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Zbl 1138.26010
Brown, R.C.
Some weighted sum and product inequalities in $L^p$ spaces and their applications. (English)
[J] Banach J. Math. Anal. 2, No. 2, 42-58, electronic only (2008). ISSN 1735-8787/e

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.

MSC 2000:: *26D10 Inequalities involving derivatives, diff. and integral operators
47A30 Operator norms and inequalities
34B24 Sturm-Liouville theory
47E05 Ordinary differential operators

Keywords: weighted sum inequalities; weighted product inequalities; Liouville operators; limit-point conditions; relatively bounded perturbations

Citations: Zbl 0891.34062

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