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Article

Keywords:
Hardy inequality; modular inequality; weight functions
Summary:
If $P$ is the Hardy averaging operator - or some of its generalizations, then weighted modular inequalities of the form \int u \phi(Pf) \leq C\int v \phi(f) are established for a general class of functions $\phi$. Modular inequalities for the two- and higher dimensional Hardy averaging operator are also given.
References:
[1] Maria J. Carro, Hans Heinig: Modular inequalities for the Calderón operator. Tohoku Math. J. To appear. MR 1740541
[2] M. DeGuzmán: Real Variable Methods in Fourier analysis. Univ. Complutense de Madrid, Fac. Mat., 1977.
[3] P. Drábek H. P. Heinig A. Kufner: Higher dimensional Hardy inequality. Internat. Ser. Numer. Math. 123 (1997), 3-16. MR 1457264
[4] G. H. Hardy J. E. Littlewood G. Pólya: Inequalities. Cambridge, 1934. MR 0046395
[5] H. P. Heinig R. Kerman M. Krbec: Weighted exponential inequalities. Preprint, vol. 79, Math. Inst., Acad. Science, Praha, 1992, pp. 30. MR 1828685
[6] Hans P. Heinig, Qinsheng Lai: Weighted modular inequalities for Hardy-type operators on monotone functions. Preprint. MR 1756661
[7] Qinsheng Lai: Weighted modular inequalities for Hardy-type operators. J. London Math. Soc. To appear. MR 1710168
[8] N. Levinson: Generalizations of an inequality of Hardy. Duke J. Math. 31 (1964), 389-394. DOI 10.1215/S0012-7094-64-03137-0 | MR 0171885 | Zbl 0126.28101
[9] B. Opic A. Kufner: Hardy type inequalities. Pitman Series 219, Harlow, 1990. MR 1069756
[10] B. Opic P. Gurka: Weighted inequalities for geometric means. Proc. Arner. Math. Soc. 120 (1994), no. 3, 771-779. DOI 10.1090/S0002-9939-1994-1169043-4 | MR 1169043
[11] E. Sawyer: Weighted inequalities for the two dimensional Hardy operator. Studia Math. 82 (1985), 1-16. MR 0809769 | Zbl 0585.42020
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