Notes on limits of Sobolev spaces and the continuity of interpolation scales

We extend lemmas by Bourgain-Brezis-Mironescu (2001), and Maz'ya-Shaposhnikova (2002), on limits of Sobolev spaces, to the setting of interpolation scales. This is achieved by means of establishing the continuity of real and complex interpolation scales at the end points. A connection to extrapolation theory is developed, and a new application to limits of Sobolev scales is obtained. We also give a new approach to the problem of how to recognize constant functions via Sobolev conditions.

     

Real interpolation method,Lorentz spaces and refined Sobolev inequalities

In this article we give a straightforward proof of refined inequalities between Lorentz spaces and Besov spaces and we generalize previous results of H. Bahouri and A. Cohen [2]. Our approach is based in the characterization of Lorentz spaces as real interpolation spaces. We will also study the sharpness and optimality of these inequalities.     

A proof of the trace theorem of Sobolev spaces on Lipschitz domains

A complete proof of the trace theorem of Sobolev spaces on Lipschitz domains has not appeared in the literature yet. The purpose of this paper is to give a complete proof of the trace theorem of Sobolev spaces on Lipschitz domains by taking advantage of the intrinsic norm on . It is proved that the trace operator is a linear bounded operator from to for .

     

Superposition operator in Sobolev spaces on domains

For an arbitrary open set we characterize all functions on the real line such that for all . New element in the proof is based on Maz'ya's capacitary criterion for the imbedding .     

Sobolev spaces on bounded symmetric domains

We obtain a formula for the Sobolev inner product in standard weighted Bergman spaces of holomorphic functions on a bounded symmetric domain in terms of the Peter–Weyl components in the Hua–Schmidt decomposition, and use it to clarify the relationship between the analytic continuation of these standard weighted Bergman spaces and the Sobolev spaces on bounded symmetric domains.     

An elementary proof of the Brezis and Mironescu theorem on the composition operator in fractional Sobolev spaces

An elementary proof of the Brezis and Mironescu theorem on the boundedness and continuity of the composition operator: is given. The proof includes the case p = 1. Received August 15, 2001; accepted October 17, 2001.     

Abstract Hardy-Sobolev spaces and interpolation

The purpose of this work is to describe an abstract theory of Hardy-Sobolev spaces on doubling Riemannian manifolds via an atomic decomposition. We study the real interpolation of these spaces with Sobolev spaces and finally give applications to Riesz inequalities.     

Approximation in Sobolev spaces by piecewise affine interpolation

Functions in a Sobolev space are approximated directly by piecewise affine interpolation in the norm of the space. The proof is based on estimates for interpolations and does not rely on the density of smooth functions.     

Multipliers between Sobolev spaces and fractional differentiation

We characterize the pointwise multipliers which maps a Sobolev space to a Sobolev space in the case |s|<r<d/2.     

A simple characterization of weighted Sobolev spaces with bounded multiplication operator

In this paper we give a simple characterization of weighted Sobolev spaces (with piecewise monotone weights) such that the multiplication operator is bounded: it is bounded if and only if the support of μ₀ is large enough. We also prove some basic properties of the appropriate weighted Sobolev spaces. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials.     

分数次Orlicz极大算子在齐型空间中的局部加权端点估计

利用齐型空间中的覆盖引理及其有界区域的二进方体分解得到了分数次Orlicz极大算子在齐型空间（X,d,μ）中的有界区域Ω上的局部加权端点估计．该工作为分数次积分交换子[b,Iα】在欧式空间R^n中的有界区域上的加权端点弱型估计推广到齐型空间奠定了基础．     

On the first stationary boundary-value problem of elasticity in weighted Sobolev spaces in exterior domains of R3

This paper solves the first boundary-value problem of elasticity both in interior and exterior domains ofR ³. The equations are set in weighted Sobolev spaces for exterior domains that describe the decay of the functions at infinity. The results established include existence, regularity, and convergence of iterations of the solution.This research was supported by the Rashi Foundation.     

Function spaces between BMO and critical Sobolev spaces

The function spaces Dk(Rn) are introduced and studied. The definition of these spaces is based on a regularity property for the critical Sobolev spaces Ws,p(Rn), where sp=n, obtained by J. Bourgain, H. Brezis, New estimates for the Laplacian, the div-curl, and related Hodge systems, C. R. Math. Acad. Sci. Paris 338 (7) (2004) 539-543 (see also J. Van Schaftingen, Estimates for L¹-vector fields, C. R. Math. Acad. Sci. Paris 339 (3) (2004) 181-186). The spaces Dk(Rn) contain all the critical Sobolev spaces. They are embedded in BMO(Rn), but not in VMO(Rn). Moreover, they have some extension and trace properties that BMO(Rn) does not have.     

Weierstrass'' Theorem in Weighted Sobolev Spaces

We characterize the set of functions which can be approximated by polynomials with the following norm

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for a big class of weights w₀, w₁, …, w_k     

Analytic regularity of Stokes flow on polygonal domains in countably weighted Sobolev spaces

We investigate the analytic regularity of the Stokes problem in a polygonal domain with straight sides and piecewise analytic data. We establish a shift theorem in weighted Sobolev spaces of arbitrary order with explicit control of the order-dependence of the constants. The shift-theorem in the framework of countably weighted Sobolev spaces implies in particular interior analyticity and Gevrey-type analytic regularity near the corners.     

Weighted Sobolev theorem in Lebesgue spaces with variable exponent

For the Riesz potential operator Iα there are proved weighted estimates

     

Sobolev spaces and the Cayley transform

The generalised Cayley transform  from an Iwasawa -group into the corresponding real unit sphere  induces isomorphisms between suitable Sobolev spaces and . We study the differential of  , and we obtain a criterion for a function to be in  .

     

Generalized Weighted Sobolev Spaces and Applications to Sobolev Orthogonal Polynomials I

In this paper we present a definition of Sobolev spaces with respect to general measures, prove some useful technical results, some of them generalizations of classical results with Lebesgue measure and find general conditions under which these spaces are complete. These results have important consequences in approximation theory. We also find conditions under which the evaluation operator is bounded.