Let $s\in \mathbb{R}$ and $0<p\leqslant \infty$. The fractional Fock–Sobolev spaces $F_{\mathscr{R}}^{s,p}$ are introduced through the fractional radial derivatives $\mathscr{R}^{s/2}$. We describe explicitly the reproducing kernels for the fractional Fock–Sobolev spaces $F_{\mathscr{R}}^{s,2}$ and then get the pointwise size estimate of the reproducing kernels. By using the estimate, we prove that the fractional Fock–Sobolev spaces $F_{\mathscr{R}}^{s,p}$ are identified with the weighted Fock spaces $F_{s}^{p}$ that do not involve derivatives. So, the study on the Fock–Sobolev spaces is reduced to that on the weighted Fock spaces.

中文翻译：

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分数 FOCK-SOBOLEV 空间

让$s\in \mathbb{R}$和$0<p\leqslant\infty$. 分数 Fock–Sobolev 空间$F_{\mathscr{R}}^{s,p}$通过分数径向导数引入$\mathscr{R}^{s/2}$. 我们明确地描述了分数 Fock–Sobolev 空间的再生核$F_{\mathscr{R}}^{s,2}$然后得到再生内核的逐点大小估计。通过使用估计，我们证明了分数 Fock–Sobolev 空间$F_{\mathscr{R}}^{s,p}$用加权 Fock 空间标识$F_{s}^{p}$不涉及衍生品。因此，对 Fock-Sobolev 空间的研究简化为对加权 Fock 空间的研究。