Random embedding of lpn{\ell_p^n} into lr
详细信息 查看全文
- 作者：1. Institut de Mathématiques de Jussieu ; Université Pierre et Marie Curie ; Paris 6 ; 175 ; rue du Chevaleret ; 75013 Paris ; France2. Université Paris-Est ; équipe d’Analyse et Mathématiques Appliquées ; 5 ; boulevard Descartes ; Champs sur Marne ; 77454 Marne-la-Vallée Cedex 2 ; France
- 关键词：Primary 60E07 – 46B20 – 46B09 – Secondary 52A21
- 刊名：Mathematische Annalen
- 出版年：2011
- 出版时间：August 2011
- 年：2011
- 卷：350
- 期：4
- 页码：953-972
- 全文大小：265.3 KB
- 参考文献：1. Bernués J., López-Valdes M.: Tail estimates and a random embedding of into . Publ. Math. Debrecen 70(1–2), 9–18 (2007)
2. Bourgain, J., Kalton, N.J., Tzafriri, L.: Geometry of finite-dimensional subspaces and quotients of L p . Geometric aspects of functional analysis (1987–88), pp. 138–175. Lecture Notes in Mathematics, 1376, Springer, Berlin (1989)
3. Bourgain J., Lindenstrauss J., Milman V.: Approximation of zonoids by zonotopes. Acta. Math. 162(1–2), 73–141 (1989)
4. Bretagnolle J., Dacunha-Castelle D.: Application de l’étude de certaines formes linéaires aléatoires au plongement d’espaces de Banach dans des espaces L p . Ann. Sci. Ecole Norm. Sup. 4(2), 437–480 (1969)
5. Bretagnolle J., Dacunha-Castelle D., Krivine J.L.: Lois stables et espaces L p . Ann. Inst. H. Poincaré 2, 231–259 (1966)
6. Dvoretzky, A.: Some results on convex bodies and Banach spaces. In: 1961 Proceedings International Symposium of Linear Spaces (Jerusalem, 1960), pp. 123–160. Jerusalem Academic Press, Jerusalem; Pergamon, Oxford (1960)
7. Elton J.: Sign-embeddings of . Trans. Am. Math. Soc. 279(1), 113–124 (1983)
8. Esseen C.G.: On the concentration function of a sum of independent random variables. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9, 290–308 (1968)
9. Friedland, O., Guédon, O.: A multidimensional Esseen type inequality (work in progress)
10. Friedland O., Sodin S.: Bounds on the concentration function in terms of the Diophantine approximation. C. R. Math. Acad. Sci. Paris 345(9), 513–518 (2007)
11. Halász G.: Estimates for the concentration function of combinatorial number theory and probability. Period. Math. Hungar. 8(3–4), 197–211 (1977)
12. Johnson W.B., Schechtman G.: Embedding into . Acta. Math. 149(1–2), 71–85 (1982)
13. Johnson W.B., Schechtman G.: Very tight embeddings of subspaces of L p , 1 ≤ p < 2, into . Geom. Funct. Anal. 13(4), 845–851 (2003)
14. Johnson, W.B., Schechtman, G.: Finite dimensional subspaces of L p . Handbook of the Geometry of Banach Spaces, vol. I, pp. 837–870. Amsterdam, North-Holland (2001)
15. Kashin, B.S.: The widths of certain finite-dimensional sets and classes of smooth functions (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 41(2):334–351, 478 (1977)
16. Ledoux, M., Talagrand, M.: Probability in Banach spaces, pp. xii+480. Springer, Berlin (1991)
17. LePage R., Woodroofe M., Zinn J.: Convergence to a stable distribution via order statistics. Ann. Probab. 9(4), 624–632 (1981)
18. Litvak A.E., Pajor A., Rudelson M., Tomczak-Jaegermann N.: Smallest singular value of random matrices and geometry of random polytopes. Adv. Math. 195, 491–523 (2005)
19. Marcus M.B., Pisier G.: Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes. Acta. Math. 152(3–4), 245–301 (1984)
20. Maurey, B.: Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces L p (in French with an English summary), pp. ii+163. Astérisque No. 11. Société Mathématique de France, Paris (1974)
21. Maurey, B.: Type, cotype and K-convexity. Handbook of the geometry of Banach spaces, vol. 2, pp. 1299–1332. North-Holland, Amsterdam (2003)
22. Maurey B., Pisier G.: Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach (French). Stud. Math. 58(1), 45–90 (1976)
23. Milman V.D.: A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies (Russian). Funkcional. Anal. i Prilozen. 5(4), 28–37 (1971)
24. Naor A., Zvavitch A.: Isomorphic embedding of , 1 < p < 2, into . Isr. J. Math. 122, 371–380 (2001)
25. Pisier G.: On the dimension of the -subspaces of Banach spaces, for 1 ≤ p < 2. Trans. Am. Math. Soc. 276(1), 201–211 (1983)
26. Rosenthal H.P.: On subspaces of L p . Ann. Math. 97, 344–373 (1973)
27. Rudelson M.: Invertibility of random matrices: norm of the inverse. Ann. Math. 168, 575–600 (2008)
28. Rudelson, M., Vershynin, R.: The smallest singular value of a random rectangular matrix. Commun. Pure Appl. Math. (2010) (to appear)
29. Schechtman G.: Special orthogonal splittings of . Isr. J. Math. 189, 337–347 (2004)
30. Szarek S.: On Kashin’s almost Euclidean orthogonal decomposition of . Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26(8), 691–694 (1978)
31. Szarek S., Tomczak-Jaegermann N.: On nearly Euclidean decomposition for some classes of Banach spaces. Compos. Math. 40(3), 367–385 (1980)
32. Talagrand M.: Embedding subspaces of L 1 into . Proc. Am. Math. Soc. 108(2), 363–369 (1990)
33. Talagrand, M.: Embedding subspaces of L p in . Geometric aspects of functional analysis (Israel, 1992–1994). Oper. Theory Adv. Appl., vol. 77, pp. 311–325. Birkh?user, Basel (1995)
34. Tao, T., Vu, V.: Additive combinatorics. Cambridge Studies in Advanced Mathematics, vol. 105, pp. xviii+512. Cambridge University Press, Cambridge (2006) - 作者单位：http://www.springerlink.com/content/h24509p5484889q7/
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Mathematics - 出版者：Springer Berlin / Heidelberg
- ISSN：1432-1807
文摘
For any 0 < p < 2 and any natural numbers N > n, we give an explicit definition of a random operator S : lpn ? \mathbbRN{S : \ell_p^n \to \mathbb{R}^N} such that for every 0 < r < p < 2 with r ≤ 1, the operator Sr = S : lpn ? lrN{S_r = S : \ell_p^n \to \ell_r^N} satisfies with overwhelming probability that ||Sr|| ||(Sr)| Im S-1|| ￡ C(p,r)n/(N-n){\|S_r\| \, \|(S_r)_{| {\rm Im}\, S}^{-1}\| \le C(p,r)^{n/(N-n)}}, where C(p, r) > 0 is a real number depending only on p and r. One of the main tools that we develop is a new type of multidimensional Esseen inequality for studying small ball probabilities.