International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 50, Pages 2695-2704
doi:10.1155/S0161171204303145
On the sublinear operators factoring through Lq
Department of Mathematics, M'sila University, P.O. Box 166, Ichbilia, M'sila 28105, Algeria
Received 15 March 2003
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.