TY - JOUR

T1 - Inequalities associated with dilations

AU - Ozawa, Tohru

AU - Sasaki, Hironobu

N1 - Funding Information:
The authors are grateful to the referee for useful comments and information of the reference [2]. The second author was supported by the Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.

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

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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 -