Remarks on Gagliardo–Nirenberg type inequality with critical Sobolev space and BMO
Remarks on Gagliardo–Nirenberg type inequality with critical Sobolev space and BMO
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DOI：
10.1007/s00209-007-0258-5
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发表时间：
2008-08
影响因子：
0.8
通讯作者：
H. Kozono;H. Wadade
中科院分区：
文献类型：
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作者：
H. Kozono;H. Wadade
在临界微分阶数=n/r的齐次Sobolev空间中，我们考虑了广义Gagliardo-Nirenberg不等式，它描述了对所有q≦q<；∞，其中1；l；p；p；∞和1&l；r<；∞的嵌入.我们建立了该嵌入常数的最优生长速率ASQ→∞。特别地，我们以这样一种方式将极限端点r=∞理解为BMO空间，其中常数Cn仅依赖于n。作为应用，我们明确指出著名的John-Nirenberg不等式是我们估计的结果。进一步，利用∞范数和范数与n/r的对数建立了BMO-界，这可视为Brezis-Galouet-Wainger不等式的推广。
We consider the generalized Gagliardo–Nirenberg inequality inin the homogeneous Sobolev spacewith the critical differential orders=n/r, which describes the embedding such asfor allqwithp≦q< ∞, where 1 <p< ∞ and 1 <r< ∞. We establish the optimal growth rate asq→ ∞ of this embedding constant. In particular, we realize the limiting end-pointr= ∞ as the space ofBMOin such a way thatwith the constantCndepending only onn. 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 theL∞-bound is established by means of theBMO-norm and the logarithm of the-norm withs>n/r, which may be regarded as a generalization of the Brezis–Gallouet–Wainger inequality.