On the Luzin $N$-property and the uncertainty principle for Sobolev mappings

Abstract

We say that a mapping $v:{ℝ}^n\to {ℝ}^d$ satisfies the $\left(\tau ,\sigma \right)$-$N$-property if ${\mathcal{\mathscr{H}}}^{\sigma }\left(v\left(E\right)\right)=0$ whenever ${\mathcal{\mathscr{H}}}^{\tau }\left(E\right)=0$, where ${\mathcal{\mathscr{H}}}^{\tau }$ means the Hausdorff measure. We prove that every mapping $v$ of Sobolev class $W_p^k\left({ℝ}^n,{ℝ}^d\right)$ with $kp>n$ satisfies the $\left(\tau ,\sigma \right)$-$N$-property for every $0<\tau \ne {\tau }_{\ast }:=n-\left(k-1\right)p$ with

$$\sigma =\sigma \left(\tau \right):=\left\{\begin{array}{cc}\tau \hfill & \text{ if }\tau >{\tau }_{\ast },\hfill \\ p\tau ∕\left(kp-n+\tau \right)\hfill & \text{ if }0<\tau <{\tau }_{\ast }.\hfill \end{array}\right.$$

We prove also that for $k>1$ and for the critical value $\tau ={\tau }_{\ast }$ the corresponding $\left(\tau ,\sigma \right)$-$N$-property fails in general. Nevertheless, this $\left(\tau ,\sigma \right)$-$N$-property holds for $\tau ={\tau }_{\ast }$ if we assume in addition that the highest derivatives ${\nabla }^{k}v$ belong to the Lorentz space $L_{p,1}\left({ℝ}^n\right)$ instead of $L_p$.

We extend these results to the case of fractional Sobolev spaces as well. Also, we establish some Fubini-type theorems for $N$-Nproperties and discuss their applications to the Morse–Sard theorem and its recent extensions.