Weighted harmonic Bergman kernel on half-spaces. (English) Zbl 1102.31004

Let \(P_z(w)=\frac{2}{n\omega_n}(z_n+w_n)| z-\bar{w}| ^{-2}\) be an extended Poisson kernel for the upper half-plane \(H\) in \(\mathbb R^n,\; n\geq 2\), and \(R_{\alpha} (z,w)=C_{\alpha}D^{\alpha+1}P_z(w),\; \alpha>-1,\; C_{\alpha}=\text{const}\), where \(D^{\alpha}\) is the Liouville fractional derivative.
Let \(b_{\alpha}^p\) stand for the weighted harmonic Bergman space in \(H\) with the norm \[ \| u\| =\left(\int_H | u(z)| ^p z_n^{\alpha} \,dz \right)^{1/p},\quad p\geq 1. \] The authors’ principal result is the representation \[ u(z)=\int_H u(w)R_{\alpha}(z,w)w_n^{\alpha} \,dw, \] implying that \(R_{\alpha}(z,\cdot )\) is Bergman’s kernel for \(b^2_{\alpha},\; \alpha >-1\).


31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
30D55 \(H^p\)-classes (MSC2000)
32A36 Bergman spaces of functions in several complex variables
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
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