# Inequalities associated with dilations

Tohru Ozawa, Hironobu Sasaki

研究成果: Article

8 引用 (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).

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

### Fingerprint

Dilation
Hardy Inequality
Operator
Vanish
Semigroup
Infinity
Generator

### ASJC Scopus subject areas

• Mathematics(all)
• Applied Mathematics

### これを引用

Inequalities associated with dilations. / Ozawa, Tohru; Sasaki, Hironobu.

：: Communications in Contemporary Mathematics, 巻 11, 番号 2, 04.2009, p. 265-277.

研究成果: Article

Ozawa, Tohru ; Sasaki, Hironobu. / Inequalities associated with dilations. ：: Communications in Contemporary Mathematics. 2009 ; 巻 11, 番号 2. pp. 265-277.
@article{1deb481e59794e83ba38638be6f51647,
title = "Inequalities associated with dilations",
abstract = "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).",
keywords = "Generator of semi-group of dilations, Hardy's inequality, Inequalities, Poincar{\'e}'s inequality, Sobolev's inequality",
author = "Tohru Ozawa and Hironobu Sasaki",
year = "2009",
month = "4",
doi = "10.1142/S0219199709003351",
language = "English",
volume = "11",
pages = "265--277",
journal = "Communications in Contemporary Mathematics",
issn = "0219-1997",
publisher = "World Scientific Publishing Co. Pte Ltd",
number = "2",

}

TY - JOUR

T1 - Inequalities associated with dilations

AU - Ozawa, Tohru

AU - Sasaki, Hironobu

PY - 2009/4

Y1 - 2009/4

N2 - 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).

AB - 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).

KW - Generator of semi-group of dilations

KW - Hardy's inequality

KW - Inequalities

KW - Poincaré's inequality

KW - Sobolev's inequality

UR - http://www.scopus.com/inward/record.url?scp=65349102901&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=65349102901&partnerID=8YFLogxK

U2 - 10.1142/S0219199709003351

DO - 10.1142/S0219199709003351

M3 - Article

AN - SCOPUS:65349102901

VL - 11

SP - 265

EP - 277

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

SN - 0219-1997

IS - 2

ER -