### Abstract

Some properties of distributions f satisfying x · ∇ f ∈ L^{p} (ℝ^{n}), 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 ∈ L^{p} (ℝ^{n}) and |x|^{n/p}|f(x)| vanishes at infinity, then f belongs to L^{p} (ℝ^{n}). One of the Sobolev type inequalities is shown to be equivalent to the Hardy inequality in L^{2} (ℝ^{n}).

Original language | English |
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Pages (from-to) | 265-277 |

Number of pages | 13 |

Journal | Communications in Contemporary Mathematics |

Volume | 11 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2009 Apr |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications in Contemporary Mathematics*,

*11*(2), 265-277. https://doi.org/10.1142/S0219199709003351

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

Research output: Contribution to journal › Article

*Communications in Contemporary Mathematics*, vol. 11, no. 2, pp. 265-277. https://doi.org/10.1142/S0219199709003351

}

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 -