AbstractIn 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.
Sufficient and Necessary Conditions for Holder’s Inequality in Weighted Orlicz Spaces