Published online by Cambridge University Press: 01 April 2008
Let an almost everywhere positive function Φ be convex for p>1 and p<0, concave for p∈(0,1), and such that Axp≤Φ(x)≤Bxp holds on for some positive constants A≤B. In this paper we derive a class of general integral multidimensional Hardy-type inequalities with power weights, whose left-hand sides involve instead of , while the corresponding right-hand sides remain as in the classical Hardy’s inequality and have explicit constants in front of integrals. We also prove the related dual inequalities. The relations obtained are new even for the one-dimensional case and they unify and extend several inequalities of Hardy type known in the literature.
The research of the first author was supported by the Swedish Institute under the Guest Fellowship Programme 210/05529/2005. The research of the third author was supported by the Croatian Ministry of Science, Education and Sports, under Research Grant 058-1170889-1050.