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Some weighted sum and product inequalities in ${L}^{p}$ spaces and their applications. (English) Zbl 1138.26010
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 ${L}^{p}$ 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.
##### MSC:
 26D10 Inequalities involving derivatives, differential and integral operators 47A30 Operator norms and inequalities 34B24 Sturm-Liouville theory 47E05 Ordinary differential operators