Riesz Theorem and Riesz-Fejér inequality for weighted harmonic Bergman spaces with applications to Möbius invariant spaces

The aim of this paper is twofold. First, we establish a Riesz conjugate theorem for weighted harmonic Bergman spaces. More precisely, we prove that if $f=u+iv$ is a harmonic $K$-quasiregular mapping in $\mathbb{D}$ and the real part $u$ belongs to the weighted harmonic Bergman space $a_{\alpha }^p$, $0<p<\mathrm{\infty }$, then the imaginary part $v$ also belongs to the same space, together with a quantitative norm estimate. Moreover, for $1<p<\mathrm{\infty }$, the corresponding constant is shown to be independent of the weight parameter $\alpha$. Second, we establish Riesz–Fejér inequalities for weighted harmonic Bergman spaces for $1<p<\mathrm{\infty }$. In the special case $p=2$, we further improve the corresponding constant by using the Hilbert space structure and orthogonality techniques. As applications of our main results, we establish Riesz conjugate theorems and Riesz–Fejér inequalities for the Möbius invariant spaces $Q(n,p,\alpha )$ introduced by Zhu [Illinois J. Math. 51 (2007), pp. 977–1002] and their harmonic counterparts $Q_h(n,p,\alpha )$ introduced by Sun, Liu, and Wang [Potential Anal. 65 (2026), Article no. 12].