×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI