The self improving property of the Jacobian determinant in Orlicz spaces

Let $P$ be an increasing function on $[0,infty[$ satisfying the divergence condition [int_1^inftyrac{P(t)}{t^2},dt=infty,.] We find a function $A$ diverging at $infty$ and positive exponents $alpha_1$, $alpha_2$, so that, for every mapping $f$ with distortion $K$ satisfying $exp(P(K))in L^1loc$, the Jacobian determinant $J_f$ has the property [J_f A(J_f)^{-alpha_2}in L^1locimplica J_f A(J_f)^{alpha_1}in L^1loc,.] We also show optimality of $A$, in the sense that it cannot be substituted by any function whose logarithm grows faster than $log A$ at infinity. Moreover, we show that the divergence condition cannot be dropped. This constitutes a far reaching generalization of the so-called self-improving property of the Jacobian determinant, which can be traced back to the work of Gehring [Acta Math. 130 (1973), 265--277].