### 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 |

### 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

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## Cite this

Ozawa, T., & Sasaki, H. (2009). Inequalities associated with dilations.

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