加权调和伯格曼空间的里斯定理和里斯 - 费耶尔不等式及其在莫比乌斯不变空间中的应用
Riesz Theorem and Riesz-Fejér inequality for weighted harmonic Bergman spaces with applications to Möbius invariant spaces
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中文总结 AI 辅助
本文为加权调和伯格曼空间建立里斯共轭定理与里斯 - 费耶尔不等式，给出范数估计及常数性质，还将结果应用于莫比乌斯不变空间及其调和对应空间，通过特定方法改进特殊情况常数。
中文摘要 AI 辅助
本文有两个目的。首先，为加权调和伯格曼空间建立一个里斯共轭定理。具体而言，证明若\(f = u + iv\)是\(\mathbb{D}\)中的调和\(K -\)拟正则映射且实部\(u\)属于加权调和伯格曼空间\(a_\alpha^p\)，\(0 < p < \infty\)，则虚部\(v\)也属于同一空间，并给出定量范数估计。对于\(1 < p < \infty\)，相应常数与权重参数\(\alpha\)无关。其次，为\(1 < p < \infty\)的加权调和伯格曼空间建立里斯 - 费耶尔不等式。在\(p = 2\)的特殊情况，利用希尔伯特空间结构和正交性技术进一步改进相应常数。作为主要结果的应用，为朱引入的莫比乌斯不变空间\(Q(n,p,\alpha)\)及其调和对应空间\(Q_h(n,p,\alpha)\)建立里斯共轭定理和里斯 - 费耶尔不等式。
英文摘要
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 $a_α^p$, $0<p<\infty$, 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 $α$. Second, we establish Riesz--Fejér 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--Fejér inequalities for the Möbius invariant spaces $Q(n,p,α)$ introduced by Zhu [Illinois J. Math. 51 (2007), pp. 977--1002] and their harmonic counterparts $Q_h(n,p,α)$ introduced by Sun, Liu, and Wang [Potential Anal. 65 (2026), Article no. 12].
