Remarks on the Gagliardo-Nirenberg Type Inequality in the Besov and the Triebel-Lizorkin Spaces in the Limiting Case
Remarks on the Gagliardo-Nirenberg Type Inequality in the Besov and the Triebel-Lizorkin Spaces in the Limiting Case
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关于极限情况下 Besov 和 Triebel-Lizorkin 空间中的 Gagliardo-Nirenberg 型不等式的评论
DOI：
10.1007/s00041-009-9069-x
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发表时间：
2009
影响因子：
1.2
通讯作者：
H. Wadade
中科院分区：
文献类型：
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作者：
H. Wadade
AbstractWe consider the generalized Gagliardo-Nirenberg inequality in
$\Bbb{R}^{n}$
including homogeneous Besov space
$\dot{B}^{s}_{r,\rho}(\Bbb{R}^{n})$
with the critical order s=n/r, which describes the continuous embedding such as
$L^{p}(\Bbb{R}^{n})\cap\dot{B}^{n/r}_{r,\rho}(\Bbb{R}^{n})\subset L^{q}(\Bbb{R}^{n})$
for all q with p≦q<∞, where 1≦p≦r<∞ and 1<ρ≦∞. Indeed, the following inequality holds: $$\|u\|_{L^{q}(\Bbb{R}^{n})}\leqq C\,q^{1-1/\rho}\|u\|_{L^{p}(\Bbb{R}^{n})}^{p/q}\|u\|_{\dot{B}^{n/r}_{r,\rho}(\Bbb{R}^{n})}^{1-p/q},$$ where C is a constant depending only on r. In this inequality, we have the exact order 1−1/ρ of divergence to the power q tending to the infinity. Furthermore, as a corollary of this inequality, we obtain the Gagliardo-Nirenberg inequality with the homogeneous Triebel-Lizorkin space
$\dot{F}^{n/r}_{r,\rho}(\Bbb{R}^{n})$
, which implies the usual Sobolev imbedding with the critical Sobolev space
$\dot{H}^{n/r}_{r}(\Bbb{R}^{n})$
. Moreover, as another corollary, we shall prove the Trudinger-Moser type inequality in
$\dot{B}^{n/r}_{r,\rho}(\Bbb{R}^{n})$
.
AbstractWe consider the generalized Gagliardo-Nirenberg inequality in
$\Bbb{R}^{n}$
including homogeneous Besov space
$\dot{B}^{s}_{r,\rho}(\Bbb{R}^{n})$
with the critical order s=n/r, which describes the continuous embedding such as
$L^{p}(\Bbb{R}^{n})\cap\dot{B}^{n/r}_{r,\rho}(\Bbb{R}^{n})\subset L^{q}(\Bbb{R}^{n})$
for all q with p≦q<∞, where 1≦p≦r<∞ and 1<ρ≦∞. Indeed, the following inequality holds: $$\|u\|_{L^{q}(\Bbb{R}^{n})}\leqq C\,q^{1-1/\rho}\|u\|_{L^{p}(\Bbb{R}^{n})}^{p/q}\|u\|_{\dot{B}^{n/r}_{r,\rho}(\Bbb{R}^{n})}^{1-p/q},$$ where C is a constant depending only on r. In this inequality, we have the exact order 1−1/ρ of divergence to the power q tending to the infinity. Furthermore, as a corollary of this inequality, we obtain the Gagliardo-Nirenberg inequality with the homogeneous Triebel-Lizorkin space
$\dot{F}^{n/r}_{r,\rho}(\Bbb{R}^{n})$
, which implies the usual Sobolev imbedding with the critical Sobolev space
$\dot{H}^{n/r}_{r}(\Bbb{R}^{n})$
. Moreover, as another corollary, we shall prove the Trudinger-Moser type inequality in
$\dot{B}^{n/r}_{r,\rho}(\Bbb{R}^{n})$
.
Trudinger-Moser不等式最佳常数的上界及其在Gagliardo-Nierenberg不等式中的应用
DOI：
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发表时间：
2006
期刊：
Indiana Univ. Math. J. 56
影响因子：
--
作者：
H.Kozono;T.Sato;H.Wadade
通讯作者：
H.Wadade
影响因子：
0.8
作者：
H. Kozono;H. Wadade
通讯作者：
H. Kozono;H. Wadade
H.Kozono：“Besov 空间中的临界 Sobolev 不等式和一些半线性演化方程的正则性准则”Math.Z.. 239。（2002）
DOI：
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发表时间：
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影响因子：
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