Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-20T06:42:54.731Z Has data issue: false hasContentIssue false
  • English
  • Français

A General Integral Inequality for the Derivative of an Equimeasurable Rearrangement

Published online by Cambridge University Press:  20 November 2018

G. F. D. Duff
G. F. D. Duff*
Affiliation:
Chelsea College, University of London; Stanford University, Stanford, California
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The theory of non-increasing (decreasing) equimeasurable rearrangements of functions was introduced by Hardy and Littlewood [6] in connection with their studies of fractional integrals and integral operators. Elementary properties of equimeasurable decreasing rearrangements are given in the monograph [7] of Hardy, Littlewood, and Polya on inequalities, while a more recent treatment is Okikiolu [9, § 5.4].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 4 , 01 August 1976 , pp. 793 - 804
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Bliss, G. A., An integral inequality, J. Lond. Math. Soc. 5 (1930), 40–46.Google Scholar
2. Chandrasekhar, S., Introduction to the study of stellar structure (Univ. of Chicago Press, 1939, reprinted Dover, 1957).Google Scholar
3. Duff, G. F. D., Differences, derivatives and decreasing rearrangements, Can. J. Math. 19 (1967), 1153–1178.Google Scholar
4. Duff, G. F. D., Integral inequalities for equimeasurable rearrangements, Can. J. Math. 22 (1970), 408–430.Google Scholar
5. Fowler, R. H., Further studies of Enden's and similar differential equations, Quart. J. Math. (Oxford Series) 2 (1931), 259–288.Google Scholar
6. Hardy, G. H. and Littlew∞d, J. E., Some properties of fractional integrals, I, Math. Zeit. 27 (1928) 565–606.Google Scholar
7. Hardy, G. H. and Littlew∞d, J. E., Notes on the theory of series (XII); On certain inequalities connected with the calculus of variations, J. Lond. Math. Soc. 5 (1930), 34–39.Google Scholar
8. Hardy, G. H. and Littlew∞d, J. E., A maximal theorem with function-theoretic application, Acta. Math. 5 (1930), 81–116.Google Scholar
9. Hardy, G. H., Littlew∞d, J. E. and Polya, G., Inequalities (Cambridge U.P. 1934).Google Scholar
10. Gradshteyn, I. S. and Ryshik, I. M., (trans. A. Jeffery), Tables of integrals, series and products (Academic Press, New York (1965).Google Scholar
11. Okikiolu, G. O., Aspects of the theory of bounded integral operators in Lp spaces (Academic Press, New York, 1971).Google Scholar
12. Polya, G. and G. Szegô, Isoperimetric inequalities in mathematical physics, Annals of Math. Studies £7 (Princeton, 1951).Google Scholar
13. Sobolev, S., On a theorem of functional analysis, Math. Sbornik (N.S.) 4 (1938), 471–497.Google Scholar
14. Talenti, G., Best constant in Sobolev inequality, Institute Matherriatico Ulisse Dini, Firenze, typescript report, 1974, 32 p.Google Scholar