Capozzoli, Costantino:
Sufficient Conditions for Integrability of Distortion Function Kf 1
Bollettino dell'Unione Matematica Italiana Serie 9 2 (2009), fasc. n.3, p. 699-710, (English)
pdf (127 Kb), djvu (106 Kb). | MR 2569298 | Zbl 1191.46027
Sunto
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 $p > 1$.
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