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Reduction of Opial-type inequalities to norm inequalities
Author(s):
Gord
Sinnamon
Journal:
Proc. Amer. Math. Soc.
132
(2004),
375-379.
MSC (2000):
Primary 26D15
Posted:
September 5, 2003
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Abstract:
Weighted Opial-type inequalities are shown to be equivalent to weighted norm inequalities for sublinear operators and for nearly positive operators. Examples involving the Hardy-Littlewood maximal function and the nonincreasing rearrangement are presented.
References:
-
- 1.
- C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, Vol. 129, Academic Press, Boston, MA, 1988. MR 89e:46001
- 2.
- S. Bloom, First and second order Opial inequalities, Studia Math. 126 (1997), 27-50. MR 98g:26016
- 3.
- R. C. Brown, A. M. Fink, and D. B. Hinton, Some Opial, Lyapunov, and De la Valée Poussin inequalities with nonhomogeneous boundary conditions, J. Inequal. Appl. 5 (2000), 11-37. MR 2000m:34073
- 4.
- G. Sinnamon, Weighted Hardy and Opial-type inequalities, J. Math. Anal. Appl. 160 (1991), 434-445. MR 92f:26037
- 5.
- V. D. Stepanov, The weighted Hardy's inequality for nonincreasing functions, Trans. Amer. Math. Soc., 338 (1993), 173-186. MR 93j:26012
- 6.
- A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, Orlando, FL, 1986. MR 91m:42001
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Additional Information:
Gord
Sinnamon
Affiliation:
Department of Mathematics, University of Western Ontario, London, Ontario, N6A 5B7, Canada
Email:
sinnamon@uwo.ca
DOI:
10.1090/S0002-9939-03-07184-3
PII:
S 0002-9939(03)07184-3
Keywords:
Opial inequality,
Opial-type inequality,
weight
Received by editor(s):
July 19, 2002
Posted:
September 5, 2003
Additional Notes:
Supported by the Natural Sciences and Engineering Research Council of Canada
Communicated by:
Jonathan M. Borwein
Copyright of article:
Copyright
2003,
American Mathematical Society
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