Riesz Theorem and Riesz-Fej\'er inequality for weighted harmonic Bergman spaces with applications to M\"obius 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 $a_\alpha^p$, $0<p<\infty$, then the imaginary part $v$ also belongs to the same space, together with a quantitative norm estimate. Moreover, for $1<p<\infty$, the corresponding constant is shown to be independent of the weight parameter $\alpha$. Second, we establish Riesz--Fej\'er inequalities for weighted harmonic Bergman spaces for $1<p<\infty$. 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--Fej\'er inequalities for the M\"obius 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].