Generalized Hausdorff dimension distortion in Euclidean spaces under Sobolev mappings

Abstract We investigate how the integrability of the derivatives of Orlicz–Sobolev mappings defined on open subsets of R n affect the sizes of the images of sets of Hausdorff dimension less than n. We measure the sizes of the image sets in terms of generalized Hausdorff measures.

Dimension gap under Sobolev mappings

Abstract We prove an essentially sharp estimate in terms of generalized Hausdorff measures for the images of boundaries of Holder domains under continuous Sobolev mappings, satisfying suitable Orlicz–Sobolev conditions. This estimate marks a dimension gap, which was first observed in [2] for conformal mappings.

Generalized Dimension Distortion under Mappings of Sub-Exponentially Integrable Distortion

We prove a dimension distortion estimate for mappings of sub-exponentially integrable distortion in Euclidean spaces, which is essentially sharp in the plane.

Regularity and modulus of continuity of space-filling curves

We study critical regularity assumptions on space-filling curves that possess certain modulus of continuity. The bounds we obtain are essentially sharp, as demonstrated by an example. peerReviewed

Generalized dimension distortion under Sobolev mappings

Generalized dimension distortion under planar Sobolev homeomorphisms

We prove essentially sharp dimension distortion estimates for planar Sobolev-Orlicz homeomorphisms.

Generalized dimension estimates for images of porous sets under monotone Sobolev mappings

We give an essentially sharp estimate in terms of generalized Hausdorff measures for images of porous sets under monotone Sobolev mappings, satisfying suitable Orlicz-Sobolev conditions.