Factorization and extension of positive homogeneous polynomials
Volume 221 / 2014
Studia Mathematica 221 (2014), 87-99
MSC: Primary 47H60.
DOI: 10.4064/sm221-1-5
Abstract
We study the following problem: Given a homogeneous polynomial from a sublattice of a Banach lattice to a Banach lattice, under which additional hypotheses does this polynomial factorize through $L_p$-spaces involving multiplication operators? We prove that under some lattice convexity and concavity hypotheses, for polynomials certain vector-valued norm inequalities and weighted norm inequalities are equivalent. We combine these results and prove a factorization theorem for positive homogeneous polynomials which is a variant of a celebrated factorization theorem for linear operators due to Maurey and Rosenthal. Our main application is a Hahn–Banach extension theorem for positive homogeneous polynomials between Banach lattices.
