Characterizations of harmonic quasiregular mappings in function spaces

Abstract

We study conjugate-type phenomena for complex-valued harmonic quasiregular mappings in the unit disk across three function space families: $Q(n,p,\hat{I}\pm )$, $F(p,q,s)$, 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 $Q_h(1,p,\hat{I}\pm )$ (with $\hat{I}\pm >-1$ and $\hat{I}\pm +1<p<\hat{I}\pm +2$), 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 $(K,K^{\prime })$-quasiregular mappings, yielding explicit estimates with an additional inhomogeneous term involving $K^{\prime }$. 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 $\hat{I}{\pm }_K$ of the family of harmonic quasiconformal mappings.

20 pages, comments are welcome