Characterizations of harmonic quasiregular mappings in function spaces

TL;DR

This paper investigates the properties of harmonic quasiregular mappings in the unit disk across various function spaces, establishing conjugate relations, stability results, and membership criteria with explicit bounds depending on quasiregularity constants.

Contribution

It provides new conjugate and stability results for harmonic quasiregular mappings in multiple function spaces, with explicit $K$-dependent bounds and criteria for membership in harmonic $M$- and $F$-scales.

Findings

Real and imaginary parts of harmonic quasiregular mappings belong to the same function space with $K$-dependent bounds.

Sharp $K$-dependence established for stability in the harmonic $F$-scale.

Criteria for membership in harmonic $M$- and $F$-scales for normalized harmonic quasiconformal mappings.

Abstract

We study conjugate-type phenomena for complex-valued harmonic quasiregular mappings in the unit disk across three function space families: $Q(n,p,\alpha)$, $F(p,q,s)$, 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 $Q_h(1,p,\alpha)$ (with $\alpha>-1$ and $\alpha+1<p<\alpha+2$), 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 $\alpha_K$ of the family of harmonic quasiconformal mappings.