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 languageEnglish
    Pages (from-to)265-277
    Number of pages13
    JournalCommunications in Contemporary Mathematics
    Volume11
    Issue number2
    DOIs
    Publication statusPublished - 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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