Sobolev Spaces with Muckenhoupt Weights, Singularities and Inequalities

2008 ◽  
Vol 15 (2) ◽  
pp. 263-280
Author(s):  
Dorothee D. Haroske
Keyword(s):  
Sobolev Spaces ◽  
Weighted Spaces ◽  
Muckenhoupt Weights ◽  
Growth Envelopes

Abstract We use the recently introduced concept of growth envelopes to characterize weighted spaces of type , where 𝑤 belongs to some Muckenhoupt 𝐴𝑝 class, and discuss some applications.

scholarly journals A note on optimal Hermite interpolation in Sobolev spaces

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Guiqiao Xu ◽  
Xiaochen Yu
Keyword(s):  
Weight Function ◽  
Sobolev Spaces ◽  
Lagrange Interpolation ◽  
Hermite Interpolation ◽  
Weighted Spaces ◽  
Interpolation Algorithms ◽  
Interpolation Systems

AbstractThis paper investigates the optimal Hermite interpolation of Sobolev spaces $W_{\infty }^{n}[a,b]$ W ∞ n [ a , b ] , $n\in \mathbb{N}$ n ∈ N in space $L_{\infty }[a,b]$ L ∞ [ a , b ] and weighted spaces $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p< \infty $ 1 ≤ p < ∞ with ω a continuous-integrable weight function in $(a,b)$ ( a , b ) when the amount of Hermite data is n. We proved that the Lagrange interpolation algorithms based on the zeros of polynomial of degree n with the leading coefficient 1 of the least deviation from zero in $L_{\infty }$ L ∞ (or $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p<\infty $ 1 ≤ p < ∞ ) are optimal for $W_{\infty }^{n}[a,b]$ W ∞ n [ a , b ] in $L_{\infty }[a,b]$ L ∞ [ a , b ] (or $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p<\infty $ 1 ≤ p < ∞ ). We also give the optimal Hermite interpolation algorithms when we assume the endpoints are included in the interpolation systems.

Regularity of the solutions for elliptic problems on nonsmooth domains in ℝ3, Part I: countably normed spaces on polyhedral domains

1997 ◽  
Vol 127 (1) ◽  
pp. 77-125 ◽  
Author(s):  
Benqi Guo ◽  
Ivo Babuška
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Three Dimensional ◽  
Elliptic Problems ◽  
Regularity Theory ◽  
Weighted Spaces ◽  
Regularity Of Solutions ◽  
The Finite Element Method ◽  
Normed Spaces ◽  
Nonsmooth Domains

This is the first of a series of three papers devoted to the regularity of solutions of elliptic problems on nonsmooth domains in ℝ3. The present paper introduces various weighted spaces and countably weighted spaces in the neighbourhood of edges and vertices of polyhedral domains, and it concentrates on exploring the structure of these spaces, such as the embeddings of weighted Sobolev spaces, the relation between weighted Sobolev spaces and weighted continuous function spaces, and the relations between the weighted Sobolev spaces and countably weighted Sobolev spaces in Cartesian coordinates and in the spherical and cylindrical coordinates. These well-defined spaces are the foundation for the comprehensive study of the regularity theory of elliptic problems with piecewise analytic data in ℝ3, which are essential for the design of effective computation and the analysis of the h – p version of the finite element method for solving elliptic problems in three-dimensional nonsmooth domains arising from mechanics and engineering.

scholarly journals Entropy numbers of embeddings of function spaces with Muckenhoupt weights, III. Some limiting cases

2011 ◽  
Vol 9 (2) ◽  
pp. 129-178 ◽  
Author(s):  
Dorothee D. Haroske ◽  
Leszek Skrzypczak
Keyword(s):  
Singular Point ◽  
Polynomial Growth ◽  
Function Spaces ◽  
Sequence Spaces ◽  
Weighted Spaces ◽  
Muckenhoupt Weights ◽  
Asymptotic Estimates ◽  
Entropy Numbers ◽  
Wavelet Bases ◽  
Compact Embeddings

We study compact embeddings for weighted spaces of Besov and Triebel-Lizorkin type where the weight belongs to some MuckenhouptApclass. This extends our previous results [25] to more general weights of logarithmically disturbed polynomial growth, both near some singular point and at infinity. We obtain sharp asymptotic estimates for the entropy numbers of this embedding. Essential tools are a discretisation in terms of wavelet bases, as well as a refined study of associated embeddings in sequence spaces and interpolation arguments in endpoint situations.

Sobolev Spaces

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Sobolev Spaces ◽  
Embedding Theorems ◽  
Approximation Numbers ◽  
Selection Of

This chapter presents a selection of some of the most important results in the theory of Sobolev spacesn. Special emphasis is placed on embedding theorems and the question as to whether an embedding map is compact or not. Some results concerning the k-set contraction nature of certain embedding maps are given, for both bounded and unbounded space domains: also the approximation numbers of embedding maps are estimated and these estimates used to classify the embeddings.

Sign in / Sign up

Export Citation Format

Share Document