On the luzin N-property and the uncertainty principle for Sobolev mappings

We say that a mapping v from Rn to Rd satisfies the (tau, sigma) -N-property if H^sigma( v (E)) = 0 whenever H^tau (E) = 0, where H^tau means the Hausdorff measure. We prove that every mapping v of Sobolev class W_p^k (Rn,Rd ) with kp > n satisfies the (tau, sigma)-N-property for every 0 < tau different from tau*: = n - (k -1)p. We prove also that for k > 1 and for the critical value tau= tau* the corresponding (tau, sigma)-N-property fails in general. Nevertheless, this (tau, sigma)-N-property holds for tau = tau* if we assume in addition that the highest derivatives (Nabla^k)v belong to the Lorentz space L_p,1(Rn ) 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-properties and discuss their applications to the Morse-Sard theorem and its recent extensions.