Sharp Riesz conjugate functions theorems for quasiregular mappings

One of the celebrated results by Riesz \cite{Rie} is the Riesz conjugate functions theorem for analytic functions in the complex plane $\mathbb{C}$. The study on the Riesz conjugate functions theorem for functions in higher dimensional spaces has attracted much attention. Fefferman and Stein \cite{FS-1972} established the Riesz conjugate functions theorem for the Cauchy-Riemann systems in the upper half real space $\mathbb{R}^{n+1}_{+}$. Astala and Koskela \cite{AS-2} investigated the Riesz conjugate functions theorem for quasiconformal mappings of the unit ball $\mathbf{B}^{n}$ in $\mathbb{R}^n$, and posed an open problem which is as follows: Does there exist a quasiconformal analog for the Riesz theorem on conjugate functions? The purpose of this paper is to develop some methods to study this topic further, in particular, Astala-Koskela's open problem. First, we prove a sharp Riesz conjugate functions theorem for a class of quasiregular mappings of $\mathbf{B}^{n}$ for all $n\geq 2$ which satisfy the so-called Heinz's nonlinear differential inequality. As a direct consequence of this result, we find that the answer to Astala-Koskela's open problem is affirmative for harmonic quasiregular mappings of $\mathbf{B}^{n}$ for all $n\geq 2$. Second, we obtain a sharp Riesz conjugate functions theorem for invariant harmonic $K$-quasiregular mappings of $\mathbf{B}^{n}$ for all $n\geq 2$ which shows that the answer to Astala-Koskela's open problem is affirmative for these mappings. At last, we introduce the family of $\kappa$-pluriharmonic mappings of the unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$, and establish a sharp Riesz conjugate functions theorem for these mappings for all $n\geq 1$. Consequently, we generalize and improve all main results by Liu and Zhu \cite{L-Z}.