A general approach to weighted Lp rellich type inequalities related to greiner operator

KÖMBE İ., YENER A.

Communications on Pure and Applied Analysis, cilt.18, sa.2, ss.869-886, 2019 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 18 Sayı: 2
  • Basım Tarihi: 2019
  • Doi Numarası: 10.3934/cpaa.2019042
  • Dergi Adı: Communications on Pure and Applied Analysis
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.869-886
  • Anahtar Kelimeler: Greiner operator, weighted Rellich inequality, remainder term
  • İstanbul Ticaret Üniversitesi Adresli: Evet

Özet

In this paper we exhibit some sufficient conditions that imply general weighted Lp Rellich type inequality related to Greiner operator without assuming a priori symmetric hypotheses on the weights. More precisely, we prove that given two nonnegative functions a and b, if there exists a positive supersolution ν of the Greiner operator Δκ such that Δκ(a|Δκν|p-2Δκν)≥bνp-1 almost everywhere in R2n+1; then a and b satisfy a weighted Lp Rellich type inequality. Here, p > 1 and Δκ = Σn j=1(x2 j+y2 j) is the sub-elliptic operator generated by the Greiner vector fields xj{equation presented} where (z,l)=(x,y,l)∈ R2n+1=Rn×Rn×R,|Z|={equation presented} and k ≥ 1. The method we use is quite practical and constructive to obtain both known and new weighted Rellich type inequalities. On the other hand, we also establish a sharp weighted Lp Rellich type inequality that connects first to second order derivatives and several improved versions of two-weight Lp Rellich type inequalities associated to the Greiner operator Δκ on smooth bounded domains Ω in R2n+1.