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Journal of Function Spaces and Applications
Volume 4 (2006), Issue 2, Pages 163-191

Three weights higher order Hardy type inequalities

1Eurasian National University, Munaytpasov St., 5, 010000 Astana, Kazakhstan
2Luleå University of Technology, Department of Mathematics, SE – 971 87 Luleå, Sweden

Received 1 May 2005

Academic Editor: Vladimir Stepanov

Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We investigate the following three weights higher order Hardy type inequality (0.1) gq,u  CDρkgp,v where Dρi denotes the following weighted differential operator: {dig(t)dti,i=0,1,...,m1,dimdtim(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<pq<. 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.