C~β(■)在C~α(■)(0<α<β<1)的非稠密性

Nondensity of C~β(■)在C~α(■)(0<α<β<1)

下为Banacb空间.在偏微分方程的研究中,Holder空间是一类十分有用的函数空间。揭示Holder空间的性质无疑具有一定意义。众所周知,C~∞(Ω)在L′(Ω)、C^K(Ω)(K为正整数)稠密。但本工作将指出对C^n(Ω),C~∞(Ω)已不再是它的稠密子空间,更具体地说,将给出C~β(Ω)在C^n(Ω)(0<α<β<1)的闭子空间的特征,并证明它在C^n(Ω)不稠密。

The author shows that closed subspaoe of Cp(Ω) in C(Ω)(0<α<β<1)consists of ail of function in C(Ω) which is retriction of some function f∈Cn0(B) where B is a ball such that 'α-Holder modulus' [f(x+h) -f(z)]/ |k| is continuous function for x, and uniformly continuous with respect to h. Some functions which belong to are found and nandensity of Cp(Ω) in C(Ω) follows immediately.