Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 427-442

DIMENSION DISTORTION OF IMAGES OF SETS UNDER SOBOLEV MAPPINGS

Stanislav Hencl and Petr Honzík

Charles University, Department of Mathematical Analysis
Sokolovská 83, 186 00 Prague 8, Czech Republic; hencl 'at' karlin.mff.cuni.cz

Charles University, Department of Mathematical Analysis
Sokolovsk&aacte; 83, 186 00 Prague 8, Czech Republic; honzik 'at' gmail.com

Abstract. Let f : Rⁿ → 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 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.

2010 Mathematics Subject Classification: Primary 46E35, 28A78.

Key words: Sobolev mapping, Hausdorff dimension.

Reference to this article: S. Hencl and P. Honzík: Dimension distortion of images of sets under Sobolev mappings. Ann. Acad. Sci. Fenn. Math. 40 (2015), 427-442.

Full document as PDF file

doi:10.5186/aasfm.2015.4026