变指数勒贝格空间中A-范数等于O-范数.pdf

_______________________________________________________________________________ Amemiya Norm Equals Orlicz Norm in Variable Exponent Lebesgue Spaces∗ Xianling Fan Department of Mathematics, Lanzhou University, Lanzhou, P. R. of China 730000 fanxl@ Abstract We prove that the Amemiya norm equals the Orlicz norm in variable exponent Lebesgue spaces. 1 Introduction In an Orlicz space (or generalized Orlicz space) LΦ (Ω) defined via an Orlicz function Φ(t) (or Φ(x, t)), three norms, which are called Luxemburg norm ·Φ , Orlicz norm o A ·Φ and Amemiya norm ·Φ respectively, are often used (see [3,5,6,7,8]). It is well known that in the Orlicz space LΦ (Ω), the Orlicz norm and the Amemiya norm are equal if Φ(t) is an N -function (see [5,6,8]). Recently, Hudzik and Maligranda [3] have o A is also true when Φ(t) is a general Orlicz func- proved that the equality ·Φ = ·Φ tion, i.e., Φ(t) is a nonnegative convex function vanishing at zero (but not identically 0 or ∞ on (0, ∞)). It is not hard to believe that for the generalized Orlicz spaces, when o A Φ(x, t) satisfies some additional conditions, the equality ·Φ = ·Φ also holds, but