On the inclusion of some Lorentz spaces

Abstract

Let (X, Sigma, mu) be a measure space. It is well known that l(p)(X) subset of or equal to l(q) (X) whenever 0 < p less than or equal to q less than or equal to infinity. Subramanian [12] characterized all positive measures it on (X, E) for which L-p(mu) subset of or equal to L-q (mu) whenever 0 < p less than or equal to q less than or equal to infinity and Romero [10] completed and improved some results of Subramanian [12]. Miamee [6] considered the more general inclusion L-p(mu) subset of or equal to L-q (nu) where mu and nu are two measures on (X, Sigma). Let L(p(1), q(1))(X, mu) and L(p(2), q(2))(X, nu) be two Lorentz spaces,where 0 < p(1), p(2) < infinity and 0 < q(1), q(2) less than or equal to infinity. In this work we generalized these results to the Lorentz spaces and investigated that under what conditions L(p(1), q(1)) (X, mu) subset of or equal to L(p(2), q(2)) (X, nu) for two different measures mu and nu on (X, Sigma).