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Mixed Norm Type Hardy Inequalities

Published online by Cambridge University Press:  20 November 2018

Alberto Fiorenza ,
Babita Gupta  and
Pankaj Jain
Alberto Fiorenza
Affiliation:
Dipartimento di Costruzioni e Metodi Matematici in Architettura, Universitá di Napoli, I-80134 Napoli, Italy, andIstituto per le Applicazioni del Calcolo “Mauro Picone”, sezione di Napoli, Consiglio Nazionale delle Ricerche, I-80131 Napoli, Italye-mail: fiorenza@unina.it
Babita Gupta
Affiliation:
Department of Mathematics, Shivaji College (University of Delhi), Raja Garden, Delhi 110 027, Indiae-mail: babita.gupta@hotmail.com
Pankaj Jain
Affiliation:
Department of Mathematics, Deshbandhu College (University of Delhi), Kalkaji, New Delhi 110 019, Indiae-mail: pankajkrjain@hotmail.com
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Abstract

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

Keywords

26D1026D15Hardy inequalityreverse Hardy inequalitymixed normHardy–Steklov operator
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 4 , 01 December 2011 , pp. 630 - 644
Copyright
Copyright © Canadian Mathematical Society 2011

References

[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] 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] Benedek, A. and Panzone, R., The spaces Lp with mixed norm.. Duke Math. J. 28(1961), 301–324. doi:10.1215/S0012-7094-61-02828-9Google Scholar
[4] Bennett, G., Factorizing the classical inequalities. Mem. Amer. Math. Soc. 120(1996), no. 576.Google Scholar
[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] Heinig, H. P. and Sinnamon, G., Mapping properties of integral averaging operators. Studia Math. 129(1998), no. 2, 157–177.Google Scholar
[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.2003.09.036Google Scholar
[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] 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-2004-053-5Google Scholar
[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] Kufner, A. and Persson, L.-E., Weighted inequalities of Hardy type. World Scientific, River Edge, NJ, 2003.Google Scholar
[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:10.2307/27642033Google Scholar
[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] Maligranda, L., Orlicz spaces and interpolation. Seminários de Matemática, 5, Departamento de matematica, universidade Estadual de Campinas, Brazil, 1989.Google Scholar
[15] Maligranda, L., Calderón–Lozanovskĭ construction for mixed norm spaces. Acta Math. Hungar. 103(2004), no. 4, 279–302. doi:10.1023/B:AMHU.0000028829.15720.02Google Scholar
[16] Prokhorov, D. V. Weighted Hardy's inequalities for negative indices.. Publ. Mat. 48(2004), 423–443.Google Scholar
[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] 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] 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