Scaled Enflo type is equivalent to Rademacher type

Transcription

1 Scaled Enflo tye is equivalent to Radeacher tye Manor Mendel California Institute of Technology Assaf Naor Microsoft Research Abstract We introduce the notion of scaled Enflo tye of a etric sace, and show that for Banach saces, scaled Enflo tye is equivalent to Radeacher tye. Introduction Recall that a Banach sace is said to have Radeacher tye > 0 see [7]) if there exists a constant T < such that for every x,..., x n, ε x T x, ) where here, and in what follows, denotes the exectation with resect to uniforly chosen ε = ε,..., ε n ) {, } n. The infiu over all constants T for which ) holds is denoted T ). Motivated by the search for concrete versions of Ribe s theore [2] for various fundaental local roerties of Banach saces see the discussion in [2, 9, 8]), several researchers roosed non-linear notions of tye, which ake sense in the setting arbitrary etric saces see [5,, ]). In articular, following Enflo [5] we say that a etric sace M, d M ) has Enflo tye if there exists a constant K such that for every n N and every f : {, } n M, d M f ε), f ε)) T d M f ε,..., ε, ε, ε +,..., ε n ), f ε,..., ε, ε, ε +,..., ε n ) ). 2) For Banach saces ) follows fro 2) by considering the function ε n ε x. The question whether in the category of Banach saces Radeacher tye ilies Enflo tye was osed by Enflo in [5], and in full generality reains oen. In [] Pisier showed that if a Banach sace has Radeacher then it has Enflo tye for every < see also the work of Bourgain, Milan and Wolfson [] for a siilar result which holds for a another notion of non-linear tye). In [0] it was shown that for UMD Banach saces see [4]) Radeacher tye is equivalent to Enflo tye. Motivated by our recent work on etric cotye [8], we introduce below the notion of scaled Enflo tye of a etric sace which is, in a sense, oosite to the notion of etric cotye defined in [8]), and show that for Banach saces, scaled Enflo tye is equivalent to Radeacher tye. This settles the long standing roble of finding a urely etric forulation of the notion of tye though Enflo s roble described above reains oen). Modulo soe of the results of [8], the roof of our ain theore is very sile. Definition. Scaled Enflo tye). Let M, d M ) be a etric sace and > 0. We say that M has scaled Enflo tye with constant τ if for every integer n there exists an even integer such that for every f : M, d M f x + ) ) 2 ε, f x) dµx) τ d M f x + e ), f x) ) dµx), ) where µ is the unifor robability easure on, and {e } n is the standard basis of Rn. The infiu over all constants τ for which ) holds is denoted τ M).

2 Theore.2. Let be a Banach sace and [, 2]. Then has Radeacher tye if and only if has scaled Enflo tye. More recisely, 2π T ) τ ) 5T ). 2 Proof of Theore.2 We start by showing that scaled Enflo tye ilies Radeacher tye. Lea 2.. Let be a Banach sace and [, 2]. Then T ) 2πτ ). Proof. Let be a Banach sace and assue that τ ) < for soe [, 2]. Fix τ > τ ), v,..., v n, and let be an even integer. Define f : by f x,..., x n ) = n e 2πix v. Then f ) x + e f x) dµx) = e 2πi ) v 2π v, 4) and f x + ) 2 ε f x) dµx) = 2 e 2πix v dµx). 5) We recall the contraction rincile see [6]), which states that for every a,..., a n R, Thus, e 2πix v dµx) = ε a v e 2πi x + ε ) 4 )v ) ax a n dµx) = ε v. ε e 2πix v dµx) 2 E ε ε v. 6) Cobining 4), 5) and 6) yields the required result. Let be a Banach sace with tye, an integer divisible by 4, and k an odd integer. Fix f : and ε {, } n. Define A k) f : by Lea 2.2. For and every f : A k) f x) = f x + z). z k,k) n 2Z) n A k) f x) f x) dµx) k ) n f x + e ) f x) dµx). Proof. For every t R let st) be the sign of t with convention that s0) = 0). For every z, f x + z) f x) z z f x + l= t= ) z t e t + l sz ) e f ) x + z t e t + l ) sz ) e. t= 2

