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Weighted Sobolev spaces and embedding theorems
Functional Analysis · Submitted 24 Mar 2007
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities.
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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 the corresponding weighted Sobolev space. Such classes of homeomorphisms induce bounded composition operators for weighted Sobolev spaces. With the help of these homeomorphism classes the embedding problem for non-smooth domains is reduced to the corresponding classical embedding problem for smooth domains. Examples of domains with anisotropic H\"older singularities demonstrate sharpness of our machinery comparatively with known results.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In the present paper we study embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt A_p-condition. | contribution 0.81 | 0.81 | 0.06 | asserted 0.95 |
| s1 | Sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities. | result 0.96 | 0.94 | 0.57 | asserted 1.00 |
| s2 | 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 the corresponding weighted Sobolev space. | method 0.99 | 0.63 | 0.08 | asserted 0.99 |
| s3 | Such classes of homeomorphisms induce bounded composition operators for weighted Sobolev spaces. | method 0.46 | 0.56 | 0.09 | asserted 1.00 |
| s4 | With the help of these homeomorphism classes the embedding problem for non-smooth domains is reduced to the corresponding classical embedding problem for smooth domains. | method 0.93 | 0.86 | 0.25 | asserted 1.00 |
| s5 | Examples of domains with anisotropic H\"older singularities demonstrate sharpness of our machinery comparatively with known results. | result 0.79 | 0.89 | 0.12 | asserted 1.00 |
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Original abstract
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 the corresponding weighted Sobolev space. Such classes of homeomorphisms induce bounded composition operators for weighted Sobolev spaces. With the help of these homeomorphism classes the embedding problem for non-smooth domains is reduced to the corresponding classical embedding problem for smooth domains. Examples of domains with anisotropic H\"older singularities demonstrate sharpness of our machinery comparatively with known results.
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Since the density of $ℓ_μ$ is an $A_2$ weight, the weighted Sobolev space $W^1, 2(Ω, ℓ_μ)$ is a Hilbert space and $C^∞(Ω, ℝ)$ is dense in $W^1, 2(Ω, ℓ_μ)$ (see theorem 1 in [ref] ).
A dynamic Laplacian for identifying Lagrangian coherent structures on weighted Riemannian manifolds · Weighted Sobolev spaces · 0.98
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