Dimension of images of subspaces under Sobolev mappings

Abstrakt

Let m< n < alpha < p {= n and let f is an element of W^{1,P}(R^n, R^k) be p-quasicontinuous. We find an optimal value of beta(n, m, p, alpha) such that for H^beta a.e. y is an element of (0, 1)(n-m) the Hausdorff dimension of f((0, 1)^m x {y}) is at most alpha.

We construct an example to show that the value of the optimal 11 does not increase once p goes below the critical case p < alpha.

Klíčová slova

• Sobolev mapping
• Hausdorff dimension