International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 50, Pages 2695-2704

On the sublinear operators factoring through Lq

Lahcène Mezrag and Abdelmoumene Tiaiba

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.


Let 0<pq+. 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 Xu˜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=Mguu˜ 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}uT is latticially bounded. Without this condition this equivalence is not true in general.