Lorentz norm inequalities for the Hardy operator involving suprema

Prokhorov, Dmitry

doi:10.1090/S0002-9939-2012-10976-1

Lorentz norm inequalities for the Hardy operator involving suprema
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Proc. Amer. Math. Soc. 140 (2012), 1585-1592 Request permission

Abstract:

The weighted Lorentz norm inequalities for the Hardy operator involving suprema are characterized.

References

Kenneth F. Andersen and Benjamin Muckenhoupt, Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72 (1982), no. 1, 9–26. MR 665888, DOI 10.4064/sm-72-1-9-26
Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
María J. Carro and Javier Soria, Weighted Lorentz spaces and the Hardy operator, J. Funct. Anal. 112 (1993), no. 2, 480–494. MR 1213148, DOI 10.1006/jfan.1993.1042
Huann Ming Chung, Richard A. Hunt, and Douglas S. Kurtz, The Hardy-Littlewood maximal function on $L(p,\,q)$ spaces with weights, Indiana Univ. Math. J. 31 (1982), no. 1, 109–120. MR 642621, DOI 10.1512/iumj.1982.31.31012
David E. Edmunds, Petr Gurka, and Luboš Pick, Compactness of Hardy-type integral operators in weighted Banach function spaces, Studia Math. 109 (1994), no. 1, 73–90. MR 1267713
Amiran Gogatishvili, Bohumír Opic, and Luboš Pick, Weighted inequalities for Hardy-type operators involving suprema, Collect. Math. 57 (2006), no. 3, 227–255. MR 2264321
Amiran Gogatishvili and Luboš Pick, A reduction theorem for supremum operators, J. Comput. Appl. Math. 208 (2007), no. 1, 270–279. MR 2347749, DOI 10.1016/j.cam.2006.10.048
M. L. Gol′dman, H. P. Heinig, and V. D. Stepanov, On the principle of duality in Lorentz spaces, Canad. J. Math. 48 (1996), no. 5, 959–979. MR 1414066, DOI 10.4153/CJM-1996-050-3
Elena Lomakina and Vladimir Stepanov, On the compactness and approximation numbers of Hardy-type integral operators in Lorentz spaces, J. London Math. Soc. (2) 53 (1996), no. 2, 369–382. MR 1373067, DOI 10.1112/jlms/53.2.369
Dmitry V. Prokhorov, Inequalities for Riemann-Liouville operator involving suprema, Collect. Math. 61 (2010), no. 3, 263–276. MR 2732371, DOI 10.1007/BF03191232
Eric Sawyer, Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator, Trans. Amer. Math. Soc. 281 (1984), no. 1, 329–337. MR 719673, DOI 10.1090/S0002-9947-1984-0719673-4
Vladimir D. Stepanov, Weighted norm inequalities for integral operators and related topics, Nonlinear analysis, function spaces and applications, Vol. 5 (Prague, 1994) Prometheus, Prague, 1994, pp. 139–175. MR 1322312