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On the Principle of Duality in Lorentz Spaces

Published online by Cambridge University Press:  20 November 2018

M. L. Gol'Dman ,
H. P. Heinig  and
V. D. Stepanov
M. L. Gol'Dman
Affiliation:
Moscow Institute of Radiotechnology, Electronics and Automation Krupskaja 8-1-187 Moscow 117311 Russia email: e-mail: seulydia@glas.apc.org
H. P. Heinig
Affiliation:
Department of Mathematics and Statistics McMaster University, Hamilton, Ontario L8S 4K1 email: e-mail:Heinig@mcmaster.ca
V. D. Stepanov
Affiliation:
Computer Center of the Far-Eastern Branch of the Russian Academy of Sciences Shelest 118-205 Khabarovsk 6800042 email: e-mail:1600@as.khabarovsk.su
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Abstract

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characterization of the spaces dual to weighted Lorentz spaces are given by means of reverse Hölder inequalities (Theorems 2.1, 2.2). This principle of duality is then applied to characterize weight functions for which the identity operator, the Hardy-Littlewood maximal operator and the Hilbert transform are bounded on weighted Lorentz spaces.

Keywords

47B3842B2046E3026D15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 5 , 01 October 1996 , pp. 959 - 979
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. Andersen, K.F., Weighted generalized Hardy inequalities for non-increasing functions, Canad. J., Math. 43(1991), 1121–1135.Google Scholar
2. Ariño, M. and Muckenhoupt, B., Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for non-increasing functions, Trans. Amer. Math., Soc. 320(1990), 727–735.Google Scholar
3. Bennett, C. and Sharpley, R., Interpolation of Operators, Pure Appl. Math. 129, Acad. Press, 1988.Google Scholar
4. Bergh, J. and Löfström, J., Interpolation Spaces. An Introduction, Springer Verlag, New York 1976.Google Scholar
5. Sh, M.. Braverman, On a class of operators, J. London Math. Soc., (2) 47(1993), 119–128.Google Scholar
6. Carro, M.J. and Soria, J., Weighted Lorentz spaces and the Hardy operator, J. Funct. Anal., (2) 112(1993), 480–494.Google Scholar
7. Carro, M.J., Boundedness of some integral operators, Canad. J. Math., (6) 45(1993), 1155–1166.Google Scholar
8. Gol'dman, M.L., Functions spaces and their applications, Patrice Lumumba Univ., (1991), 35–67.Google Scholar
9. Gol'dman, M.L., On integral inequalities on a cone of functions with monotonicity properties, Soviet Math. Dokl. (2)44(1992), 581–587.Google Scholar
10. Gol'dman, M.L., Function spaces, differential operators and nonlinear analysis, Teubner Texte, Math. 133(1993), 274–279.Google Scholar
11. Halperin, I., Function spaces, Canad. J., Math. 5(1953), 273–288.Google Scholar
12. Heinig, H.P. and Stepanov, V.D., Weighted Hardy inequalities for increasing functions, Canad. J., Math. 45(1993), 104–116.Google Scholar
13. Krein, S.G., Yu. Petunin, I. and Semenov, E.M., Interpolation of linear operators, Trans. Amer. Math. Soc, Providence, Rhode Island, 1982.Google Scholar
14. Lorentz, G.G., On the theory of spaces A, Pacific J., Math. 1(1951), 411-129.Google Scholar
15. Maz'ja, V.G., Sobolev Spaces, Springer Verlag, Berlin, 1985.Google Scholar
16. Oskolkov, K.I., Approximation properties of summable functions on sets of full measure, Math. USSR Sb., (4) 32(1977), 489–517.Google Scholar
17. Sawyer, E.T., Boundedness of classical operators in classical Lorentz spaces, Studia, Math. 96(1990), 145–158.Google Scholar
18. Sinnamon, G., Spaces defined by level functions and their duals, Studia Math., (1) 111(1994), 19–52.Google Scholar
19. Stein, E.M., Note on the class L log!, Studia, Math. 32(1969), 301–310.Google Scholar
20. Stein, E.M. and Weiss, G., Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, 1971.Google Scholar
21. Stepanov, V.D., On integral operators on the cone of monotone functions and embeddings of the Lorentz spaces, Soviet Math., Dokl. 43(1991), 620–623.Google Scholar
22. Stepanov, V.D., Weighted inequalities for a class ofVolterra convolution operators, J. London Math. Soc., (2) 45(1992), 232–242.Google Scholar
23. Stepanov, V.D., On weighted estimates for a class of integral operators, Siberian Math., J. 34(1993), 755–766.Google Scholar
24. Stepanov, V.D., The weighted Hardy's inequality for non-increasing functions, Trans. Amer. Math., Soc. 338(1993), 173–186.Google Scholar
25. Stepanov, V.D., Integral operators on the cone of monotone functions, J. London Math. Soc., (2) 48(1993), 465–487.Google Scholar
26. Triebel, H., Interpolation theory, function spaces, differential operators, Deutscher Verl. Wiss., Berlin, 1978.Google Scholar
27. Zygmund, A., Trigonmetric series, vol. I, Cambridge Univ. Press, 1959.Google Scholar