Estimates of gradient and of Jacobian of harmonic mappings defined in the unit disk

Abstract

Let $F:\mathbb T\to \gamma$ be a bounded measurable function of the unit circle $\mathbb T$ onto a rectifiable Jordan curve $\gamma$ with the length $|\gamma |$, and let $w=P[F]$ be its harmonic extension to the unit disk $\mathbb U$. By using the arc length parametrization of $\gamma$ we obtain the following results: (i) If $F$ is a quasi-homeomorphism and $1\le p<2$, the $L^p$-norm of the Hilbert-Schmidt norm of the gradient of $w$ is bounded as follows: $\|D(w)\|_{p}\le \frac {|\gamma |}{4\sqrt 2}(\frac {16}{\pi (2-p)})^{1/p}$. (ii) If $F$ is $p$-Lipschitz continuous and $\gamma$ is Dini smooth, then the Jacobian of $w$ is bounded in $\mathbb U$ by a constant $C(p,\gamma )$. The first result is an extension of a recent result of Verchota and Iwaniec, and Martin and Sbordone, while the second result is an extension of a classical result of Martio where $\gamma =\mathbb T$.