On the Hausdorff Dimension of Continuous Functions Belonging to Hölder and Besov Spaces on Fractal d-Sets
On the Hausdorff Dimension of Continuous Functions Belonging to Hölder and Besov Spaces on Fractal d-Sets
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分形 d 集上属于 Hölder 和 Besov 空间的连续函数的 Hausdorff 维数
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发表时间：
2010
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通讯作者：
A. Caetano
中科院分区：
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作者：
Abel Carvalho;A. Caetano
研究了分形d集上Hölder和Besov空间（可积性p≥1）中函数图的Hausdorff维数。用平滑度参数s∈(0,1)表示，得到尖锐上界min{d+1−s，d/s}。特别是，当从d≥s传递到d<s时，从d+1 - s到d/s的行为会发生变化，这意味着即使是高度非光滑的函数定义在立方体上，当被限制在稀薄的分形上时，也没有那么粗糙的图。
The Hausdorff dimension of the graphs of the functions in Hölder and Besov spaces (in this case with integrability p≥1) on fractal d-sets is studied. Denoting by s∈(0,1] the smoothness parameter, the sharp upper bound min{d+1−s,d/s} is obtained. In particular, when passing from d≥s to d<s there is a change of behaviour from d+1−s to d/s which implies that even highly nonsmooth functions defined on cubes in ℝn have not so rough graphs when restricted to, say, rarefied fractals.