### Abstract

We consider the generalized Gagliardo-Nirenberg inequality in ℝ^{n} in the homogeneous Sobolev space H^{s, rn} with the critical differential order s = n/r, which describes the embedding such as L^{p}ℝ^{n}∩ H^{n/r,r}ℝ^{n} L^{q}ℝ^{n} 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|| _{L}^{q}ℝ^{n}C_{n}q||u||_{L} ^{p}ℝ^{n}p}{q}}||u||BMO^{1}p}{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 H^{s, r} -norm with s > n/r, which may be regarded as a generalization of the Brezis-Gallouet- Wainger inequality.

Original language | English |
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Pages (from-to) | 935-950 |

Number of pages | 16 |

Journal | Mathematische Zeitschrift |

Volume | 259 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2008 Aug 1 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematische Zeitschrift*,

*259*(4), 935-950. https://doi.org/10.1007/s00209-007-0258-5