Sufficient Conditions for Integrability of Distortion Function Kf 1

Capozzoli, Costantino

Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 3, pp. 699-710

Résumé

Assume that $\Omega$, $\Omega'$ are planar domains and $f \colon \Omega \xrightarrow{\text{onto}} \Omega'$ is a homeomorphism belonging to Sobolev space $W_{\text{loc}}^{1,1}(\Omega; \mathbb{R}^{2})$ with finite distortion. We prove that if the distortion function $K_{f}$ of $f$ satisfies the condition $\operatorname{dist}_{\text{EXP}} (K_{f}, L^{\infty}) 1$, then the distortion function $K_{f^{-1}}$ of $f^{-1}$ belongs to $L^{1}_{\text{loc}}(\Omega')$. We show that this result is sharp in sense that the conclusion fails if $\operatorname{dist}_{\text{EXP}} (K_{f}, L^{\infty}) = 1$. Moreover, we prove that if the distortion function $K_{f}$ satisfies the condition $\operatorname{dist}_{\text{EXP}} (K_{f}, L^{\infty}) = \lambda$ for some $\lambda > 0$, then $K_{f^{-1}}$ belongs to $L^{p}_{\text{loc}}(\Omega')$ for every $p \in (0, \frac{1}{2\lambda})$. As special case of this result we show that if the distortion function $K_{f}$ satisfies the condition $\operatorname{dist}_{\text{EXP}} (K_{f}, L^{\infty}) = 0$, then $K_{f^{-1}}$ belongs to intersection of $L^{p}_{\text{loc}}(\Omega')$ for all $p > 1$.