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Weighted Hardy Inequalities for Increasing Functions

Published online by Cambridge University Press:  20 November 2018

H. P. Heinig  and
V. D. Stepanov
H. P. Heinig
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S4K1
V. D. Stepanov
Affiliation:
Department of Mathematics, University of Alabama, Tuscaloosa, Alabama 35487-0350, USA
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Abstract

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The purpose of this paper is to characterize the weight functions for which the Hardy operator , with non-decreasing function ƒ, is bounded between various weighted Lp -spaces for a wide range of indices. Our characterizations complement for the most part those of E. T. Sawyer [11] and V. D. Stepanov [15] for the Hardy operator of non-increasing function.

Keywords

26D1542B25weighted inequalitiesHardy's inequality

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 1 , 01 February 1993 , pp. 104 - 116
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Andersen, K.F., Weighted generalized Hardy inequalities for nonincreasing functions, Canad. J. Math. 43(1991), 1121–1135.Google Scholar
2. Arino, M. and Muckenhoupt, B., Maximal functions on classical Lorentz spaces and Hardy s inequality with weights for non-increasing functions, Trans. Amer. Math. Soc. (2) 320(1990), 727–735.Google Scholar
3. Boyd, D.W., The Hilbert transform on rearrangement-invariant spaces, Canad. J. Math. 19(1967), 599–616.Google Scholar
4. Krein, S.G., Petunin, Yu. I. and Semenov, E.M., Interpolation of Linear Operators, Transi. Amer. Math. Monographs 54, A.M.S. Providence R.I., 1982.Google Scholar
5. Lai, Shanzhong, Weighted norm inequalities for general operators on monotone functions, preprint.Google Scholar
6. Lai, Shanzhong, Weak and strong type inequalities with weights for general operators on monotone functions, preprint.Google Scholar
7. Martin-Reyes, F.J. and Sawyer, E., Weighted inequalities for Riemann-Liouville fractional integrals of order one and greater, Proc. Amer. Math. Soc. 106(1989), 727–733.Google Scholar
8. Muckenhoupt, B., Hardy's inequality with weights, Studia Math. 44(1972), 31–38.Google Scholar
9. Neugebauer, C.J., Weighted norm inequalities for averaging operators of monotone functions, to appear.Google Scholar
10. Opic, B. and Kufner, A., Hardy type inequalities, Pitman Research Notes in Math. Sen, John Wiley, N.Y., 1990.Google Scholar
11. Sawyer, E.T., Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96(1990), 149–158.Google Scholar
12. Sinnamon, G., Operators on Lebesgue spaces with general measures, Ph.D. Thesis, McMaster University, 1987.Google Scholar
13. Stepanov, V.D., Two weighted estimates for the Riemann-Liouville integrals, Math. Ustav (39), Prague Czech., preprint, 1988.Google Scholar
14. Stepanov, V.D., Weighted inequalities for a class ofVolterra convolution operators, J. London Math. Soc. (2) 45(1992), 232–242.Google Scholar
15. Stepanov, V.D., The weighted Hardy s inequality for non-increasing functions, Trans. Amer. Math. Soc, to appear.Google Scholar