Equivalent Normings of Spaces of Functions of Variable Smoothness
Equivalent Normings of Spaces of Functions of Variable Smoothness
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作者：
O. V. Besov
对于定义在R上的函数的Banach空间B p，q和F ap，q，其变光滑性a = a（x）由它们的差的性质决定，本文用它们的Fourier变换的光滑并矢分解的加权范数建立了等价的赋范.本文研究了n维欧氏空间Rn上函数的Banach空间Ba p，q和F ap，q，1 < p，q < ∞.函数的光滑性由其差分作为点x ∈ Rn的函数的加权积分范数的有限性和差分步长决定。也就是说，对于1 < p，q < ∞，我们考虑空间B p，q = { u：u∈Lp（R，loc），|B p，q = [ ∞ ∑ k=1 sup| H| ≤1 μ g/ml（2h）u| Lp（R）[1 q + α 0 u]| Lp（R）<∞ }，
For the Banach spaces B p,q and F a p,q of functions defined on R whose variable smoothness a = a(x) is determined by the behavior of their differences, equivalent normings are established in terms of weighted norms of smooth dyadic decompositions of their Fourier transforms. In this paper, we study the Banach spaces Ba p,q and F a p,q, 1 < p, q < ∞, of functions defined on the n-dimensional Euclidean space Rn. The smoothness properties of functions are determined by the finiteness of the weighted integral norms of their differences considered as functions of a point x ∈ Rn and the difference step. Namely, for 1 < p, q < ∞, we consider the spaces B p,q = { u : u∈Lp(R, loc), ‖u|B p,q‖= [ ∞ ∑ k=1 sup |h|≤1 ‖ak∆(2h)u|Lp(R)‖ ] 1 q + ‖a0u|Lp(R)‖<∞ } ,