Hardy spaces of harmonic quasiconformal mappings and Baernstein's theorem

TL;DR

The paper addresses Baernstein-type extremal problems for harmonic $K$-quasiconformal mappings in geometric subclasses of ${\mathcal{S}}_H^0(K)$ and derives integral-mean estimates for their analytic and co-analytic parts. By leveraging the Baernstein star-function and subordination techniques, the authors establish sharp-type bounds $M_p(r,h')\le M_p(r,H_k)$ and $M_p(r,g')\le M_p(r,G_k)$, with refined bounds for subclasses via $\mathscr{H}_k$ and $\mathscr{G}_k$, and obtain integral-mean controls for $f=h+\overline{g}$. They then determine the ranges of $p$ for which these geometric classes embed into the harmonic Hardy spaces $h^p$, improving Nowak’s results (e.g., $p<1$ for ${\\mathcal{K}}_H$, and $p<1/2$ for ${\\mathcal{C}}_H$ and related classes), and contrast these with the Astala–Koskela bound $p<1/(2K)$. The work illuminates how harmonic Baernstein-type extremals behave under quasiconformal distortion and raises open questions about sharpness and potential broader inclusions $({\\mathcal{S}}_H^0(K)\subset h^p$ for $p$ near $1/2$).

Abstract

Let $\mathcal{S}_H^0(K)$, $K\ge 1$, be the class of normalized $K$-quasiconformal harmonic mappings in the unit disk. We obtain Baernstein type extremal results for the analytic and co-analytic parts of functions in the geometric subclasses of $\mathcal{S}_H^0(K)$. We then apply these results to obtain integral means estimates for the respective classes. Furthermore, we find the range of $p>0$ such that these geometric classes of harmonic quasiconformal mappings are contained in the Hardy space $h^p$, thereby refining some earlier results of Nowak.

Theorems & Definitions (18)

• Theorem A
• Theorem B
• Theorem 1
• Theorem 2
• Remark 1
• Corollary 1
• Corollary 2
• Theorem 3
• Remark 2
• Definition 1