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Sobolev spaces, dimension, and random series. (English) Zbl 0938.28001
The author investigates dimension-increasing properties of mappings in the Sobolev space $$W^{1,p}({\mathbb R}^n)$$, $$p>n\geq 1$$. He establishes exact relations between the Hausdorff dimensions of $$E\subseteq{\mathbb R}^n$$ and the image $$f(E)$$, for some random function $$f\in W^{1,p}({\mathbb R}^n)$$. Similar packing dimension results are also obtained.

MSC:
 28A78 Hausdorff and packing measures 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 60G50 Sums of independent random variables; random walks 60G57 Random measures
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