On comparison of the coarse embeddability into a Hilbert space and into other Banach spaces

M. Gromov [8] suggested to use uniform embeddings into a Hilbert space or into a uniformly convex space as a tool for solving some of the well-known problems. G. Yu [20] and G. Kasparov and G. Yu [10] have shown that this is indeed a very powerful tool. G. Yu in [20] used the condition of embeddability into a Hilbert space; G. Kasparov and G. Yu [10] used the condition of embeddability into a general uniformly convex space. These results made it interesting to compare coarse embeddability into a Hilbert space with the coarse embeddability into other Banach spaces. Results of this type were obtained in the papers [9], [14], [15], [17], and [19], where coarse embeddability of Banach spaces into each other was studied. In [9] it was shown that `p (p > 2) in not coarsely embeddable into `2, in [19] this result was strengthened to a characterization of quasi-Banach spaces which are coarsely embeddable into a Hilbert space. In [15] it was shown that cotype is an obstruction for coarse embeddability of Banach spaces. In [14, Remark 5.10] it was proved that Lq embeds coarsely into Lp for (1 ≤ q < p ≤ ∞). In [17] it was shown that `2 embeds coarsely into `p for each 1 ≤ p ≤ ∞. Because `2 is, in many respects, the ‘best’ space, and because of the Dvoretzky’s theorem (see [6] and [16]) it is natural to expect that `2 is among the most difficult spaces to embed into. We prove this for coarse embeddings of locally finite metric spaces (Theorem 1). (It is worth mentioning that the applications intended in [8] deal with even more narrow class of spaces with bounded geometry.) It is natural to try to strengthen ∗This research was initiated when the author participated in the supported by the NSF Workshop on Analysis and Probability, Texas A & M University, July 2006.