We show that useful estimates for the essential norm of pointwise multipliers acting on a function space may be easily obtained from certain functional-analytic facts related to the notions of inner variation and the measure of noncompactness. Information about relevant embedding maps enables the Dirichlet problem for the p-Laplacian to be studied, and a brief discussion is given of the generalizations of the trigonometric functions that appear naturally in this connection. The paper considers the so-called strict s-numbers, which form an important subclass of the family of all s-numbers. However, the interest to be charged by the bank will be passed on to you as an upfront discount. The results are new even in the case of Euclidean spaces. Necessary and sufficient conditions for the boundedness or compactness of T are given. One of the ways of meeting this need is by means of Banach function spaces, which were introduced in 1955 by Luxemburg.
It turns out that if a Zygmund space L p logL a is used as a base instead of L p , then in certain cases the target can be an Orlicz space of multiple exponential type. As an authoritative account of a new and rapidly developing branch of spectral theory, this work will be of great interest to research workers and students in the field and related topics. Recent work concerning the representation of compact linear operators acting between Banach spaces is discussed. We show that where cpq is an explicit constant depending only on p and q. There are applications to the p-Laplacian and similar nonlinear partial differential equations. The maps considered are either simple integral operators acting in Lebesgue spaces or Sobolev embeddings; in these cases the exact value of the strict s-numbers is determined. Among other features we only want to mention the multiplicativity of entropy numbers: let X, Y, Z be complex quasi- Banach spaces and.
On the basis of these estimates for singular integrals and maximal functions we formulate the norm convergence and a summability by linear methods of Fourier series in a two-weighted setting. It is shown that for a simple integral operator of Hardy type the singular values are the eigenvalues of a non-linear Sturm-Liouville equation and coincide with the approximation numbers of the operator. These lower bounds contradict the estimates from above that would be obtained if the behaviour of entropy numbers under real interpolation was as good as conjectured. The results are applied to give sharp two-sided estimates of the entropy numbers of some embeddings of Besov spaces. An outline is given of the work of Grubb in which her abstract theory is applied to uniformly elliptic operators generated by differential expressions A, leading to the identification of all closed realisations of A by means of boundary conditions expressed in terms of differential operators acting between function spaces defined on the boundary. Then we focus on the Hardy operator with functions u and v both identically equal to 1 and certain first-order Sobolev embeddings, and show that the generalised trigonometric functions play an essential rôle in the derivation of estimates of s-numbers of these maps. At the present time, this is the only book which focuses systematically on differential operators on spaces with variable integrability.
Concerning the embedding id B given in 1. These spaces have attracted attention not only because of their intrinsic mathematical importance as natural, interesting examples of non-rearrangement-invariant function spaces but also in view of their applications, which include the mathematical modeling of electrorheological fluids and image restoration. In this chapter we introduce the p-trigonometric functions, for 1 p ¥, and establish their fundamental properties. Operator Theory: Advances and Applications, vol 238. You may not use this site to distribute or download any material when you do not have the legal rights to do so. The first four sections deal with symmetric Sturm—Liouville operators and lead to a description of coercive sectorial operators of this type. When u and v are both identically equal to 1 and b is finite, the exact value of the norm of T is determined; it is shown that it is attained at a function expressible in terms of generalised trigonometric functions.
This work enables us to construct an example showing that the behaviour under real interpolation of entropy numbers can be even worse than in the example of 7. This paper introduces a generalisation of the notion of singular value for Hilbert space operators to more general Banach spaces. The focus of this paper is on these new developments. The links are provided solely by this site's users. The final chapter is devoted to the problems encountered when trying to represent non-compact maps. © 2014 by World Scientific Publishing Co.
The core properties of completeness, compactness, connectedness and simple connectedness are examined; compactness and connectedness are motivated in a variety of ways. Only some proofs of these results are given, but references are provided to works in which detailed expositions of such matters can be found. In fact what is proved is the analogous result when the Sobolev space is based on a member of a class of Banach function spaces that includes both Lp Ω and Lp Ω , the Lebesgue space with variable exponent p satisfying natural conditions. This provides a Banach space version of the well-known Hilbert space result of E. However, this list is by no means complete. Here the abstract results of the previous chapter are applied to give realisations of second-order elliptic operators. Esta página recoge referencias bibliográficas de materiales disponibles en los fondos de las Bibliotecas que participan en Dialnet.
However, until comparatively recently there were no similar results when the action takes place between Banach spaces. For example, consider the critical case of the Sobolev embedding theorem associated with Pohozaev, Trudinger and Yudovic, among others in which the Sobolev space is based upon L p and the target is an Orlicz space of exponential type. Moreover, the methods used in the spectral analysis of self-adjoint or normal maps are totally different from those brought into play to handle compact maps between Banach spaces. To do this, lemmas of combinatorial type are established and used to obtain lower bounds for the entropy numbers of a particular diagonal map acting between Lorentz sequence spaces. En ningún caso se trata de una página que recoja la producción bibliográfica de un autor de manera exhaustiva.
The lack of orthogonality is partially compensated by the systematic use of polar sets. To cope with the demands of the later chapters the improper Riemann integral is introduced. There are applications to the p-Laplacian and similar nonlinear partial differential equations. This paper settles a long-standing question by showing that in certain circumstances the entropy numbers of a map do not behave well under real interpolation. Representations of Linear Operators Between Banach Spaces Operator Theory: Advances and Applications by David E. There are applications to the p-Laplacian, the p-biharmonic operator and integral operators of Hardy type.