A general approach to weighted Rellich type inequalities on Carnot groups

Goldstein J. A., KÖMBE İ., YENER A.

Monatshefte fur Mathematik, vol.186, no.1, pp.49-72, 2018 (SCI-Expanded, Scopus) identifier identifier

  • Publication Type: Article / Article
  • Volume: 186 Issue: 1
  • Publication Date: 2018
  • Doi Number: 10.1007/s00605-017-1060-z
  • Journal Name: Monatshefte fur Mathematik
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.49-72
  • Keywords: Carnot groups, Weighted Rellich inequality, Remainder term
  • İstanbul Ticaret University Affiliated: Yes

Abstract

We give a simple sufficient criterion on a pair of nonnegative weight functions a and b on a Carnot group G, so that the following general weighted Lp Rellich type inequality∫Ga|ΔGu|pdx≥∫Gb|u|pdxholds for every u∈C0∞(G) and p> 1. It is worthwhile to notice that our method easily derives previously known weighted Rellich type inequalities with a sharp constant in a more adequate fashion and also enables us to obtain new ones. We also present a sharp Lp Rellich type inequality that connects first to second order derivatives and some new two-weight Rellich type inequalities with remainders on bounded domains Ω in G via a differential inequality and the improved two-weight Hardy inequality in Goldstein et al. (Discret Contin Dyn Syst 37:2009–2021, 2017).