TY - JOUR

T1 - Remarks on Gagliardo-Nirenberg type inequality with critical Sobolev space and BMO

AU - Kozono, Hideo

AU - Wadade, Hidemitsu

N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2008/8

Y1 - 2008/8

N2 - 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 Lpℝn∩ Hn/r,rℝn Lqℝn 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|| LqℝnCnq||u||L pℝnp}{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.

AB - 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 Lpℝn∩ Hn/r,rℝn Lqℝn 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|| LqℝnCnq||u||L pℝnp}{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.

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U2 - 10.1007/s00209-007-0258-5

DO - 10.1007/s00209-007-0258-5

M3 - Article

AN - SCOPUS:43749088974

VL - 259

SP - 935

EP - 950

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 4

ER -