Characterizations of harmonic quasiregular mappings in function spaces
Jihua Sun, Junming Liu, Zhi-Gang Wang
TL;DR
The paper investigates conjugate-type stability for complex-valued harmonic quasiregular mappings in the unit disk within Möbius-invariant function spaces, unifying two-derivative and non-derivative scales. It proves that if the real part $u$ of a harmonic $K$-quasiregular map $f=u+iv$ lies in $Q_h(1,p,α)$ (with $α>-1$, $α+1<p<α+2$), then the imaginary part $v$ also lies in the same space with a sharp $K$-dependent bound; an analogous result holds for the $F_h(p,q,s)$ scale and extends to harmonic $(K,K')$-quasiregular mappings with an inhomogeneous term. The paper then derives membership criteria for normalized harmonic quasiconformal maps in $M_h$ and $F_h$-scales and shows how derivatives such as $f_z$, $ar{f}_{ar{z}}$, $f_θ$, and $b f_b$ inherit these spaces, with the order parameter $α_K$ governing the admissible parameter ranges. These results yield explicit norm estimates and corollaries for Morrey, Bergman, and $Q_s$-type spaces, providing a quantitative, unified framework for conjugate-type phenomena in harmonic quasiregular mappings.
Abstract
We study conjugate-type phenomena for complex-valued harmonic quasiregular mappings in the unit disk across three function space families: $Q(n,p,α)$, $F(p,q,s)$, and the non-derivative $M(p,q,s)$. For a harmonic $K$-quasiregular mapping $f=u+iv$, we first show that if the real part $u$ belongs to $Q_h(1,p,α)$ (with $α>-1$ and $α+1<p<α+2$), the imaginary part $v$ lies in the same space with a $K$-dependent quantitative bound. An analogous stability result is established for the harmonic $F$-scale, with sharp $K$-dependence. These results are extended to harmonic $(K, K')$-quasiregular mappings, yielding explicit estimates with an additional inhomogeneous term involving $K'$. Finally, for normalized harmonic quasiconformal mappings, %$f\in\mathcal S_H(K)$, we derive membership criteria in the harmonic $M$- and $F$-scales, and obtain corresponding conclusions for their natural derivatives, with parameter ranges governed by the order $α_K$ of the family of harmonic quasiconformal mappings.