# Inequalities associated with dilations

Tohru Ozawa, Hironobu Sasaki

10 被引用数 (Scopus)

## 抄録

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).

本文言語 English 265-277 13 Communications in Contemporary Mathematics 11 2 https://doi.org/10.1142/S0219199709003351 Published - 2009 4

## ASJC Scopus subject areas

• Mathematics(all)
• Applied Mathematics