Characteristic of monotonicity of Orlicz function spaces equipped with the Orlicz norm

Paweł Foralewski ; Henryk Hudzik ; Radosław Kaczmarek ; Miroslav Krbec

Commentationes Mathematicae, Tome 53 (2013) no. 2, p. 421−432.

Résumé

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.

Détail

DOI : 10.14708/cm.v53i2.4365

Classification : 46B42, 46E30
Mots-clés : Orlicz space, Orlicz norm, Kothe space, Kothe dual, characteristic of monotonicity, strict monotonicity, point of order smoothness