
Recenzirano
This paper is devoted to the study of upper bounds for the norm of the convolution operator in Morrey spaces. The spaces Mp,qα, which cover the classical Morrey spaces, are introduced. Moreover, ...their embedding properties are investigated, and their interpolation properties are described. Young-O'Neil type inequalities in Morrey spaces are proved. New results on the boundedness of Riesz's potential in Morrey spaces are established.
Two exact embedding theorems are proved for arbitrary interpolation functors on arbitrary couples of global Morrey spaces. Since the calculation of interpolation functors on couples of global Morrey ...spaces is known only in certain cases, these theorems can replace interpolation theorems for global Morrey spaces. This fact is demonstrated by examples of functors of real interpolation Peetre, of complex interpolation Calderon’s, and a functor generated by unconditionally convergent series.
A formula is given that makes it possible to reduce the calculation of an interpolation functor on a pair of local Morrey spaces to the calculation of this functor on pairs of vector function spaces ...constructed from the ideal spaces involved in the definition of the Morrey spaces in question. It is shown that a pair of local Morrey spaces is K-monotone if and only if the pair of vector function spaces mentioned above is K-monotone. This reduction makes it possible to obtain new interpolation theorems even for classical local spaces.
We employ a new approach to show that the Calderón construction for a couple of global Morrey spaces coincides with the Morrey space with appropriate parameters only under rather strong assumptions ...on the couples of ideal spaces that parameterize the original Morrey spaces. We show that, in the case of classical examples of global Morrey spaces, these assumptions are necessary and sufficient. Applying a well-known reduction, we use the Calderón construction for a couple of global Morrey spaces to describe the spaces given by the complex interpolation method and also to prove new interpolation theorems for global Morrey spaces.
We present short proofs of the classical interpolation theorems for derivatives by Mittag–Leffler and Germay, respectively Katsnelson, Airapetjan, Dzhrbashyan and Vinogradov and present such a ...theorem for the disk algebra.
The aim of this expository article is to present recent developments in the centuries-old discussion on the interrelations between continuous and differentiable real valued functions of one real ...variable. The truly new results include, among others, the D^n- C^n interpolation theorem: For every n-times differentiable f\colon \mathbb{R}\to \mathbb{R} and perfect P\subset \mathbb{R}, there is a C^n function g\colon \mathbb{R}\to \mathbb{R} such that f\restriction P and g\restriction P agree on an uncountable set and an example of a differentiable function F\colon \mathbb{R}\to \mathbb{R} (which can be nowhere monotone) and of compact perfect \mathfrak{X}\subset \mathbb{R} such that F'(x)=0 for all x\in \mathfrak{X} while F\mathfrak{X}=\mathfrak{X}. Thus, the map \mathfrak{f}=F\restriction \mathfrak{X} is shrinking at every point though, paradoxically, not globally. However, the novelty is even more prominent in the newly discovered simplified presentations of several older results, including a new short and elementary construction of everywhere differentiable nowhere monotone h\colon \mathbb{R}\to \mathbb{R} and the proofs (not involving Lebesgue measure/integration theory) of the theorems of Jarník-- Every differentiable map f\colon P\to \mathbb{R}, with P\subset \mathbb{R} perfect, admits differentiable extension F\colon \mathbb{R}\to \mathbb{R} --and of Laczkovich-- For every continuous g\colon \mathbb{R}\to \mathbb{R} there exists a perfect P\subset \mathbb{R} such that g\restriction P is differentiable . The main part of this exposition, concerning continuity and first-order differentiation, is presented in a narrative that answers two classical questions: To what extent must a continuous function be differentiable ? and How strong is the assumption of differentiability of a continuous function ? In addition, we give an overview of the results concerning higher-order differentiation. This includes the Whitney extension theorem and the higher-order interpolation theorems related to the Ulam-Zahorski problem. Finally, we discuss the results concerning smooth functions that are independent of the standard axioms of ZFC set theory. We close with a list of currently open problems related to this subject.
The generalization of the Lions-Peetre interpolation method of means considered in the present survey is less general than the generalizations known since the 1970s. However, our level of ...generalization is sufficient to encompass spaces that are most natural from the point of view of applications, like the Lorentz spaces, Orlicz spaces, and their analogues. The spaces considered here have three parameters: two positive numerical parameters and of equal standing, and a function parameter . For these spaces can be regarded as analogues of Orlicz spaces under the real interpolation method. Embedding criteria are established for the family of spaces , together with optimal interpolation theorems that refine all the known interpolation theorems for operators acting on couples of weighted spaces and that extend these theorems beyond scales of spaces. The main specific feature is that the function parameter can be an arbitrary natural functional parameter in the interpolation. Bibliography: 43 titles.
Though deceptively simple and plausible on the face of it, Craig's interpolation theorem (published 50 years ago) has proved to be a central logical property that has been used to reveal a deep ...harmony between the syntax and semantics of first order logic. Craig's theorem was generalized soon after by Lyndon, with application to the characterization of first order properties preserved under homomorphism. After retracing the early history, this article is mainly devoted to a survey of subsequent generalizations and applications, especially of many-sorted interpolation theorems. Attention is also paid to methodological considerations, since the Craig theorem and its generalizations were initially obtained by proof-theoretic arguments while most of the applications are model-theoretic in nature. The article concludes with the role of the interpolation property in the quest for "reasonable" logics extending first-order logic within the framework of abstract model theory.
PDF
