EMBEDDING THEOREM OF THE WEIGHTED SOBOLEV–LORENTZ SPACES

2021 ◽  
pp. 1-18
Author(s):  
HONGLIANG LI ◽  
JIANMIAO RUAN ◽  
QINXIU SUN
Keyword(s):  
Lorentz Spaces ◽  
Embedding Theorem ◽  
Sobolev Embedding ◽  
Norm Inequalities ◽  
Mixed Norms

Abstract Weight criteria for embedding of the weighted Sobolev–Lorentz spaces to the weighted Besov–Lorentz spaces built upon certain mixed norms and iterated rearrangement are investigated. This gives an improvement of some known Sobolev embedding. We achieve the result based on different norm inequalities for the weighted Besov–Lorentz spaces defined in some mixed norms.

scholarly journals A direct proof of Sobolev embeddings for quasi-homogeneous Lizorkin–Triebel spaces with mixed norms

2007 ◽  
Vol 5 (2) ◽  
pp. 183-198 ◽  
Author(s):  
Jon Johnsen ◽  
Winfried Sickel
Keyword(s):  
Sobolev Spaces ◽  
Direct Proof ◽  
Embedding Theorem ◽  
Sobolev Embedding ◽  
Homogeneous Type ◽  
Sobolev Embedding Theorem ◽  
Mixed Norms ◽  
Sobolev Embeddings ◽  
Special Cases ◽  
Anisotropic Spaces

The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin–Triebel spaces (that contain theLp-Sobolev spacesHpsas special cases). The method extends to a proof of the corresponding fact for general Lizorkin–Triebel spaces based on mixedLp-norms. In this context a Nikol' skij–Plancherel–Polya inequality for sequences of functions satisfying a geometric rectangle condition is proved. The results extend also to anisotropic spaces of the quasi-homogeneous type.

scholarly journals Norm inequalities in multidimensional Lorentz spaces

2008 ◽  
Vol 103 (2) ◽  
pp. 278
Author(s):  
Boris Simonov ◽  
Sergey Tikhonov
Keyword(s):  
Trigonometric Series ◽  
Sufficient Conditions ◽  
Lorentz Spaces ◽  
Necessary And Sufficient Conditions ◽  
Double Trigonometric Series ◽  
Norm Inequalities ◽  
Necessary And Sufficient

In this paper we obtain necessary and sufficient conditions for double trigonometric series to belong to generalized Lorentz spaces, not symmetric in general. Estimates for the norms are given in terms of coefficients.

scholarly journals Growth Envelopes of Some Variable and Mixed Function Spaces

2022 ◽  
Vol 32 (3) ◽  
Author(s):  
Dorothee D. Haroske ◽  
Cornelia Schneider ◽  
Kristóf Szarvas
Keyword(s):  
Lorentz Space ◽  
Function Spaces ◽  
Lorentz Spaces ◽  
Lebesgue Spaces ◽  
Mixed Norms ◽  
Hardy Type Inequalities ◽  
Variable Lebesgue Spaces ◽  
Hardy Type ◽  
Growth Envelopes ◽  
Natural Way

AbstractWe study unboundedness properties of functions belonging to Lebesgue and Lorentz spaces with variable and mixed norms using growth envelopes. Our results extend the ones for the corresponding classical spaces in a natural way. In the case of spaces with mixed norms, it turns out that the unboundedness in the worst direction, i.e., in the direction where $$p_{i}$$ p i is the smallest, is crucial. More precisely, the growth envelope is given by $${\mathfrak {E}}_{{\mathsf {G}}}(L_{\overrightarrow{p}}(\varOmega )) = (t^{-1/\min \{p_{1}, \ldots , p_{d} \}},\min \{p_{1}, \ldots , p_{d} \})$$ E G ( L p → ( Ω ) ) = ( t - 1 / min { p 1 , … , p d } , min { p 1 , … , p d } ) for mixed Lebesgue and $${\mathfrak {E}}_{{\mathsf {G}}}(L_{\overrightarrow{p},q}(\varOmega )) = (t^{-1/\min \{p_{1}, \ldots , p_{d} \}},q)$$ E G ( L p → , q ( Ω ) ) = ( t - 1 / min { p 1 , … , p d } , q ) for mixed Lorentz spaces, respectively. For the variable Lebesgue spaces, we obtain $${\mathfrak {E}}_{{\mathsf {G}}}(L_{p(\cdot )}(\varOmega )) = (t^{-1/p_{-}},p_{-})$$ E G ( L p ( · ) ( Ω ) ) = ( t - 1 / p - , p - ) , where $$p_{-}$$ p - is the essential infimum of $$p(\cdot )$$ p ( · ) , subject to some further assumptions. Similarly, for the variable Lorentz space, it holds$${\mathfrak {E}}_{{\mathsf {G}}}(L_{p(\cdot ),q}(\varOmega )) = (t^{-1/p_{-}},q)$$ E G ( L p ( · ) , q ( Ω ) ) = ( t - 1 / p - , q ) . The growth envelope is used for Hardy-type inequalities and limiting embeddings. In particular, as a by-product, we determine the smallest classical Lebesgue (Lorentz) space which contains a fixed mixed or variable Lebesgue (Lorentz) space, respectively.

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