论文标题
Wasserstein Sobolev空间的一般类别：圆柱功能的密度，反射性，均匀的凸性和克拉克森的不等式
The general class of Wasserstein Sobolev spaces: density of cylinder functions, reflexivity, uniform convexity and Clarkson's inequalities
论文作者
论文摘要
我们表明，圆柱体的代数在Wasserstein Sobolev空间中起作用$ H^{1，Q}（\ Mathcal {p} _p（x，x，x，\ mathsf {d}），w_ {p，\ {p，\ mathsf {d}}}，\ mathfrak {d}}，\ mathfrak {m}） $（P，\ Mathsf {d}）$ - WASSERSTEIN SPACE $（\ MATHCAL {P} _p（X，X，\ Mathsf {D}），W_ {P，Mathsf {D}）$上的$（Mathsf {d}）$ - WASSERSTEIN SPACE $（\ MATHSF {d}），$（Mathsf {d}）$ - $（Mathsf {d}）$ - $（Mathsf {d}）$ - W_ {作为一个应用，我们证明，如果基本的度量空间是可分离的Banach空间$ \ Mathbb {b} $，则Wasserstein Sobolev空间是反射的（sesp。〜均匀地凸出），如果$ \ mathbb {b} $ reflexive是reflexive（sesp。最后，我们还提供了足够的条件，可以使克拉克森（Clarkson）类型不平等的有效性在Wasserstein Sobolev空间中。
We show that the algebra of cylinder functions in the Wasserstein Sobolev space $H^{1,q}(\mathcal{P}_p(X,\mathsf{d}), W_{p, \mathsf{d}}, \mathfrak{m})$ generated by a finite and positive Borel measure $\mathfrak{m}$ on the $(p,\mathsf{d})$-Wasserstein space $(\mathcal{P}_p(X,\mathsf{d}), W_{p, \mathsf{d}})$ on a complete and separable metric space $(X,\mathsf{d})$ is dense in energy. As an application, we prove that, in case the underlying metric space is a separable Banach space $\mathbb{B}$, then the Wasserstein Sobolev space is reflexive (resp.~uniformly convex) if $\mathbb{B}$ is reflexive (resp.~if the dual of $\mathbb{B}$ is uniformly convex). Finally, we also provide sufficient conditions for the validity of Clarkson's type inequalities in the Wasserstein Sobolev space.
