# Weighted Sobolev spaces and embedding theorems

@article{Goldshtein2007WeightedSS, title={Weighted Sobolev spaces and embedding theorems}, author={Vladimir Gol'dshtein and A. Ukhlov}, journal={Transactions of the American Mathematical Society}, year={2007}, volume={361}, pages={3829-3850} }

In the present paper we study embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt A p -condition. Sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities. The proposed method is based on the concept of 'generalized' quasiconformal homeomorphisms (homeomorphisms with bounded mean distortion). The choice of the homeomorphism type depends on the choice of…

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## References

SHOWING 1-10 OF 36 REFERENCES

### Weighted Sobolev spaces and capacity.

- Mathematics
- 1994

Let Ω be an open set in R and 1 < p < ∞ . In this paper we consider the theory of weighted Sobolev spaces H with weight function in Muckenhoupt’s Ap -class. Our main purpose is to provide a coherent…

### Applications of change of variables operators for exact embedding theorems

- Mathematics
- 1994

We propose here a new method for the investigation of embedding operators. It is based on an exact description of classes of homeomorphisms that induce change of variables operators on the Sobolev…

### Nonlinear Potential Theory of Degenerate Elliptic Equations

- Mathematics
- 1993

Introduction. 1: Weighted Sobolev spaces. 2: Capacity. 3: Supersolutions and the obstacle problem. 4: Refined Sobolev spaces. 5: Variational integrals. 6: A-harmonic functions. 7: A superharmonic…

### Some necessary and some sufficient conditions for the compactness of the embedding of weighted Sobolev spaces

- Mathematics
- 2003

We give some necessary conditions and sufficient conditions for the compactness of the embedding of Sobolev spaces $W^{1,p}(\Omega,w) \to L^p(\Omega,w),$ where $w$ is some weight on a domain $\Omega…

### Hölder domains and Poincaré domains

- Mathematics
- 1990

A domain D c Rd of finite volume is said to be a p-Poincare domain if there is a constant Mp(D) so that fU UDII dx d , then D is a p-Poincare domain. This answers a question of Axler and Shields…

### Weighted Sobolev Spaces

- Mathematics
- 1985

Introduction Motivation Weight Domain Hardy Inequality Part One: POWER-TYPE WEIGHTS: Some Elementary Assertions Density of Smooth Functions Imbedding Theorems Miscellaneous Part Two: GENERAL WEIGHTS:…

### Quasiconformal Mappings and Sobolev Spaces

- Mathematics
- 1990

1. Preliminary Information about Integration Theory.- 1. Notation and Terminology.- 1.1. Sets in Rn.- 1.2. Classes of Functions in Rn.- 2. Some Auxiliary Information about Sets and Functions in Rn.-…

### Continuous and compact imbeddings of weighted sobolev spaces

- Mathematics
- 1998

Continuous and compact imbeddings of weighted sobolev spaces، للحصول على النص الكامل يرجى زيارة مكتبة الحسين بن طلال في جامعة اليرموك او زيارة موقعها الالكتروني

### Compactness of the embedding operators for rough domains

- Mathematics, Computer Science
- 2000

New classes of non-smooth bounded domains D, for which the embedding operator from H^1(D) into L^2 (D) is compact are introduced, and applications to scattering by rough obstacles are mentioned.

### Nonlinear Potential Theory and Weighted Sobolev Spaces

- Mathematics
- 2000

The book systematically develops nonlinear potential theory and the Sobolev space theory covers results concerning approximation, extension, and interpolation, Sobolev and Poincare inequalities, Ma…