scholarly journals Growth Envelopes of Anisotropic Function Spaces

2008 ◽  
pp. 95-118
Author(s):  
Susana Moura ◽  
Júlio Neves ◽  
Mariusz Piotrowski
Keyword(s):  
Function Spaces ◽  
Anisotropic Function ◽  
Growth Envelopes

scholarly journals Continuity envelopes of spaces of generalised smoothness: a limiting case; embeddings and approximation numbers

2005 ◽  
Vol 3 (1) ◽  
pp. 33-71 ◽  
Author(s):  
António M. Caetano ◽  
Dorothee D. Haroske
Keyword(s):  
Asymptotic Behaviour ◽  
Function Spaces ◽  
Approximation Numbers ◽  
Local Growth ◽  
Recent Developments ◽  
Limiting Case ◽  
Growth Envelopes

Continuity envelopes for the spaces of generalised smoothnessBpq(s,Ψ)(ℝn)andFpq(s,Ψ)(ℝn)are studied in the so-called supercriticals=1+n/p, paralleling recent developments for a corresponding limiting case for local growth envelopes of spaces of such a type. In addition, the power of the concept is used in proving conditions for some embeddings between function spaces to hold, as well as in the study of the asymptotic behaviour of approximation numbers of related embeddings.

scholarly journals Non-smooth atomic decompositions of anisotropic function spaces and some applications

2007 ◽  
Vol 180 (2) ◽  
pp. 169-190 ◽  
Author(s):  
Susana D. Moura ◽  
Iwona Piotrowska ◽  
Mariusz Piotrowski
Keyword(s):  
Function Spaces ◽  
Atomic Decompositions ◽  
Anisotropic Function

On the semigroups of translations and deformations in anisotropic function spaces with uniform metric

2014 ◽  
Vol 90 (3) ◽  
pp. 663-665 ◽  
Author(s):  
V. A. Kostin ◽  
D. V. Kostin ◽  
A. V. Kostin
Keyword(s):  
Function Spaces ◽  
Anisotropic Function

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.

Anisotropic function spaces. II: Traces

1984 ◽  
Vol 10 (1) ◽  
pp. 79-96 ◽  
Author(s):  
Hans Triebel
Keyword(s):  
Function Spaces ◽  
Anisotropic Function
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