International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 49-52, Pages 2695-2704
doi:10.1155/S0161171204303145

On the sublinear operators factoring through Lq

Abstract

Let 0<p≤q≤+∞. Let T be a bounded sublinear operator from a Banach space X into an Lp(Ω,μ) and let ∇T be the set of all linear operators ≤T. In the present paper, we will show the following. Let C be a positive constant. For all u in ∇T, Cpq(u)≤C (i.e., u admits a factorization of the form X→u˜Lq(Ω,μ)→MguLq(Ω,μ), where u˜ is a bounded linear operator with ‖u˜‖≤C, Mgu is the bounded operator of multiplication by gu which is in BLr+(Ω,μ) (1/p=1/q+1/r), u=Mgu∘u˜ and Cpq(u) is the constant of q-convexity of u) if and only if T admits the same factorization; This is under the supposition that {gu}u∈∇T is latticially bounded. Without this condition this equivalence is not true in general.