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
SN - 0219-1997
VL - 11
SP - 265
EP - 277
JO - Communications in Contemporary Mathematics
JF - Communications in Contemporary Mathematics
IS - 2
ER -