Zygmund theorem for harmonic quasiregular mappings

Abstract

Let $K\ge 1$. We prove Zygmund theorem for $K-$quasiregular harmonic mappings in the unit disk $\mathbb{D}$ in the complex plane by providing a constant $C(K)$ in the inequality $$\Vert f{\Vert }_1\le C(K)(1+\Vert \mathrm{Re}(f){\log }^{+}|\mathrm{Re}f|{\Vert }_1),$$ provided that $\mathrm{Im}f(0)=0$. Moreover for a quasiregular harmonic mapping $f=(f_1,\dots ,f_n)$ defined in the unit ball $\mathbb{B}\subset {\mathbb{R}}^n$, we prove the asymptotically sharp inequality $$\Vert f{\Vert }_1-|f(0)|\le (n-1)K^2(\Vert f_1\log f_1{\Vert }_1-f_1(0)\log f_1(0)),$$ when $K\to 1$, provided that $f_1$ is positive.

arXiv comment10 pages