ON THE SUBLINEAR OPERATORS FACTORING THROUGH Lq LAHCÈNE MEZRAG and ABDELMOUMENE TIAIBA

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 ũ → Lq(Ω,μ) Mgu → Lp(Ω,μ), where ũ is a bounded linear operator with ‖ũ‖ ≤ C , Mgu is the bounded operator of multiplication by gu which is in BL+r (Ω,μ) (1/p = 1/q+1/r ), u = Mgu ◦ ũ 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.