In this paper, we will use the convex modular \(\rho ^{*}(f)\) to investigate \(\Vert f\Vert _{\Psi ,q}^{*}\) on \((L_{\Phi })^{*}\) defined by the formula \(\Vert f\Vert _{\Psi ,q}^{*}=\inf _{k>0}\frac{1}{k}s_{q}(\rho ^{*}(kf))\), which is the norm formula in Orlicz dual spaces equipped with p-Amemiya norm. The attainable points of dual norm \(\Vert f\Vert _{\Psi ,q}^{*}\) are discussed, the interval for dual norm \(\Vert f\Vert _{\Psi ,q}^{*}\) attainability is described. By presenting the explicit form of supporting functional, we get sufficient and necessary conditions for smooth points. As a result, criteria for smoothness of \(L_{\Phi ,p}~(1\le p\le \infty )\) is also obtained. The obtained results unify, complete and extended as well the results presented by a number of paper devoted to studying the smoothness of Orlicz spaces endowed with the Luxemburg norm and the Orlicz norm separately.

在本文中，我们将使用凸模\（\ RHO ^ {*}（F）\）调查\（\韦尔˚F\韦尔_ {\幽，Q} ^ {*} \）在\（（L_由公式\（\ Vert f \ Vert _ {\ Psi，q} ^ {*} = \ inf _ {k> 0} \ frac {1} {k}定义的 {\ Phi}）^ {*} \）s_ {q}（\ rho ^ {*}（kf））\），这是配备p -Amemiya范数的Orlicz对偶空间中的范数公式。讨论对偶范数\（\ Vert f \ Vert _ {\ Psi，q} ^ {*} \）的可达点， 对偶范数 \（\ Vert f \ Vert _ {\ Psi，q} ^ { *} \）描述了可达到性。通过呈现支持功能的显式形式，我们获得了光滑点的充分和必要条件。结果，光滑度的标准也可以获得\（L _ {\ Phi，p}〜（1 \ le p \ le \ infty）\）。所获得的结果统一，完善和扩展了许多专门研究赋予Luxemburg范数和Orlicz范数的Orlicz空间的光滑度的论文所给出的结果。