DIMENSION DISTORTION OF IMAGESOF SETS UNDER SOBOLEV MAPPINGS

摘要

Let f : Rn →Rk be a continuous representative of a mapping in a Sobolev space W1,p, p > n. Suppose that the Hausdorff dimension of a set M is at most α. Kaufmann [12] proved an optimal bound β = pα/(p - n + α) for the dimension of the image of M under the mapping f.We show that this bound remains essentially valid even for 1 p ≤ n and we also prove analogous bound for mappings in Sobolev spaces with higher order or even fractional smoothness.