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On a fourth-order singular integral inequality*

Published online by Cambridge University Press:  14 November 2011

A. Russell
A. Russell
Affiliation:
Department of Mathematics, University of Dundee
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Synopsis

The inequality considered in this paper is

where N is the real-valued symmetric differential expression defined by

General properties of this inequality are considered which result in giving an alternative account of a previously considered inequality

to which (*) reduces in the case p = q = 0, r = 1.

Inequality (*) is also an extension of the inequality

as given by Hardy and Littlewood in 1932. This last inequality has been extended by Everitt to second-order differential expressions and the methods in this paper extend it to fourth-order differential expressions. As with many studies of symmetric differential expressions the jump from the second-order to the fourth-order introduces difficulties beyond the extension of technicalities: problems of a new order appear for which complete solutions are not available.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 80 , Issue 3-4 , 1978 , pp. 249 - 260
Copyright
Copyright © Royal Society of Edinburgh 1978

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