Peano cubes with derivatives in a Lorentz space

Abstract

We show that for any length-compact metric space $Y$ and any $1 \lt q\leq n$, there is a continuous surjection in a suitably defined Sobolev–Lorentz space space $W^{1,n,q}([0,1]^n, Y)$. On the other hand, we show that mappings in the space $W^{1,n,1}([0,1]^n, Y)$ satisfy condition (N). This implies that the target $Y$ can be at most $n$-dimensional.