论文标题
Lipschitz空间上功能的积分表示和支持
Integral representation and supports of functionals on Lipschitz spaces
论文作者
论文摘要
我们分析了lipschitz $ \ mathrm {lip} _0（m）$在完整的度量空间$ m $上函数的空间$ \ mathrm {lip} _0（m）$上的连续线性功能之间的关系。特别是，我们描述了由措施引起的连续功能，反之亦然。对于较弱的$^\ ast $连续函数，即lipschitz-freake $ \ mathcal {f}（m）$的成员，考虑了$ m $的措施。对于一般情况，我们表明适当的设置是$ m $的统一（或塞缪尔）压实，并且与$ \ Mathcal {f}（m）$的处理是一致的。此设置还允许我们对$ \ mathrm {Lip} _0（m）^\ ast $的所有元素的支持定义，其属性与$ \ Mathcal {f}（M）$相似的属性，我们表明，当这种度量存在时，它与代表度量的支持相吻合。我们推断出$ \ mathrm {lip} _0（m）^\ ast $的成员可以表示为两个正函数的差异，将类似约旦的分解为正面和负面部分。
We analyze the relationship between Borel measures and continuous linear functionals on the space $\mathrm{Lip}_0(M)$ of Lipschitz functions on a complete metric space $M$. In particular, we describe continuous functionals arising from measures and vice versa. In the case of weak$^\ast$ continuous functionals, i.e. members of the Lipschitz-free space $\mathcal{F}(M)$, measures on $M$ are considered. For the general case, we show that the appropriate setting is rather the uniform (or Samuel) compactification of $M$ and that it is consistent with the treatment of $\mathcal{F}(M)$. This setting also allows us to give a definition of support for all elements of $\mathrm{Lip}_0(M)^\ast$ with similar properties to those in $\mathcal{F}(M)$, and we show that it coincides with the support of the representing measure when such a measure exists. We deduce that the members of $\mathrm{Lip}_0(M)^\ast$ that can be expressed as the difference of two positive functionals admit a Jordan-like decomposition into a positive and a negative part.
