Spectral inequalities involving the sums and products of functions

[摘要] In this paper, the notation≺and≺≺denote the Hardy-Littlewood-Pólyaspectral orderrelations for measurable functions defined on a fnite measure space(X,Λ,μ)withμ(X)=a, and expressions of the formf≺gandf≺≺gare calledspectral inequalities. Iff,g∈L1(X,Λ,μ), it is proven that, for someb≥0,log[b+(δfιg)+]≺≺log[b+(fg)+]≺≺log[b+(δfδg)+]wheneverlog+[b+(δfδg)+]∈L1([0,a]), hereδandιrespectively denote decreasing and increasing rearrangement. With the particular caseb=0of this result, the Hardy-Littlewood-Pólya-Luxemburg spectral inequalityfg≺≺δfδgfor0≤f,g∈L1(X,Λ,μ)is shown to be a consequence of the well-known but seemingly unrelated spectral inequalityf+g≺δf+δg(wheref,g∈L1(X,Λ,μ)), thus giving new proof for the former spectral inequality. Moreover, the Hardy-Littlewood-Pólya-Luxemburg spectral inequality is also tended to give(δfιg)+≺≺(fg)+≺≺(δfδg)+and(δfδg)−≺≺(fg)−≺≺(δfιg)−for not necessarily non-negativef,g∈L1(X,Λ,μ).

[学科分类] 数学（综合）