Some questions about complex harmonic functions

Luis E. Benítez-Babilonia, Raúl Felipe
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Abstract

In this paper, we propose composition products in the class of complex harmonic functions so that the composition of two such functions is again a complex harmonic function. From here, we begin the study of the iterations of the functions of this class showing briefly its potential to be a topic of future research. In parallel, we define and study composition operators on a Hardy type space denoted by \(HH^{2}(\mathbb {D})\) of complex harmonic functions also introduced for us in the present work. The symbols of these composition operators have of form \(\chi +\overline{\pi }\) where \(\chi ,\pi \) are analytic functions from \(\mathbb {D}\) into \(\mathbb {D}\). We also analyze the space of bounded linear operators on \(HH^{2}(\mathbb {D})\).

关于复调函数的几个问题
在本文中,我们提出了复调函数类中的组成积,从而使两个此类函数的组成再次成为复调函数。由此,我们开始研究该类函数的迭代,并简要展示了其作为未来研究课题的潜力。与此同时,我们定义并研究了复调函数的哈代类型空间上的组成算子,用 \(HH^{2}(\mathbb {D})\ 表示。这些组成算子的符号具有 \(\chi +\overline{pi }\) 的形式,其中 \(\chi ,\pi \) 是从 \(\mathbb {D}\) 到 \(\mathbb {D}\) 的解析函数。我们还分析了 \(HH^{2}(\mathbb {D})\) 上有界线性算子的空间。
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