Summary: We survey some old and new results concerning weighted norm inequalities of sum and product form and apply the theory to obtain limitpoint conditions for second order differential operators of Sturm-Liouville form defined in
spaces. We also extend results of T. G. Anderson
and D. B. Hinton
[J. Inequal. Appl. 1, No. 4, 375–400 (1997; Zbl 0891.34062
)] 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.