Smoothness of Orlicz function spaces equipped with the p-Amemiya norm

In this paper, we will use the convex modular $$\rho ^{*}(f)$$ ρ ∗ ( f ) to investigate $$\Vert f\Vert _{\Psi ,q}^{*}$$ ‖ f ‖ Ψ , q ∗ on $$(L_{\Phi })^{*}$$ ( L Φ ) ∗ defined by the formula $$\Vert f\Vert _{\Psi ,q}^{*}=\inf _{k>0}\frac{1}{k}s_{q}(\rho ^{*}(kf))$$ ‖ f ‖ Ψ , q ∗ = inf k > 0 1 k s q ( ρ ∗ ( k f ) ) , which is the norm formula in Orlicz dual spaces equipped with p-Amemiya norm. The attainable points of dual norm $$\Vert f\Vert _{\Psi ,q}^{*}$$ ‖ f ‖ Ψ , q ∗ are discussed, the interval for dual norm $$\Vert f\Vert _{\Psi ,q}^{*}$$ ‖ f ‖ Ψ , 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 $$L_{\Phi ,p}~(1\le p\le \infty )$$ L Φ , p ( 1 ≤ p ≤ ∞ ) 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.