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Dimension-invariant Sobolev imbeddings

100%
Krbec M., Schmeisser H.-J.
Banach Center Publications
|
2011
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tom 92
|
nr 1
205-217
EN
We survey recent dimension-invariant imbedding theorems for Sobolev spaces.
2
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Critical imbeddings with multivariate rearrangements

100%
Krbec M., Schmeisser H.-J.
Studia Mathematica
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2007
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tom 181
|
nr 3
255-284
EN
We are concerned with imbeddings of general spaces of Besov and Lizorkin-Triebel type with dominating mixed derivatives in the first critical case. We employ multivariate exponential Orlicz and Lorentz-Orlicz spaces as targets. We study basic properties of the target spaces, in particular, we compare them with usual exponential spaces, showing that in this case the multivariate clones are in fact better adapted to the character of smoothness of the imbedded spaces. Then we prove sharp limiting imbedding theorems and establish estimates for the multivariate growth envelope functions.
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Characteristic of monotonicity of Orlicz function spaces equipped with the Orlicz norm

81%
Foralewski P., Hudzik H., Kaczmarek R., Krbec M.
Commentationes Mathematicae
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2013
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tom 53
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nr 2
EN
We first prove that the property of strict monotonicity of a~K\"othe space \((E,\|.\|_E)\) and\slash or of its K\"othe dual \((E',\|.\|_{E'})\) can be used successfully to compare the supports of \(x\in E\backslash\{\theta\}\) and \(y\in S(E')\), where \(=\|x\|_E\). Next we prove that any element \(x\in S_{+}(E)\) with \(\mu(T\backslash\operatorname{supp} x)=0\) is a~point of order smoothness in \(E\), whenever \(E\) is an order continuous K\"othe space. Finally, we present formulas for the characteristic of monotonicity of Orlicz function spaces endowed with the Orlicz norm in the case when the generating Orlicz function does not satisfy suitable \(\Delta_2\)-condition or the measure is non-atomic infinite, and some lower and upper estimates for the characteristic of monotonicity of this spaces when the measure is non-atomic and finite.
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