论文标题
在同质的Sobolev和整个空间上的空间
On homogeneous Sobolev and Besov spaces on the whole and the half space
论文作者
论文摘要
在本文中，我们建议在$ \ mathbb {r}^n $和$ \ mathbb {r}^n _+$上提出基础构造分数订单的均匀sobolev空间。该结构完成了$ \ Mathcal {s}'_ H（\ Mathbb {r}^n）$在$ \ Mathbb {r}^n）上的均匀besov空间的构建。我们还将扩展Danchin和Mucha对$ \ Mathbb {r}^n _+$所做的处理，以及由Danchin，Hieber，Mucha和Tolksdorf在$ \ Mathbb {r}^n $ and $ \ Mathbb上构建整数订单的同质sobolev空间。讨论了真实和复杂的插值，二元性和密度的特性。还审查了跟踪结果。我们的方法主要依赖于插值理论，并在Besov空间中提供了一些已经知道的结果的简单证明。整个尺度上缺乏完整性将导致考虑交叉空间的估计值，以避免此问题。作为标准和简单的应用，我们将这些均匀功能空间中的Dirichlet和Neumann Laplacians的问题处理。
In this paper, we propose an elementary construction of homogeneous Sobolev spaces of fractional order on $\mathbb{R}^n$ and $\mathbb{R}^n_+$. This construction completes the construction of homogeneous Besov spaces on $\mathcal{S}'_h(\mathbb{R}^n)$ started by Bahouri, Chemin and Danchin on $\mathbb{R}^n$. We will also extend the treatment done by Danchin and Mucha on $\mathbb{R}^n_+$, and the construction of homogeneous Sobolev spaces of integer orders started by Danchin, Hieber, Mucha and Tolksdorf on $\mathbb{R}^n$ and $\mathbb{R}^n_+$. Properties of real and complex interpolation, duality, and density are discussed. Trace results are also reviewed. Our approach relies mostly on interpolation theory and yields simpler proofs of some already known results in the case of Besov spaces. The lack of completeness on the whole scale will lead to consideration of intersection spaces with decoupled estimates to circumvent this issue. As standard and simple applications, we treat the problems of Dirichlet and Neumann Laplacians in these homogeneous functions spaces.
