article

Title: The self-improving property of the Jacobian determinant in Orlicz spaces

Authors: Flavia Giannetti, Luigi Greco and Antonia Passarelli di Napoli

Issue: Volume 59 (2010), Issue 1, 91-114

Abstract:

$Let $P$ be an increasing function on $\left[ 0,\infty \right[$ satisfying the divergence condition \[ \int_{1}^{\infty} \frac{P(t)}{t^2} \mathrm{d}t = \infty. \] We find a function $\mathscr{A}$ diverging at $\infty$ and positive exponents $\alpha_{1}$, $\alpha_{2}$, so that, for every mapping $f$ with distortion $K$ satisfying $\mathrm{e}^{P(K)} \in L^{1}_{\mathrm{loc}}$, the Jacobian determinant $J_{f}$ has the property \[ J_{f} \mathscr{A} (J_f)^{-\alpha_{2}} \in L^{1}_{\mathrm{loc}} \implies J_{f} \mathscr{A} (J_{f})^{\alpha_{1}} \in L^{1}_{\mathrm{loc}}. \] We also show optimality of $\mathscr{A}$, in the sense that it cannot be substituted by any function whose logarithm grows faster than $\log \mathscr{A}$ 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 (see F.W. Gehring, \textit{The $L^{p}$-integrability of the partial derivatives of a quasiconformal mapping}, Acta Math. \textbf{130} (1973), 265-277).$