Hilbert-type inequalities in homogeneous cones

Abstract

We prove L-L bounds for the class of Hilbert-type operators associated with generalized powers Q in a homogeneous cone Ω. Our results extend and slightly improve earlier work from [10], where the problem was considered for scalar powers α = (α, . . . , α) and symmetric cones Ω. We give a more transparent proof, provide new examples, and briefly discuss a long standing open question regarding characterization of L boundedness for the case of vector indices α.