### Abstract

We will consider a Trudinger-Moser inequality for the critical Sobolev space H^{n/p,p}(ℝ^{n}) with the fractional derivatives in ℝ^{n} and obtain an upper bound of the best constant of such an inequality. Moreover, by changing normalization from the homogeneous norm to the inhomogeneous one, we will give the best constant in the Hubert space H ^{n/2,2}(ℝ^{n}). As an application, we will obtain some lower bound of the best constant of a Gagliardo-Nirenberg inequality. Indiana University Mathematics Journal

Original language | English |
---|---|

Pages (from-to) | 1951-1974 |

Number of pages | 24 |

Journal | Indiana University Mathematics Journal |

Volume | 55 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2006 |

Externally published | Yes |

### Fingerprint

### Keywords

- Average function
- Fractional integral
- Gagliadro-Nirenberg inequality
- Rearrangement
- Riesz potential
- Sobolev inequality
- Trudinger-Moser inequality

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Indiana University Mathematics Journal*,

*55*(6), 1951-1974. https://doi.org/10.1512/iumj.2006.55.2743

**Upper bound of the best constant of a trudinger-moser inequality and its application to A Gagliardo-Nirenberg inequality.** / Kozono, Hideo; Sato, Tokushi; Wadade, Hidemitsu.

Research output: Contribution to journal › Article

*Indiana University Mathematics Journal*, vol. 55, no. 6, pp. 1951-1974. https://doi.org/10.1512/iumj.2006.55.2743

}

TY - JOUR

T1 - Upper bound of the best constant of a trudinger-moser inequality and its application to A Gagliardo-Nirenberg inequality

AU - Kozono, Hideo

AU - Sato, Tokushi

AU - Wadade, Hidemitsu

PY - 2006

Y1 - 2006

N2 - We will consider a Trudinger-Moser inequality for the critical Sobolev space Hn/p,p(ℝn) with the fractional derivatives in ℝn and obtain an upper bound of the best constant of such an inequality. Moreover, by changing normalization from the homogeneous norm to the inhomogeneous one, we will give the best constant in the Hubert space H n/2,2(ℝn). As an application, we will obtain some lower bound of the best constant of a Gagliardo-Nirenberg inequality. Indiana University Mathematics Journal

AB - We will consider a Trudinger-Moser inequality for the critical Sobolev space Hn/p,p(ℝn) with the fractional derivatives in ℝn and obtain an upper bound of the best constant of such an inequality. Moreover, by changing normalization from the homogeneous norm to the inhomogeneous one, we will give the best constant in the Hubert space H n/2,2(ℝn). As an application, we will obtain some lower bound of the best constant of a Gagliardo-Nirenberg inequality. Indiana University Mathematics Journal

KW - Average function

KW - Fractional integral

KW - Gagliadro-Nirenberg inequality

KW - Rearrangement

KW - Riesz potential

KW - Sobolev inequality

KW - Trudinger-Moser inequality

UR - http://www.scopus.com/inward/record.url?scp=33846886977&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846886977&partnerID=8YFLogxK

U2 - 10.1512/iumj.2006.55.2743

DO - 10.1512/iumj.2006.55.2743

M3 - Article

AN - SCOPUS:33846886977

VL - 55

SP - 1951

EP - 1974

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 6

ER -