DIMENSION DISTORTION OF IMAGES OF SETS UNDER SOBOLEV MAPPINGS

Let f : R n → R k be a continuous representative of a mapping in a Sobolev space W 1,p , p > n. Suppose that the Hausdorff dimension of a set M is at most �. Kaufmann (12) proved an optimal bound � = p� p n+� 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.