Mixed Norm Type Hardy Inequalities

Fiorenza, Alberto ; Gupta, Babita ; Jain, Pankaj

Canadian mathematical bulletin, Tome 54 (2011) no. 4, pp. 630-644

Résumé

Higher dimensional mixed norm type inequalities involving certain integral operators are characterized in terms of the corresponding lower dimensional inequalities.

Détail

DOI : 10.4153/CMB-2011-022-0

Mots-clés : 26D10, 26D15, Hardy inequality, reverse Hardy inequality, mixed norm, Hardy–Steklov operator

Bibliographie

[1] [1] Appell, J. and Kufner, A., On the two dimensional Hardy operator in Lebesgue spaces with mixed norms. Analysis 15(1995), no. 1, 91–98. Google Scholar

[2] [2] Beesack, P. R. and Heinig, H. P., Hardy's inequalities with indices less than 1 . Proc. Amer. Math. Soc. 83(1981), no. 3, 532–536. Google Scholar

[3] [3] Benedek, A. and Panzone, R., The spaces Lp with mixed norm.. Duke Math. J. 28(1961), 301–324. doi:10.1215/S0012-7094-61-02828-9 Google Scholar

[4] [4] Bennett, G., Factorizing the classical inequalities. Mem. Amer. Math. Soc. 120(1996), no. 576. Google Scholar

[5] [5] Fiorenza, A., Gupta, B., and Jain, P., Compactness of integral operators in Lebesgue spaces with mixed norm. Math. Inequal. Appl. 11(2008), no. 2, 335–348. Google Scholar

[6] [6] Heinig, H. P. and Sinnamon, G., Mapping properties of integral averaging operators. Studia Math. 129(1998), no. 2, 157–177. Google Scholar

[7] [7] Jain, P. and Gupta, B., Compactness of Hardy-Steklov operator. J. Math. Anal. Appl. 288(2003), no. 2, 680–691. doi:10.1016/j.jmaa.2003.09.036 Google Scholar

[8] [8] Jain, P., Jain, P. K., and Gupta, B., On certain double sized integral operators over multidimensional cones. Proc. A. Razmadze Math. Inst. 131(2003), 39–60. Google Scholar

[9] [9] Jain, P., Jain, P. K., and Gupta, B., Compactness of Hardy type operators over star-shaped regions in ℝ N . Canad. Math. Bull. 47(2004), no. 4, 540–552. doi:10.4153/CMB-2004-053-5 Google Scholar

[10] [10] Jain, P., Jain, P. K., and Gupta, B., On certain weighted integral inequalities with mixed norm.. Ital. J. Pure Appl. Math. 18(2005), 23–32. Google Scholar

[11] [11] Kufner, A. and Persson, L.-E., Weighted inequalities of Hardy type. World Scientific, River Edge, NJ, 2003. Google Scholar

[12] [12] Kufner, A., Maligranda, L., and Persson, L.-E., The prehistory of the Hardy inequality. Amer. Math. Monthly 113(2006), no. 8, 715–732. doi:10.2307/27642033 Google Scholar

[13] [13] Kufner, A., Maligranda, L., and Persson, L.-E., The Hardy inequality. About its history and some related results. Vydavetelský Servis, Plzeň, 2007. Google Scholar

[14] [14] Maligranda, L., Orlicz spaces and interpolation. Seminários de Matemática, 5, Departamento de matematica, universidade Estadual de Campinas, Brazil, 1989. Google Scholar

[15] [15] Maligranda, L., Calderón–Lozanovskĭ construction for mixed norm spaces. Acta Math. Hungar. 103(2004), no. 4, 279–302. doi:10.1023/B:AMHU.0000028829.15720.02 Google Scholar

[16] [16] Prokhorov, D. V. Weighted Hardy's inequalities for negative indices.. Publ. Mat. 48(2004), 423–443. Google Scholar

[17] [17] Sinnamon, G., One dimensional Hardy-type inequalities in many dimensions. Proc. Roy. Soc. Edinburg Sect. A 128(1998), no. 4, 833–848. Google Scholar

[18] [18] Sinnamon, G., Hardy-type inequalities for a new class of integral operators. In: Analysis of divergence (Orono, ME, 1997), Birkhäuser Boston, Boston, MA, 1999, pp. 297–307. Google Scholar

[19] [19] Sinnamon, G., Hardy's inequality and monotonicity, In: Function spaces, differential operators and nonlinear analysis, Mathematical Institute of the Academy of Sciences of the Czech Republic, Prague, 2005, pp. 292–310. Google Scholar