Estimation of the measure of the image of the ball (Q1760547)

Let \(Q: G\rightarrow [0, +\infty]\) be locally integrable in a domain \(G\subseteq \mathbb{R}^{n}\). A homeomorphism \(f: G\rightarrow \mathbb{R}^{n}\) is said to be a ring \(Q\)-homeomorphism at \(x_{0}\in G\) with respect to the \(p\)-modulus (\(1<p\leq n\)) if \[ M_{p}(\triangle(fS_1, fS_2; fG))\leq\int_{A}Q(x)\eta^{p}|x-x_0|dx \] for each annulus \(A=A(x_0,r_1,r_2)\subseteq G\), \(0<r_1<r_2\) and \(\int_{r_1}^{r_2}\eta(r)dr\geq 1\). Let \(n\geq 2\) and \(f\) be a ring \(Q\)-homeomorphism from the unit ball \(\mathbb{B}^n\) into \(\mathbb{B}^n\) with respect to the \(p\)-modulus. In the article under review, the author presents the best possible estimates for \(m(fB(0,r))\) as follows: \[ m(fB(0,r))\leq \Omega_{n}\left(1+\frac{n-p}{p-1}\int_{r}^{1}\frac{dt}{t^{\frac{n-1}{p-1}}q^{\frac{1}{p-1}}(t)}\right)^{-\frac{n(p-1)}{n-p}} \] for \(1<p<n\), and \[ m(fB(0,r))\leq \Omega_{n}\exp\left(-n\int_{r}^{1}\frac{dt}{tq^{\frac{1}{n-1}}(t)}\right) \] for \(p=n\). As applications, the asymptotic behavior of ring \(Q\)-homeomorphisms with respect to the \(p\)-module at zero is discussed.