MULTIDIMENSIONAL HARDY-TYPE INEQUALITIES VIA CONVEXITY

Abstract

AbstractLet an almost everywhere positive function Φ be convex forp>1 andp<0, concave forp∈(0,1), and such thatAxp≤Φ(x)≤Bxpholds on$\mathbb {R}_{+}$for some positive constantsA≤B. In this paper we derive a class of general integral multidimensional Hardy-type inequalities with power weights, whose left-hand sides involve$\Phi ( \int _0^{x_1} \cdots \int _0^{x_n} f(\t )\,d\t )$instead of$[ \int _0^{x_1} \cdots \int _0^{x_n} f(\t ) \, d\t ]^p$, 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.