3 Observe that since k is odd, k, k) n 2Z) n =. Thus A k) f x) f x) dµx) f x + z) f x) dµx) z k,k) n 2Z) n z k,k) n 2Z) n k ) n z z z z k,k) n 2Z) n z f x + l= t= ) z t e t + lsz )e f f y + sz )e ) f y) dµx) f x + e ) f x) dµx). ) x + z t e t + l )sz )e dµx) t= Proof of theore.2. Fix an odd integer k N, with k < 2. As in [8], given {,..., n} we define S, k) Zn by For f : we define S, k) { y [ k, k] n : y 0 od 2 and l, y l od 2 }. E k) f x) = f ) S,k) x) = µs, k)) In [8] see equation 9) there) it is shown that for every x and ε {, } n, ) n k A k) f x + ε) A k) f x ε) ) = k + f x + y)dµy). 7) µs, k)) S,k) ε [ E k) where, by inequalities 4) and 42) in [8], for every ε {, } n, Thus, for every T > T ), { } ax Ux, ε) dµx), Vx, ε) dµx) 8 n 2 Z n k ) n ) k k + T T 6 A k) f x + ε) A k) f x ε) dµx) ε [ E k) E k) T + 24 n 2 k f x + e ) E k) f x e ) ] f x + e ) E k) f x e ) ] + Ux, ε) + Vx, ε), dµx) + 24 n 2 k f x + e ) E k) f x e ) dµx) + 24 n 2 f x + e ) f x e ) dµx) + 24 n 2 ) k k f x + e ) f x). f x + e ) f x) dµx) f x + e ) f x) dµx) f x + e ) f x) dµx) 8) f x + e ) f x) dµx), 9) where in 8) we used the fact that E k) is an averaging oerator, and hence has nor.

4 On the other hand f x + ) Z n 2 ε f x) dµx) Ak) f x + ) Z n 2 ε A k) f x) dµx) + f x + ) 2 ε A k) f x + ) 2 ε dµx) + E ε A k) f x) f x) dµx) 4 4 ) Eε ) Eε ) + k 4 5 T t= /4 A k) f x + 2tε) A k) f x + 2t 2)ε) dµx) + 2 E ε A k) f x + ε) A k) f x ε) dµx) + 2 ) n ) 6 T + 24 n 2 k ) + 2kn) n A k) f x) f x) dµx) f x + e ) f x) dµx) 0) f x + e ) f x) dµx) ) f x + e ) f x) dµx), 2) where in 0) we used Lea 2.2, in ) we used 8), and 2) is true if 4n 2 / k, which is a valid choice 2n / of k if n 2/. Reark 2.. If a etric sace has Enflo tye then it also has scaled Enflo tye. This follows fro a straightforward odification of Lea 2.4 in [8]. We do not know if scaled Enflo tye ilies Enflo tye. In the category of Banach saces, a ositive answer to this question would show that Enflo tye is equivalent to Radeacher tye, resolving ositively Enflo s roble [5]. We do know that for Banach saces, scaled Enflo tye ilies Enflo tye for all <, and that scaled Enflo tye and Enflo tye coincide for UMD Banach saces. Reark 2.4. The idea of scaling by 2 in the definition of scaled Enflo tye originates fro the definition of etric cotye introduced in [8], which involves a siilar scaling rocedure. In the case of non-linear tye it is ossible that this scaling is not necessary, i.e. that Enflo tye is equivalent to Radeacher tye. However, as shown in [8], in the context of etric cotye the scaling is necessary- we refer to [8] for ore details. References [] K. Ball. Markov chains, Riesz transfors and Lischitz as. Geo. Funct. Anal., 22):7 72, 992. [2] J. Bourgain. The etrical interretation of suerreflexivity in Banach saces. Israel J. Math., 562):222 20, 986. [] J. Bourgain, V. Milan, and H. Wolfson. On tye of etric saces. Trans. Aer. Math. Soc., 294):295 7, 986. [4] D. L. Burkholder. Martingales and Singular integrals in Banach saces. In Johnson, W. B. and Lindenstrauss, J. ed.), Handbook of the geoetry of Banach saces. Volue. Asterda: North-Holland [5] P. Enflo. On infinite-diensional toological grous. In Séinaire sur la Géoétrie des Esaces de Banach ), ages Ex. No. 0,. École Polytech., Palaiseau, 978. [6] M. Ledoux and M. Talagrand. Probability in Banach saces, volue 2 of Ergebnisse der Matheatik und ihrer Grenzgebiete ) [Results in Matheatics and Related Areas )]. Sringer-Verlag, Berlin, 99. Isoerietry and rocesses. [7] B. Maurey. Tye, cotye and K-convexity. In Johnson, W. B. and Lindenstrauss, J. ed.), Handbook of the geoetry of Banach saces. Volue 2. Asterda: North-Holland

5 [8] M. Mendel and A. Naor. Metric cotye. Prerint, [9] A. Naor, Y. Peres, O. Schra, and S. Sheffield. Markov chains in sooth Banach saces and Groov hyerbolic etric saces. Prerint, [0] A. Naor and G. Schechtan. Rearks on non linear tye and Pisier s inequality. J. Reine Angew. Math., 552:2 26, [] G. Pisier. Probabilistic ethods in the geoetry of Banach saces. In Probability and analysis Varenna, 985), volue 206 of Lecture Notes in Math., ages Sringer, Berlin, 986. [2] M. Ribe. On uniforly hoeoorhic nored saces. Ark. Mat., 4:27 244,