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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 \(p<\alpha\).

28A78 Hausdorff and packing measures
26A30 Singular functions, Cantor functions, functions with other special properties
Full Text: DOI
[1] Z.M. Balogh, R. Monti, J.T. Tyson, Frequency of Sobolev and quasiconformal dimension distortion, preprint, 2010. · Zbl 1266.28003
[2] Gehring, F.W.; Väisälä, J., Hausdorff dimension and quasiconformal mappings, J. London math. soc., 6, 2, 504-512, (1973) · Zbl 0258.30020
[3] Kaufman, R., Sobolev spaces, dimension, and random series, Proc. amer. math. soc., 128, 2, 427-431, (2000) · Zbl 0938.28001
[4] Mattila, P., Geometry of sets and measures in Euclidean spaces, Cambridge studies in advanced mathematics, vol. 44, (1995), Cambridge University Press
[5] Ziemer, W.P., Weakly differentiable functions, Graduate texts in mathematics, vol. 120, (1989), Springer-Verlag · Zbl 0177.08006
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