Conjugate type properties of harmonic $(K,K^{\mathrm{\prime }})$-quasiregular mappings

The main purpose of this paper is to investigate conjugate type properties for harmonic $(K,K^{\mathrm{\prime }})$-quasiregular mappings, where $K\ge 1$ and $K^{\mathrm{\prime }}\ge 0$ are constants. We first establish a Riesz type conjugate function theorem for such mappings, which generalizes and refines several existing results. Additionally, we derive an asymptotically sharp constant for a Riesz type theorem pertaining to a specific class of $K$-quasiregular mappings. Furthermore, we obtain Kolmogorov type and Zygmund type theorems for harmonic $(K,K^{\mathrm{\prime }})$-quasiregular mappings.