scholarly journals Spaces of Sobolev type with positive smoothness on ℝn, embeddings and growth envelopes

2009 ◽  
Vol 7 (3) ◽  
pp. 251-288 ◽  
Author(s):  
Cornelia Schneider
Keyword(s):  
Besov Spaces ◽  
Analytical Methods ◽  
Lorentz Spaces ◽  
Sobolev Type ◽  
Growth Envelopes

We characterize Triebel-Lizorkin spaces with positive smoothness onℝn, obtained by different approaches. First we present three settingsFp,qs(ℝn),Fp,qs(ℝn),ℑp,qs(ℝn)associated to definitions by differences, Fourier-analytical methods and subatomic decompositions. We study their connections and diversity, as well as embeddings between these spaces and into Lorentz spaces. Secondly, relying on previous results obtained for Besov spaces𝔅p,qs(ℝn), we determine their growth envelopes𝔈G(Fp,qs(ℝn))for0≺p≺∞,0≺q≤∞,s≻0, and finally discuss some applications.

scholarly journals Optimal Sobolev Type Inequalities in Lorentz Spaces

2013 ◽  
Vol 39 (3) ◽  
pp. 265-285 ◽  
Author(s):  
Daniele Cassani ◽  
Bernhard Ruf ◽  
Cristina Tarsi
Keyword(s):  
Lorentz Spaces ◽  
Sobolev Type

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.

scholarly journals Generalized Lorentz Spaces and Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Hatem Mejjaoli
Keyword(s):  
Besov Spaces ◽  
Lorentz Spaces ◽  
Strichartz Estimates ◽  
Schrodinger Equations ◽  
Sobolev Inequalities ◽  
Schrödinger Equations ◽  
Dunkl Operators

We define and study the Lorentz spaces associated with the Dunkl operators onℝd. Furthermore, we obtain the Strichartz estimates for the Dunkl-Schrödinger equations under the generalized Lorentz norms. The Sobolev inequalities between the homogeneous Dunkl-Besov spaces and generalized Lorentz spaces are also considered.

Growth envelopes of Besov spaces on fractal h -sets

2012 ◽  
Vol 286 (5-6) ◽  
pp. 550-568 ◽  
Author(s):  
António M. Caetano
Keyword(s):  
Besov Spaces ◽  
Growth Envelopes

scholarly journals Besov spaces with positive smoothness on Rn, embeddings and growth envelopes

2009 ◽  
Vol 161 (2) ◽  
pp. 723-747 ◽  
Author(s):  
Dorothee D. Haroske ◽  
Cornelia Schneider
Keyword(s):  
Besov Spaces ◽  
Growth Envelopes

Stereo modeling of HVEM specimens by serial tilting and computer graphics

1986 ◽  
Vol 44 ◽  
pp. 34-35
Author(s):  
J.R. McIntosh ◽  
D.L. Stemple ◽  
William Bishop ◽  
G.W. Hannaway
Keyword(s):  
Computer Graphics ◽  
Analytical Methods ◽  
Photographic Film ◽  
Complex Structures ◽  
Multiple Objects ◽  
Multiple Images ◽  
3 Dimensional

EM specimens often contain 3-dimensional information that is lost during micrography on a single photographic film. Two images of one specimen at appropriate orientations give a stereo view, but complex structures composed of multiple objects of graded density that superimpose in each projection are often difficult to decipher in stereo. Several analytical methods for 3-D reconstruction from multiple images of a serially tilted specimen are available, but they are all time-consuming and computationally intense.

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