No Access

Quantitative nonlinear embeddings into Lebesgue sequence spaces

Florent P. Baudier

Mathematics Department, Texas A&M University, College Station, TX 77843, USA

https://doi.org/10.1142/S1793525316500011Cited by:2

Abstract

In this paper fundamental nonlinear geometries of Lebesgue sequence spaces are studied in their quantitative aspects. Applications of this work are a positive solution to the strong embeddability problem from $ℓ_{q}$ into $ℓ_{p}$ ( $0 < p < q \leq 1$ ) and new insights on the coarse embeddability problem from $L_{q}$ into $ℓ_{q}$ , $q > 2$ . Relevant to geometric group theory purposes, the exact $ℓ_{p}$ -compression of $ℓ_{2}$ is computed. Finally coarse deformation of metric spaces with property A and locally compact amenable groups is investigated.

Keywords:

AMSC: 46B20, 46B85, 46T99, 20F65

References

1. F. Albiac, Nonlinear structure of some classical quasi-Banach spaces and $F$ -spaces, J. Math. Anal. Appl. 340 (2008) 1312–1325. Crossref, Google Scholar
2. F. Albiac and F. Baudier, Embeddability of snowflaked metrics with applications to the nonlinear geometry of the spaces $L_{p}$ and $ℓ_{p}$ for $0 < p < \infty$ , J. Geom. Anal. 25 (2015) 1–24. Crossref, Google Scholar
3. G. Arzhantseva, C. Druţu and M. Sapir, Compression functions of uniform embeddings of groups into Hilbert and Banach spaces, J. Reine Angew. Math. 633 (2009) 213–235. Google Scholar
4. P. Assouad, Plongements lipschitziens dans $R^{n}$ , Bull. Soc. Math. France 111 (1983) 429–448. (French with English Summary). Crossref, Google Scholar
5. T. Austin, Amenable groups with very poor compression into Lebesgue spaces, Duke Math. J. 159 (2011) 187–222. Crossref, Google Scholar
6. F. Baudier, Embeddings of proper metric spaces into Banach spaces, Hous. J. Math. 38 (2012) 209–223. Google Scholar
7. Y. Benyamini, The uniform classification of Banach spaces, Texas functional analysis seminar 1984–1985 (Austin, Tex.), 1985, pp. 15–38. Google Scholar
8. Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Vol. 1, American Mathematical Society Colloquium Publications, Vol. 48 (Amer. Math. Soc. 2000). Google Scholar
9. J. Bourgain, Remarks on the extension of Lipschitz maps defined on discrete sets and uniform homeomorphisms, Geom. Aspects of Funct. Anal. 1987 (1985/86) 157–167. Google Scholar
10. S. Campbell and G. Niblo, Hilbert space compression and exactness of discrete groups, J. Funct. Anal. 222 (2005) 292–305. Crossref, Google Scholar
11. M. Dadarlat and E. Guentner, Constructions preserving Hilbert space uniform embeddability of discrete groups, Trans. Amer. Math. Soc. 355 (2003) 3253–3275 (electronic). Crossref, Google Scholar
12. P. Enflo, On the nonexistence of uniform homeomorphisms between $L_{p}$ -spaces, Ark. Mat. 8 (1969) 103–105. Crossref, Google Scholar
13. A. Erschler, On isoperimetric profiles of finitely generated groups, Geom. Dedicata 100 (2003) 157–171. Crossref, Google Scholar
14. M. Gromov, Entropy and isoperimetry for linear and non-linear group actions, Groups Geom. Dyn. 2 (2008) 499–593. Crossref, Google Scholar
15. E. Guentner and J. Kaminker, Exactness and uniform embeddability of discrete groups, J. London Math. Soc. (2) 70 (2004) 703–718. Crossref, Google Scholar
16. U. Haagerup and A. Przybyszewska, Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces, arXiv:math/0606794v1. Google Scholar
