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

We say that a mapping v: ℝⁿ → ℝ^d satisfies the (τ, σ ) -N-property if H^σ( v (E)) = 0 whenever H^τ (E) = 0, where Hτ means the Hausdorff measure. We prove that every mapping v of Sobolev class W _p ^k (ℝⁿ,ℝ^d ) with kp > n satisfies the (τ, σ )-N-property for every 0 < τ ≠ τ_*: = n - (k -1)p with We prove also that for k > 1 and for the critical value τ = τ^* the corresponding (τ, σ )-N-property fails in general. Nevertheless, this (τ, σ )-N-property holds for τ = τ^* if we assume in addition that the highest derivatives ∇^kv 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-Nproperties and discuss their applications to the Morse-Sard theorem and its recent extensions.