We say that a mapping satisfies the --property if whenever , where means the Hausdorff measure. We prove that every mapping of Sobolev class with satisfies the --property for every with
We prove also that for and for the critical value the corresponding --property fails in general. Nevertheless, this --property holds for if we assume in addition that the highest derivatives belong to the Lorentz space instead of .
We extend these results to the case of fractional Sobolev spaces as well. Also, we establish some Fubini-type theorems for -Nproperties and discuss their applications to the Morse–Sard theorem and its recent extensions.
"On the Luzin $N$-property and the uncertainty principle for Sobolev mappings." Anal. PDE 12 (5) 1149 - 1175, 2019. https://doi.org/10.2140/apde.2019.12.1149