Factorization and extension of positive homogeneous polynomials

Andreas Defant¹ ; Mieczysław Mastyło²

¹Institut für Mathematik Carl von Ossietzky Universität Postfach 2503 D-26111 Oldenburg, Germany
²Faculty of Mathematics and Computer Science A. Mickiewicz University and Institute of Mathematics Polish Academy of Sciences (Poznań branch) Umultowska 87 61-614 Poznań, Poland

Studia Mathematica, Tome 221 (2014) no. 1, pp. 87-99

Résumé

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.

Détail

DOI : 10.4064/sm221-1-5

Keywords: study following problem given homogeneous polynomial sublattice banach lattice banach lattice under which additional hypotheses does polynomial factorize through p spaces involving multiplication operators prove under lattice convexity concavity hypotheses polynomials certain vector valued norm inequalities weighted norm inequalities equivalent combine these results prove factorization theorem positive homogeneous polynomials which variant celebrated factorization theorem linear operators due maurey rosenthal main application hahn banach extension theorem positive homogeneous polynomials between banach lattices