Embeddings and associated spaces of Copson-Lorentz spaces

2016/12/12 by Křepela, Martin
#46E30 #47G10 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1612.03725

Abstract

Let m,p,q∈(0,∞) and let u,v,w be nonnegative weights. We characterize validity of the inequality (∫₀^∞ w(t) (f^*(t))^q dt )^\frac 1q ≤ C (∫₀^∞ v(t) (∫_t^∞ u(s) (f^*(s))^m ds )^\frac pm dt )^\frac 1p for all measurable functions f defined on ℝⁿ and provide equivalent estimates of the optimal constant C>0 in terms of the weights and exponents. The obtained conditions characterize the embedding of the Copson-Lorentz space CL^m,p(u,v), generated by the functional ‖f‖_CL^m,p(u,v) := (∫₀^∞ v(t) (∫_t^∞ u(s) (f^*(s))^m ds )^\frac pm dt )^\frac 1p, into the Lorentz space Λ^q(w). Moreover, the results are applied to describe the associated space of the Copson-Lorentz space CL^m,p(u,v) for the full range of exponents m,p∈(0,∞).

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