Dimension of images of subspaces under Sobolev mappings

Abstract

Let $m<\alpha <p⩽n$ 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 $p<\alpha$.