17. W. B. Johnson and N. L. Randrianarivony, $l_{p} (p > 2)$ does not coarsely embed into a Hilbert space, Proc. Amer. Math. Soc. 134 (2006) 1045–1050 (electronic). Crossref, Google Scholar
18. N. J. Kalton, Coarse and uniform embeddings into reflexive spaces, Quart. J. Math. (Oxford) 58 (2007) 393–414. Crossref, Google Scholar
19. N. J. Kalton, The nonlinear geometry of Banach spaces, Rev. Mat. Complut. 21 (2008) 7–60. Crossref, Google Scholar
20. N. J. Kalton and N. L. Randrianarivony, The coarse Lipschitz structure of $ℓ_{p} \oplus ℓ_{q}$ , Math. Ann. 341 (2008) 223–237. Crossref, Google Scholar
21. S. Khot and N. Vishnoi, The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into $ℓ_{1}$ , in Proc. of the 46th Annual IEEE Conference on Foundations of Computer Science, IEEE (2005), pp. 53–62. Google Scholar
22. M. Kraus, Coarse and uniform embeddings between Orlicz sequence spaces, Mediterr. J. Math. 11 (2014) 653–666. Crossref, Google Scholar
23. G. Lancien, A short course on nonlinear geometry of Banach spaces, in Topics in Functional and Harmonic Analysis, Theta Ser. Adv. Math., Vol. 14 (Theta, 2013), pp. 77–101. Google Scholar
24. J. R. Lee and Moharrami, Bilipschitz snowflakes and metrics of negative type, STOC10, 2010. Google Scholar
25. J. R. Lee and A. Naor, $L_{p}$ metrics on the Heisenberg group and the Goemans–Linial conjecture, FOCS06 (2006). Google Scholar
26. M. Mendel and A. Naor, Euclidean quotients of finite metric spaces, Adv. Math. 189 (2004) 451–494. Crossref, Google Scholar
27. M. Mendel and A. Naor, Metric cotype, Ann. Math. (2) 168 (2008) 247–298. Crossref, Google Scholar
28. A. Naor and Y. Peres, Embeddings of discrete groups and the speed of random walks, Int. Math. Res. Not. IMRN (2008) Art. ID rnn 076, 34. Google Scholar
29. P. Nowak, On coarse embeddability into $l_{p}$ -spaces and a conjecture of Dranishnikov, Fund. Math. 189 (2006) 111–116. Crossref, Google Scholar
30. P. Nowak, On exactness and isoperimetric profiles of discrete groups, J. Funct. Anal. 243 (2007) 323–344. Crossref, Google Scholar
31. P. Nowak and G. Yu, Large Scale Geometry, EMS Textbooks in Mathematics (European Math. Soc., 2012). Crossref, Google Scholar
32. A. Y. Olshanskii and D. V. Osin, A quasi-isometric embedding theorem for groups, Duke Math. J. 162 (2013) 1621–1648. Crossref, Google Scholar
33. M. I. Ostrovskii, Coarse embeddability into Banach spaces, Topology Proc. 33 (2009) 163–183. Google Scholar
34. M. I. Ostrovskii, Embeddability of locally finite metric spaces into Banach spaces is finitely determined, Proc. Amer. Math. Soc. 140 (2012) 2721–2730. Crossref, Google Scholar
35. J.-P. Pier, Amenable Locally Compact Groups, Pure and Applied Mathematics (New York) (John Wiley & Sons, 1984). Google Scholar
36. J.-L. Tu, Remarks on Yu’s “property A” for discrete metric spaces and groups, Bull. Soc. Math. France 129 (2001) 115–139. Crossref, Google Scholar
37. A. Vershik, Amenability and approximation of infinite groups, Selecta Math. Sov. 2 (1982) 311–330. Google Scholar
38. G. Yu, The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000) 201–240. Crossref, Google Scholar

Vol. 08, No. 01

Metrics

History

Received 17 May 2013

Revised 31 March 2015

Accepted 31 March 2015

Published: 14 May 2015

Keywords

PDF download

System Upgrade on Tue, Oct 25th, 2022 at 2am (EDT)

Quantitative nonlinear embeddings into Lebesgue sequence spaces

Abstract

References

Recommended