Some weighted sum and product inequalities in L^p spaces and their applications

Abstract

We survey some old and new results concerning weighted norm inequalities of sum and product form and apply the theory to obtain limit-point conditions for second order differential operators of Sturm-Liouville form defined in $L^p$ spaces. We also extend results of Anderson and Hinton by giving necessary and sufficient criteria that perturbations of such operators be relatively bounded. Our work is in part a generalization of the classical Hilbert space theory of Sturm-Liouville operators to a Banach space setting.