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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 {\it T. G. Anderson} and {\it 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.
26D10Inequalities involving derivatives, differential and integral operators
47A30Operator norms and inequalities
34B24Sturm-Liouville theory
47E05Ordinary differential operators
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