On the Inclusion of Some Lorentz Spaces

Abstract

Let (X, Σ, μ) be a measure space. It is well known that l p(X) ⊆ lq(X) whenever 0 < p ≤ q ≤ ∞. Subramanian [12] characterized all positive measures μ on (X, Σ) for which Lp(μ) ⊆ Lq(μ) whenever 0 < p ≤ q ≤ ∞ and Romero [10] completed and improved some results of Subramanian [12]. Miamee [6] considered the more general inclusion Lp(μ) ⊆Lq(v) where μ and v are two measures on (X, Σ). Let L(p<inf>1</inf>,q<inf>1</inf>)(X, μ) and L(p<inf>2</inf>, q<inf>2</inf>)(X, v) be two Lorentz spaces,where 0 < p<inf>1</inf>,p<inf>2</inf> < ∞ and 0 <q<inf>1</inf>,q<inf>2</inf> ≤ ∞. In this work we generalized these results to the Lorentz spaces and investigated that under what conditions L(p<inf>1</inf>,q<inf>1</inf>)(X,μ) ⊆ L(p<inf>2</inf>,q<inf>2</inf>)(X, v) for two different measures μ and v on (X, Σ).