$BV$对偶中带号测度的刻画及相关的等距同构
Characterizations of signed measures in the dual of $BV$ and related isometric isomorphisms
摘要：
我们刻画了$BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$中的所有（带符号）测度，其中$BV_{\frac{n}{n-1}}(\mathbb{R}^n)$定义为$L^{\frac{n}{n-1}}(\mathbb{R}^n)$中所有函数$u$的空间，使得$Du$是有限向量值测度。我们还表明，$BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$和$BV(\mathbb{R}^n)^*$是等距同构，其中$BV(\mathbb{R}^n)$被定义为空间的所有功能$u$在$L^{1}(\mathbb{R}^n)$，使$Du$是一个有限的向量值的措施。由于我们的特点，一个老问题提出了Meyers Ziemer [MZ]解决了通过构建一个局部可积函数$f$，$f$属于$BV(\mathbb{R}^n)^{*}$，但$|f|$不。证明了$BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$中的测度与齐次Sobolev空间$\dot W^{1,1}(\mathbb{R}^n)$的对偶空间$\dot W^{1,1}(\mathbb{R}^n)^*$中的测度在等距同构意义下是一致的.对于具有Lipschitz边界的有界开集$Ω$，刻画了对偶空间$BV_0(Ω)^*$中的测度.本文的目的之一是精确定义$BV_0(Ω)$，即在$Ω$的边界上迹为零的有界变差函数空间。我们表明，在$BV_0(Ω)^*$的措施与$W^{1,1}_0(Ω)^*$的措施相一致。最后，还刻画了$BV(Ω)^*$中的有限测度类。
英文摘要：
We characterize all (signed) measures in $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$, where $BV_{\frac{n}{n-1}}(\mathbb{R}^n)$ is defined as the space of all functions $u$ in $L^{\frac{n}{n-1}}(\mathbb{R}^n)$ such that $Du$ is a finite vector-valued measure. We also show that $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$ and $BV(\mathbb{R}^n)^*$ are isometrically isomorphic, where $BV(\mathbb{R}^n)$ is defined as the space of all functions $u$ in $L^{1}(\mathbb{R}^n)$ such that $Du$ is a finite vector-valued measure. As a consequence of our characterizations, an old issue raised in Meyers-Ziemer [MZ] is resolved by constructing a locally integrable function $f$ such that $f$ belongs to $BV(\mathbb{R}^n)^{*}$ but $|f|$ does not. Moreover, we show that the measures in $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$ coincide with the measures in $\dot W^{1,1}(\mathbb{R}^n)^*$, the dual of the homogeneous Sobolev space $\dot W^{1,1}(\mathbb{R}^n)$, in the sense of isometric isomorphism. For a bounded open set $Ω$ with Lipschitz boundary, we characterize the measures in the dual space $BV_0(Ω)^*$. One of the goals of this paper is to make precise the definition of $BV_0(Ω)$, which is the space of functions of bounded variation with zero trace on the boundary of $Ω$. We show that the measures in $BV_0(Ω)^*$ coincide with the measures in $W^{1,1}_0(Ω)^*$. Finally, the class of finite measures in $BV(Ω)^*$ is also characterized.
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