单位球和Siegel域上的齐次多解析核
Homogeneously polyanalytic kernels on the unit ball and the Siegel domain
摘要：
本文证明了由方程组$\overline{D}^{(k_1,\ldots,k_n)} f=0$与$k_1+\cdots+k_n=m$定义的全阶齐次多解析函数$m$可以表示为变量$\overline{z_1},\ldots,\overline{z_n}$的全次数$<m$，并具有一定的解析系数.利用雅可比多项式的再生性质，我们建立了这类函数的加权平均值性质。在此基础上，我们给出了一个一般的方法，通过变量的加权变量变换来变换再生核。应用这些工具，我们计算了$\mathbb{C}^n$中单位球上和Siegel域上齐次多解析函数的Bergman空间的再生核.对于一维情况，Koshelev（1977），Pessoa（2014），Hachadi和Youssfi（2019）获得了类似的结果。
英文摘要：
We prove that the homogeneously polyanalytic functions of total order $m$, defined by the system of equations $\overline{D}^{(k_1,\ldots,k_n)} f=0$ with $k_1+\cdots+k_n=m$, can be written as polynomials of total degree $<m$ in variables $\overline{z_1},\ldots,\overline{z_n}$, with some analytic coefficients. We establish a weighted mean value property for such functions, using a reproducing property of Jacobi polynomials. After that, we give a general recipe to transform a reproducing kernel by a weighted change of variables. Applying these tools, we compute the reproducing kernel of the Bergman space of homogeneously polyanalytic functions on the unit ball in $\mathbb{C}^n$ and on the Siegel domain. For the one-dimensional case, analogous results were obtained by Koshelev (1977), Pessoa (2014), Hachadi and Youssfi (2019).
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