Inequalities associated with dilations

Tohru Ozawa, Hironobu Sasaki

Research output: Contribution to journalArticle

8 Citations (Scopus)

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).

Original language English 265-277 13 Communications in Contemporary Mathematics 11 2 https://doi.org/10.1142/S0219199709003351 Published - 2009 Apr

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Dilation
Hardy Inequality
Operator
Vanish
Semigroup
Infinity
Generator

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

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

In: Communications in Contemporary Mathematics, Vol. 11, No. 2, 04.2009, p. 265-277.

Research output: Contribution to journalArticle

Ozawa, Tohru ; Sasaki, Hironobu. / Inequalities associated with dilations. In: Communications in Contemporary Mathematics. 2009 ; Vol. 11, No. 2. pp. 265-277.
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