Characterizations of harmonic quasiregular mappings in function spaces

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 $(K,K^{\mathrm{\prime }})$-quasiregular mappings, yielding explicit estimates with an additional inhomogeneous term involving $K^{\mathrm{\prime }}$. 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 ${\alpha }_K$ of the family of harmonic quasiconformal mappings.