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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.

Dimension distortion of images of sets under Sobolev mappings