Inequalities associated with dilations

Tohru Ozawa*, Hironobu Sasaki

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)


Some properties of distributions f satisfying x · ∇ f ∈ Lp (ℝn), 1 ≤ p < ∞, are studied. The operator x · ∇ is the generator of a semi-group of dilations. We first give Sobolev type inequalities with respect to the operator x · ∇. Using the inequalities, we also show that if $f \in L-\rm loc ^p (\mathbb R^n)$, x · ∇ f ∈ Lp (ℝn) and |x|n/p|f(x)| vanishes at infinity, then f belongs to Lp (ℝn). One of the Sobolev type inequalities is shown to be equivalent to the Hardy inequality in L2 (ℝn).

Original languageEnglish
Pages (from-to)265-277
Number of pages13
JournalCommunications in Contemporary Mathematics
Issue number2
Publication statusPublished - 2009 Apr


  • Generator of semi-group of dilations
  • Hardy's inequality
  • Inequalities
  • Poincaré's inequality
  • Sobolev's inequality

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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