On non-surjective coarse isometries between Banach spaces. (English) Zbl 1423.46014
Summary: Assume that \(X\), \(Y\) are real Banach spaces, \(Y\) has uniform convexity of type \(p\) (\(\geq 1)\), and \(f:X\rightarrow Y\) is a standard coarse isometry. In this paper, we show that, if
\[
\int^\infty_1\frac{\varepsilon_f(s)^{\frac{1}{p}}}{s^{1+\frac{1}{p}}}\,ds<\infty,
\]
then there is a linear isometry \(U:X\rightarrow Y\) so that \[
\| f(x)-Ux\| =o(\| x\|)\text{ as }\| x\|\to\infty,
\]
where \(\varepsilon_f:\mathbb{R}^+ \to \mathbb{R}^+\) is defined by
\[
\varepsilon_f(t)=\sup \{\left| \| f(x)-f(y)\| -\| x-y\| \right| : x,y\in X,\,\| x-y\| \geq t\}.
\]
Representation properties of coarse isometries in free ultrafilter limits on \(\mathbb{N}\) are also discussed.
MSC:
| 46B04 | Isometric theory of Banach spaces |
| 46B20 | Geometry and structure of normed linear spaces |
| 46B85 | Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science |
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