Sobolev Embedding Theorems

2021 ◽  
pp. 119-137
Keyword(s):  
Sobolev Embedding ◽  
Embedding Theorems

scholarly journals Heat Equations Associated with Weinstein Operator and Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Hatem Mejjaoli
Keyword(s):  
Besov Spaces ◽  
Sobolev Embedding ◽  
Heat Semigroup ◽  
Heat Equations ◽  
Embedding Theorems

We establish a characterization for the homogeneous Weinstein-Besov spaces via the Weinstein heat semigroup. Next, we obtain the generalized Sobolev embedding theorems.

Weighted Sobolev spaces and exterior problems for the Helmholtz equation

1987 ◽  
Vol 410 (1839) ◽  
pp. 373-383 ◽  
Keyword(s):  
Hilbert Space ◽  
Helmholtz Equation ◽  
Sobolev Spaces ◽  
Existence And Uniqueness ◽  
Weighted Sobolev Spaces ◽  
Radiation Condition ◽  
Sobolev Embedding ◽  
Embedding Theorems ◽  
Uniqueness Of Solutions ◽  
Exterior Problems

Weighted Sobolev spaces are used to settle questions of existence and uniqueness of solutions to exterior problems for the Helmholtz equation. Furthermore, it is shown that this approach can cater for inhomogeneous terms in the problem that are only required to vanish asymptotically at infinity. In contrast to the Rellich–Sommerfeld radiation condition which, in a Hilbert space setting, requires that all radiating solutions of the Helmholtz equation should satisfy a condition of the form ( ∂ / ∂ r − i k ) u ∈ L 2 ( Ω ) , r = | x | ∈ Ω ⊂ R n , it is shown here that radiating solutions satisfy a condition of the form ( 1 + r ) − 1 2 ( ln ( e + r ) ) − 1 2 δ u ∈ L 2 ( Ω ) , 0 < δ < 1 2 , and, moreover, such solutions satisfy the classical Sommerfeld condition u = O ( r − 1 2 ( n − 1 ) ) , r → ∞ . Furthermore, the approach avoids many of the difficulties usually associated with applications of the Poincaré inequality and the Sobolev embedding theorems.

scholarly journals Improved Sobolev Embedding Theorems for Vector-Valued Functions

2014 ◽  
Vol 57 (2) ◽  
pp. 245-295
Author(s):  
Takashi Ichinose ◽  
Yoshimi Saitō
Keyword(s):  
Sobolev Embedding ◽  
Embedding Theorems ◽  
Vector Valued Functions ◽  
Vector Valued

scholarly journals Fourier Multipliers on Triebel-Lizorkin-Type Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-37 ◽  
Author(s):  
Dachun Yang ◽  
Wen Yuan ◽  
Ciqiang Zhuo
Keyword(s):  
Fourier Multipliers ◽  
Sobolev Embedding ◽  
Embedding Theorems ◽  
Hausdorff Spaces ◽  
Type Spaces ◽  
Hörmander’S Condition

The authors study the mapping properties of Fourier multipliers, with symbols satisfying some generalized Hörmander's condition, on Triebel- Lizorkin-type spaces and Triebel-Lizorkin-Hausdorff spaces. To this end, the authors first establish a new characterization of these spaces via some generalized (weighted)gλ∗functions, which essentially improves the known result for Triebel-Lizorkin spaces even whenτ=0. Applying this new characterization, the authors then obtain the boundedness of Fourier multipliers on Triebel-Lizorkin-type spaces and Triebel-Lizorkin-Hausdorff spaces, which also give a new proof of the Sobolev embedding theorems for these spaces.

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