We prove $L^p-L^q$ bounds for the class of Hilbert-type operators associated with generalized powers $Q^a$ in a homogeneous cone $\Omega$. Our results extend and slightly improve earlier work from [16], where the problem was considered for scalar powers $\mathbf a = (\alpha, \dots, \alpha)$ and symmetric cones. We give a more transparent proof, provide new examples, and briefly discuss the open question regarding characterization of $L^p$ boundedness for the case of vector indices $\mathbf a$. Some applications are given to boundedness of Bergman projections in the tube domain over $\Omega$.

中文翻译：

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齐锥中的希尔伯特型不等式

我们证明了齐次圆锥$ \ Omega $中与广义幂$ Q ^ a $相关的Hilbert型算子的类的$ L ^ pL ^ q $界。我们的结果扩展了并从[16]略微改善了早期的工作，在该工作中，考虑了标量幂$ \ mathbf a =（\ alpha，\ dots，\ alpha）$和对称圆锥体的问题。我们给出了更透明的证明，提供了新的示例，并简要讨论了在矢量索引$ \ mathbf a $的情况下有关$ L ^ p $有界性的特征的开放性问题。一些应用给出了在$ \ Omega $上的管域中Bergman投影的有界性。