SPECTRAL INEQUALITIES INVOLVING THE SUMS AND PRODUCTS OF FUNCTIONS

In this paper, the notation ≺ and ≺≺ denote the Hardy-Littlewood-Polya spectral order relations for measurable functions defined on a finite measure space (X,Λ,μ) with μ(X)=a, and expressions of the form f≺g and f≺≺g are called spectral inequalities. If f,gεL^1(X,Λ,μ), it is proven that, for some b≥0, log[b+(δ_fι_g)^+]≺≺log[b+(fg)^+]≺≺log[b+(δ_fδ_g)^+] whenever log^+[b+(δ_fδ_g)^+]εL^1([0,a]), here δ and ι respectively denote decreasing and increasing rearrangement. With the particular case b=0 of this result, the Hardy-Littlewood-Polya-Luxemburg spectral inequality fg≺≺δ_fδ_g for 0≤f, gεL^1(X,Λ,μ)is shown to be a consequence of the well-known but seemingly unrelated spectral inequality f+g≺δ_f+δ_g (where f,gεL^1(X,Λ,μ)), thus giving new proof for the former spectral inequality. Moreover, the Hardy-Littlewood-Polya-Luxemburg spectral inequality is also tended to give (δ_fι_g)^+≺≺(fg)^+≺≺(δ_fδ_g)^+ and (δ_fδ_g)^-≺≺(fg)^-≺≺(δ_fι_g)^- for not necessarily non-negative f,gεL^1(X,Λ,μ).

並列關鍵字

Equimeasurable Rearrangements ； Spectral Inequalities ； Convex functions ； discrete measure ； non-atomic measure ； martingale convergence theorem

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