Lahcène Mezrag and Abdelmoumene Tiaiba

Copyright © 2004 Lahcène Mezrag and Abdelmoumene Tiaiba. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.