The Exact Value of Normal Structure Coefficients and WCS coefficients in a Class of Orlicz Function Spaces

Ya. Q. Yan

Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 4, pp. 89-92

Résumé

Let $\Phi$ be an $N$-function. Then the normal structure coefficients $N$ and the weakly convergent sequence coefficients $WCS$ of the Orlicz function spaces $L^\Phi[0,1]$ generated by $\Phi$ and equipped with the Luxemburg and Orlicz norms have the following exact values. (i) If $F_\Phi(t)=t\varphi(t)/\Phi(t)$ is decreasing and $1$ (where $C_\Phi=\lim_{t\to+\infty}t\varphi(t)/\Phi(t)$), then $$ N(L^{(\Phi)}[0,1])=N(L^{\Phi}[0,1])=WCS(L^{(\Phi)}[0,1])=WCS(L^{\Phi}[0,1])=2^{1-1/C_\Phi}. $$ (ii) If $F_\Phi(t)$ is increasing and $C_\Phi>2$, then $$ N(L^{(\Phi)}[0,1])=N(L^{\Phi}[0,1])=WCS(L^{(\Phi)}[0,1])=WCS(L^{\Phi}[0,1])=2^{1/C_\Phi}. $$

Détail

Keywords: Orlicz space
Mots-clés : WCS coefficient, normal structure coefficient.