To the injectivity problem for quasiregular space mappings (Q1594335)

Let \(u\) be a locally integrable function in a domain \(D\) of \(\mathbb{R}^n\) and let \(u_r(x)\) denote the mean value of \(u\) in the ball \(B(x,r)\) in \(D\). The function \(u\) is of vanishing mean oscillation (VMO) at \(x_0\) if the mean value of \(u- u_r(x_0)\) over \(B(x_0, r)\) approaches zero as \(r\to 0\). Suppose that \(f:D\to \mathbb{R}^n\) is a non-constant quasiregular mapping, i.e. \(f\) is continuous, in \(W^{1,n}_{\text{loc}}(D)\) and satisfies \(|f'(x)|^n\leq KJ(x,f)\) a.e. in \(D\) for some \(K<\infty\). The normalized Jacobian matrix of \(f\) at \(x\) is then defined as \(M(x)= f'(x)/J(x,f)\); a non-constant quasiregular mapping has \(J(x,f)> 0\) a.e. in \(D\). A rather weak regularity condition for \(M\) guarantees the local injectivity of \(f\) in space: If \(n>2\) and if \(M\) is VMO at \(x_0\), then \(f\) is locally homeomorphic at \(x_0\). A similar result holds if the BMO norm of \(M\) is small: For each \(n> 2\) there is \(\delta> 0\) such that if the BMO norm of \(M\) in \(D\) is smaller than \(\delta\), then \(f\) is locally homeomorphic in \(D\). The proofs of these results have appeared in [\textit{O. Martio}, \textit{V. Ryazanov} and \textit{M. Vuorinen}: Math. Nach. 205, 149-161 (1999; Zbl 0935.30016)]. It is still an open question if a \(C^1\)-quasiregular mapping in \(\mathbb{R}^n\), \(n> 2\), is a local homeomorphism.