Three weights higher order Hardy type inequalities

Abstract

We investigate the following three weights higher order Hardy type inequality (0.1) ‖g‖q,u≤  C‖Dρkg‖p,v where Dρi denotes the following weighted differential operator: {dig(t)dti,i=0,1,...,m−1,di−mdti−m(p(t)dmg(t)dtm),i=m,m+1,...,k, for a weight function ρ(⋅). A complete description of the weights u, v and ρ so that (0.1) holds was given in [4] for the case 1<p≤q<∞. Here the corresponding characterization is proved for the case 1<q<p<∞. The crucial step in the proof of the main result is to use a new Hardy type inequality (for a Volterra type operator), which we first state and prove.