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

Bansah, Justice S.; Sehba, Benoît F.

Abstr. Appl. Anal. 2017, Article ID 6829093, 13 p. (2017).

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.

MSC:

44A15	Special integral transforms (Legendre, Hilbert, etc.)
33B15	Gamma, beta and polygamma functions
42B25	Maximal functions, Littlewood-Paley theory

References:

[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) · JFM 51.0179.06
[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 · doi:10.1007/978-94-011-3562-7
[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 · doi:10.1006/jmaa.2001.7525
[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 · doi:10.1007/pl00004861
[9]	Bényi, Á.; Oh, C., Best constants for certain multilinear integral operators, Journal of Inequalities and Applications, 2006, (2006) · Zbl 1116.26011 · doi:10.1155/jia/2006/28582
[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 · doi:10.7153/mia-06-58
[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 · doi:10.7153/mia-08-25
[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