论文标题
在大的$ \ ell_1 $ -sums of Lipschitz的空间和应用
On large $\ell_1$-sums of Lipschitz-free spaces and applications
论文作者
论文摘要
我们证明，Banach空间上无Lipschitz的空间$ x $密度$κ$，由$ \ nathcal {f}（x）$表示，与其$ \ ell_1 $ -sum $ \ weft（\bigoplus_κκ-\ nathcal {f}（x）$ ell均与$ \ ell_1 $ -sum $ \ weft（\bigoplus_κ-这提供了Kaufmann先前结果的扩展，在不可分割的Banach空间的背景下。此外，我们获得了实价Lipschitz函数的空间的完整分类，这些函数在$ \ Mathcal {l} _p $ -space上消失了$ 0 $。更确切地说，我们确定，对于每$ 1 \ leq p \ leq \ infty $，如果$ x $是$ \ natcal {l} _p $ - 密度$ $ $ $ $ $ $ $ $ $ \ \ m athrm {lip} _0（x）$是isomorphic to isomorphic to isomorphic to isomorphic as ismorphic as ismorphic ask Mathrm Mathrm ip} $ p <\ infty $，或$ \ mathrm {lip} _0（c_0（κ））$如果$ p = \ infty $。
We prove that the Lipschitz-free space over a Banach space $X$ of density $κ$, denoted by $\mathcal{F}(X)$, is linearly isomorphic to its $\ell_1$-sum $\left(\bigoplus_κ\mathcal{F}(X)\right)_{\ell_1}$. This provides an extension of a previous result from Kaufmann in the context of non-separable Banach spaces. Further, we obtain a complete classification of the spaces of real-valued Lipschitz functions that vanish at $0$ over a $\mathcal{L}_p$-space. More precisely, we establish that, for every $1\leq p\leq \infty$, if $X$ is a $\mathcal{L}_p$-space of density $κ$, then $\mathrm{Lip}_0(X)$ is either isomorphic to $\mathrm{Lip}_0(\ell_p(κ))$ if $p<\infty$, or $\mathrm{Lip}_0(c_0(κ))$ if $p=\infty$.
