As is known, Luzin's N-property plays an important role in real analysis and its numerous applications. In particular, the proof of this property for mappings with bounded distortions was an important milestone in the successful construction of the theory of such mappings in the classical works of Yu. G. Reshetnyak. The report discusses analogues of the N-property with respect to Hausdorff measures of general form (with an arbitrary gauge function) for mappings of Sobolev-Orlicz classes also of general form, under the only condition that this Sobolev-Orlicz class be naturally embedded in the space of continuous functions. Estimates for the distortions of the Hausdorff measures are proved, and counterexamples demonstrating the optimality of the obtained estimates are given. Due to the flexibility and richness of mappings of the Orlicz-Sobolev classes, a number of new effects were revealed that were not observed in previous works (where the talk was about "ordinary" Sobolev classes and Hausdorff measures generated by power functions).
The report is based on a joint article with Andrea Cianchi (University of Florence) and Jan Kristensen (University of Oxford).
