Remarks on Gagliardo-Nirenberg type inequality with critical Sobolev space and BMO

Hideo Kozono, Hidemitsu Wadade

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33 Citations (Scopus)

Abstract

We consider the generalized Gagliardo-Nirenberg inequality in ℝn in the homogeneous Sobolev space Hs, rn with the critical differential order s = n/r, which describes the embedding such as Lpn∩ Hn/r,rn Lqn for all q with p q < ∞, where 1 < p < ∞ and 1 < r < ∞. We establish the optimal growth rate as q → ∞ of this embedding constant. In particular, we realize the limiting end-point r = ∞ as the space of BMO in such a way that ||u|| LqnCnq||u||L pnp}{q}}||u||BMO1p}{q}} with the constant C n depending only on n. As an application, we make it clear that the well known John-Nirenberg inequality is a consequence of our estimate. Furthermore, it is clarified that the L -bound is established by means of the BMO-norm and the logarithm of the Hs, r -norm with s > n/r, which may be regarded as a generalization of the Brezis-Gallouet- Wainger inequality.

Original languageEnglish
Pages (from-to)935-950
Number of pages16
JournalMathematische Zeitschrift
Volume259
Issue number4
DOIs
Publication statusPublished - 2008 Aug 1

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ASJC Scopus subject areas

  • Mathematics(all)

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