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SPACESANDTHEIRAPPLICATIONS BanachJ.Math.Anal.2(2008),no.2,42–58 SOMEWEIGHTEDSUMANDPRODUCTINEQUALITIESINL B J M A

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Banach J. Math. Anal. 2 (2008), no. 2, 42–58

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anach

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ournal of

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athematical

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nalysis ISSN: 1735-8787 (electronic)

http://www.math-analysis.org

SOME WEIGHTED SUM AND PRODUCT INEQUALITIES IN L

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SPACES AND THEIR APPLICATIONS

R. C. BROWN

This paper is dedicated to Professor Joseph E. Peˇcari´c Submitted by Th. M. Rassias

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 inLpspaces. We also extend results of Anderson and Hinton by giving necessary and sufficient criteria that perturbations of such operators be rela- tively bounded. Our work is in part a generalization of the classical Hilbert space theory of Sturm-Liouville operators to a Banach space setting.

Department of Mathematics, University of Alabama-Tuscaloosa, AL 35487- 0350, USA

E-mail address: [email protected]

Date: Received: 12 April 2008; Accepted 21 April 2008.

2000Mathematics Subject Classification. Primary: 26D10, 47A30, 34B24; Secondary 47E05.

Key words and phrases. Weighted sum inequalities, weighted product inequalities, Sturm Liouville operators, limit-point conditions, relatively bounded perturbations.

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