Hilbert-type inequalities in homogeneous cones

Abstract

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 $\Omega$.