Boundedness criteria and norm of some multilinear Hilbert-type operators. (English) Zbl 1470.44003

Summary: We consider two families of multilinear Hilbert-type operators for which we give exact relations between the parameters so that they are bounded. We also find the exact norm of these operators.


44A15 Special integral transforms (Legendre, Hilbert, etc.)
33B15 Gamma, beta and polygamma functions
42B25 Maximal functions, Littlewood-Paley theory
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[1] Hardy, G. H., Note on a theorem of Hilbert concerning series of positive terms, Proceedings of the London Mathematical Society, 23, 2, 45-46, (1925)
[2] Hardy, G. H.; Littlewood, J. E.; Polya, G., Inequalities, (1952), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0047.05302
[3] Mitrinović, D. S.; Pečarić, J. E.; Fink, A. M., Inequalities Involving Functions and Their Integrals and Derivatives. Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and Its Applications, 53, (1991), Boston, Mass, USA: Kluwer Academic Publishers, Boston, Mass, USA · Zbl 0744.26011
[4] Yang, B. C., On a general Hardy-Hilbert’s inequality with a best value, Chinese Annals of Mathematics, 21, 4, 401-408, (2000) · Zbl 0991.26010
[5] Bicheng, Y., On Hardy–Hilbert’s Integral Inequality, Journal of Mathematical Analysis and Applications, 261, 1, 295-306, (2001) · Zbl 0989.26012
[6] Yang, B., On the extended Hilbert’s integral inequality, JIPAM. Journal of Inequalities in Pure and Applied Mathematics, 5, 4, article 96, (2004) · Zbl 1082.26019
[7] Bansah, J.; Sehba, B. F., Boundedness of a family of Hilbert-type operators and of its Bergman-type analogue · Zbl 1370.47029
[8] Bekollé, D.; Bonami, A.; Peloso, M. M.; Ricci, F., Boundedness of Bergman projections on tube domains over light cones, Mathematische Zeitschrift, 237, 1, 31-59, (2001) · Zbl 0983.32001
[9] Bényi, Á.; Oh, C., Best constants for certain multilinear integral operators, Journal of Inequalities and Applications, 2006, (2006) · Zbl 1116.26011
[10] Yang, B.; Rassias, T. M., On the way of weight coefficient and research for the Hilbert-type inequalities, Mathematical Inequalities & Applications, 6, 4, 625-658, (2003) · Zbl 1046.26012
[11] Bicheng, Y.; Brnetić, I.; Krnić, M.; Pečarić, J., Generalization of HILbert and HARdy-HILbert integral inequalities, Mathematical Inequalities & Applications, 8, 2, 259-272, (2005) · Zbl 1078.26019
[12] Fu, Z.; Grafakos, L.; Lu, S.; Zhao, F., Sharp bounds for \(m\)-linear Hardy and HILbert operators, Houston Journal of Mathematics, 38, 1, 225-244, (2012) · Zbl 1248.42020
[13] Yang, B. C., A multiple Hardy-Hilbert’s integral inequality, Chinese Annals of Mathematics A, 24, 6, 743-750, (2003) · Zbl 1072.26021
[14] Yang, B., On a new multiple extension of Hilbert’s integral inequality, Journal of Inequalities in Pure and Applied Mathematics, 6, 2, article no. 39, (2005) · Zbl 1078.26018
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