论文标题
Lipschitz Free $ p $ -ppaces $ \ mathcal {f} _p（\ mathbb {z}^d）$和$ \ mathcal {f} _p（\ mathbb {r}^d）$ for $ 0 <p \ le 1 $
Structure of the Lipschitz free $p$-spaces $\mathcal{F}_p(\mathbb{Z}^d)$ and $\mathcal{F}_p(\mathbb{R}^d)$ for $0<p\le 1$
论文作者
论文摘要
我们在本文中的目的是为Lipschitz的理论提供$ 0 <p \ le 1 $的免费$ p $ - 空格，欧几里得空间上的$ \ mathbb {r}^d $和$ \ mathbb {z}^d $。 To that end, on one hand we show that $\mathcal{F}_p(\mathbb{R}^d)$ admits a Schauder basis for every $p\in(0,1]$, thus generalizing the corresponding result for the case $p=1$ achieved in [P. Hájek and E. Pernecká, On Schauder bases in Lipschitz-free spaces, J. Math. Anal. Appl. 416 （2014年，第629-646页，在for中提出的问题$ \ Mathcal {f} _p（\ Mathbb {r}^d）$及其同构空间$ \ mathcal {f} _p（[0,1]^d）$的基础公式。 $ \ ell_ {1} $不会扩展到$ p <1 $的情况，也就是说，$ \ mathcal {f} _ {p}（\ mathbb {z}）$在$ 0 <p <p <p <1 $时都不是等词至$ \ ell_p $。
Our aim in this article is to contribute to the theory of Lipschitz free $p$-spaces for $0<p\le 1$ over the Euclidean spaces $\mathbb{R}^d$ and $\mathbb{Z}^d$. To that end, on one hand we show that $\mathcal{F}_p(\mathbb{R}^d)$ admits a Schauder basis for every $p\in(0,1]$, thus generalizing the corresponding result for the case $p=1$ achieved in [P. Hájek and E. Pernecká, On Schauder bases in Lipschitz-free spaces, J. Math. Anal. Appl. 416 (2014), no. 2, 629--646] and answering in the positive a question that was raised in [F. Albiac, J. L. Ansorena, M. Cúth, and M. Doucha, Embeddability of lp and bases in Lipschitz free $p$-spaces for $0 < p \le 1$, J. Funct. Anal. 278 (2020), no. 4, 108354, 33]. Explicit formulas for the bases of both $\mathcal{F}_p(\mathbb{R}^d)$ and its isomorphic space $\mathcal{F}_p([0,1]^d)$ are given. On the other hand we show that the well-known fact that $\mathcal{F}(\mathbb{Z})$ is isomorphic to $\ell_{1}$ does not extend to the case when $p<1$, that is, $\mathcal{F}_{p}(\mathbb{Z})$ is not isomorphic to $\ell_p$ when $0<p<1$.
