Geometric properties of Orlicz spaces equipped with $p$-Amemiya norms − results and open questions

Marek Wisła. "Geometric properties of Orlicz spaces equipped with $p$-Amemiya norms − results and open questions." Commentationes Mathematicae 55.2 (2015): null. <http://eudml.org/doc/292521>.

@article{MarekWisła2015,
abstract = {The classical Orlicz and Luxemburg norms generated by an Orlicz function $\Phi $ can be defined with the use of the Amemiya formula [H. Hudzik and L. Maligranda, Amemiya norm equals Orlicz norm in general, Indag. Math. 11 (2000), no. 4, 573-585]. Moreover, in this article Hudzik and Maligranda suggested investigating a family of p-Amemiya norms defined by the formula $\Vert x\Vert _\{\Phi ,p\}=\inf _\{k>0\} \frac\{1\}\{k\} (1+I_\Phi ^p(kx))^\{1/p\}$, where $1\le p\le \infty $ (under the convention: $(1+u^\infty )^\{1/\infty \}=\lim _\{p\rightarrow \infty \}(1+u^p)^\{1/p\}=\max \{1,u\}$ for all $u\ge 0$). Based on this idea, a number of papers have been published in the past few years. In this paper, we present some major results concerning the geometric properties of Orlicz spaces equipped with p-Amemiya norms. In the last section, a more general case of Amemiya type norms is investigated. A few open questions concerning this theory will be stated as well.},
author = {Marek Wisła},
journal = {Commentationes Mathematicae},
keywords = {rotundity; non-squareness; uniform monotonicity; dominated best approximation problem; Amemiya type norm},
language = {eng},
number = {2},
pages = {null},
title = {Geometric properties of Orlicz spaces equipped with $p$-Amemiya norms − results and open questions},
url = {http://eudml.org/doc/292521},
volume = {55},
year = {2015},
}

TY - JOUR
AU - Marek Wisła
TI - Geometric properties of Orlicz spaces equipped with $p$-Amemiya norms − results and open questions
JO - Commentationes Mathematicae
PY - 2015
VL - 55
IS - 2
SP - null
AB - The classical Orlicz and Luxemburg norms generated by an Orlicz function $\Phi $ can be defined with the use of the Amemiya formula [H. Hudzik and L. Maligranda, Amemiya norm equals Orlicz norm in general, Indag. Math. 11 (2000), no. 4, 573-585]. Moreover, in this article Hudzik and Maligranda suggested investigating a family of p-Amemiya norms defined by the formula $\Vert x\Vert _{\Phi ,p}=\inf _{k>0} \frac{1}{k} (1+I_\Phi ^p(kx))^{1/p}$, where $1\le p\le \infty $ (under the convention: $(1+u^\infty )^{1/\infty }=\lim _{p\rightarrow \infty }(1+u^p)^{1/p}=\max {1,u}$ for all $u\ge 0$). Based on this idea, a number of papers have been published in the past few years. In this paper, we present some major results concerning the geometric properties of Orlicz spaces equipped with p-Amemiya norms. In the last section, a more general case of Amemiya type norms is investigated. A few open questions concerning this theory will be stated as well.
LA - eng
KW - rotundity; non-squareness; uniform monotonicity; dominated best approximation problem; Amemiya type norm
UR - http://eudml.org/doc/292521
ER -