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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
##### Keywords:
Sobolev mapping; Hausdorff dimension
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##### References:
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