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Reproducing kernels related to the complex sphere. (English) Zbl 1004.46024
The authors study four reproducing kernel Hilbert spaces (RKHS) of complex-valued functions and describe their relationships. In $$C^{n+1}$$ $$\widetilde{B}[r]$$ denotes the closed Lie ball, and $$\widetilde{B}(r)$$ the open Lie ball of radius $$r>0$$, and $$\widetilde{S}_{\lambda}$$ denotes the complex sphere of radius $$\lambda\in C$$. Set $$\widetilde{S}_{\lambda}[r]=\widetilde{S}_{\lambda}\cap \widetilde{B}[r]$$, and $$\widetilde{S}_{\lambda,r}=\partial \widetilde{S}_{\lambda}[r]$$ the boundary of $$\widetilde{S}_{\lambda}[r]$$ when $$|\lambda|\leq r$$. The four RKHS are defined as follows. $${h}^2_{\lambda}(\widetilde{B}(r))$$, the completion of the space of germs of complex harmonic functions on $$\widetilde{B}[r]$$ with respect to (w.r.t.) the form $(f,g)_{\widetilde{S}_{\lambda,r}}=\int_{\widetilde{S}_{\lambda,r}}f(z)\overline{g(z)} dz;$ $$H^2(\widetilde{S}_{\lambda}(r))$$, the Hardy space of holomorphic functions on $$\widetilde{S}_{\lambda}(r)$$; $${\mathcal E}^2(\widetilde{S}_0;\lambda,r)$$, the image of $${h}^2_{\lambda}(\widetilde{B}(r))$$ under the conical Fourier transform; and $${\mathcal E}^2_{\Delta-\lambda^2}(\widetilde{E};r)$$, the image of $$H^2(\widetilde{S}_{\lambda}(r))$$ under the Fourier transform. This study continues previous works by the authors, for instance [J.-M. Bony (ed.), et al., New trends in microlocal analysis. Tokyo: Springer, 39-58 (1997; Zbl 0877.35082)].

##### MSC:
 4.6e+23 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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