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Zbl 1126.30014
Hencl, Stanislav; Koskela, Pekka; Zhong, Xiao
Mappings of finite distortion: reverse inequalities for the Jacobian. (English)
[J] J. Geom. Anal. 17, No. 2, 253-274 (2007). ISSN 1050-6926; ISSN 1559-002X/e

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 $f : \Omega \rightarrow \Bbb{R}^n$ is of finite distortion if $f$ belongs to the space $W^{1,1}_{\text{loc}} (\Omega)$, the Jacobian determinant $J(x, f)$ is locally integrable and there is a measurable function $K(x) \geq 1$, finite a.e., such that $\vert Df(x)\vert ^n \leq Kx) J(x,f)$ a.e. To obtain counterparts for the higher integrability for the derivative of a mapping of finite distortion, the condition $\exp(\beta K) \in L^1_{\text{loc}}(\Omega)$ for some $\beta > 0$ is used. Under this condition it is shown that for $n \geq 3$, $\exp(\log^s \log(e + 1/J(x,f)) \in L^1_{\text{loc}}(\Omega)$ for some $s = s(n) > 1$. For $n = 2$ a stronger result, involving the multiplicity of the mapping, holds: For $\gamma > 0$ and $\Omega' \subset\subset \Omega$ there is $\beta = \beta(\gamma, N(f, \Omega'))$ such that $\log^{\gamma}(e + 1/J(x,f)) \in L^1(\Omega')$ provided that the aforementioned condition holds for this $\beta$. The paper also contains reverse type inequalities for $1/J(x, f)$ and $L^1$--integrability results for the inverses of weights. The conclusions from the reverse inequalities for the intgrebility of $1/J(x,f)$ are optimal but examples show that integrability could be improved by other methods.
[Olli Martio (Helsinki)]

MSC 2000:: *30C65 Quasiconformal mappings in R$\sp n$ and other generalizations
26B10 Implicit function theorems, etc. (several real variables)

Keywords: Finite distortion; Jacobian; Reverse inequality

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