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Dimension of images of subspaces under Sobolev mappings. (English) Zbl 1245.28006
Summary: Let \(m<\alpha<p\leq n\) and let f\(\in W^{1,p}(\mathbb{R}^n,\mathbb{R}^k)\) be \(p\)-quasicontinuous. We find an optimal value of \(\beta(n,m,p,\alpha)\) such that for \({\mathcal H}^\beta\) a.e. \(y\in(0,1)^{n-m}\) the Hausdorff dimension of \(f((0,1)^m\times\{y\})\) is at most \(\alpha\). We construct an example to show that the value of the optimal \(\beta\) does not increase once \(p\) goes below the critical case \(p<\alpha\).

MSC:
28A78 Hausdorff and packing measures
26A30 Singular functions, Cantor functions, functions with other special properties
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