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Mappings of finite distortion: reverse inequalities for the Jacobian. (English) Zbl 1126.30014
Reverse Hölder inequalities play an essential role in the higher integrability properties of the derivative of quasiconformal and, more generally, quasiregular mappings. Generalizations of these inequalities to mappings of finite distortion are considered. A mapping $f:{\Omega }\to {ℝ}^{n}$ is of finite distortion if $f$ belongs to the space ${W}_{\text{loc}}^{1,1}\left({\Omega }\right)$, the Jacobian determinant $J\left(x,f\right)$ is locally integrable and there is a measurable function $K\left(x\right)\ge 1$, finite a.e., such that ${|Df\left(x\right)|}^{n}\le Kx\right)J\left(x,f\right)$ a.e. To obtain counterparts for the higher integrability for the derivative of a mapping of finite distortion, the condition $exp\left(\beta K\right)\in {L}_{\text{loc}}^{1}\left({\Omega }\right)$ for some $\beta >0$ is used. Under this condition it is shown that for $n\ge 3$, $exp\left({log}^{s}log\left(e+1/J\left(x,f\right)\right)\in {L}_{\text{loc}}^{1}\left({\Omega }\right)$ for some $s=s\left(n\right)>1$. For $n=2$ a stronger result, involving the multiplicity of the mapping, holds: For $\gamma >0$ and ${{\Omega }}^{\text{'}}\subset \subset {\Omega }$ there is $\beta =\beta \left(\gamma ,N\left(f,{{\Omega }}^{\text{'}}\right)\right)$ such that ${log}^{\gamma }\left(e+1/J\left(x,f\right)\right)\in {L}^{1}\left({{\Omega }}^{\text{'}}\right)$ provided that the aforementioned condition holds for this $\beta$. The paper also contains reverse type inequalities for $1/J\left(x,f\right)$ and ${L}^{1}$–integrability results for the inverses of weights. The conclusions from the reverse inequalities for the intgrebility of $1/J\left(x,f\right)$ are optimal but examples show that integrability could be improved by other methods.
##### MSC:
 30C65 Quasiconformal mappings in ${ℝ}^{n}$ and other generalizations 26B10 Implicit function theorems, Jacobians, transformations with several real variables
##### Keywords:
Finite distortion; Jacobian; Reverse inequality
##### References:
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