ariv:0902.955v [math.fa] Feb 2009 Haar tye and Carleson Constants Stefan Geiss October 30, 208 Abstract Paul F.. Müller For a collection E of dyadic intervals, a Banach sace, and,2] we assume the uer l estimates x I h I / I / c, L where and h I denotes the L normalized Haar function suorted on I. We determine the minimal requirement on the size of E so that these estimates imly that is of Haar tye. The characterization is given in terms of the Carleson constant of E. 2000 Mathematics Subject Classification: 46B07, 46B20 Introduction Let be a Banach sace. We fix a non-emty collection of dyadic intervals E and assume the uer l estimates h I x I I / L c ) for finitely suorted ) and some,2], where h I is the L normalized Haar function suorted on I. The consequences for, one may Research of both authors suorted in art by FWF Pr. Nr. P50907-N0.
draw from ), deend on the size and structure of the collection E. For instance, if E is a collection of airwise disjoint dyadic intervals, then any Banach sace satisfies ), hence it does not imose any restriction on. If, on the other hand, we choose E to be the collection of all dyadic intervals, then the uer l estimates ) simly state that is of Haar tye ; due to imortant work of G. Pisier [6] the latter condition is equivalent to certain renorming roerties of the Banach sace. In this aer we ask how massive a collection E has to be so that ) imlies that is of Haar tye. We give the answer to this question in terms of the Carleson constant defined by [E] : su I J E, J I J. 2) The roof is based on the following well-known dichotomy: either E can be decomosed into finitely many collections consisting of almost disjoint dyadic intervals or E contains large and densely acked blocks of dyadic intervals, with arbitrary high degree of condensation. Initially we encountered the roblem treated here in connection with our efforts to obtain a vector valued version of E. M. Semenov s characterization of bounded oerators rearranging the Haar system. See [7] and [3]. 2 Preliminaries In the following we equi the unit interval [0,) with the Lebesgue measure denoted by. Let D denote the collection of dyadic intervals in [0,), i.e. I D rovided that there exist m 0 and k 2 m such that and let I [k )/2 m,k/2 m ), D n : {I D : I 2 n } where n 0. The L normalized Haar function suorted on I D is denoted by h I, i.e. h I in the right half of I and h I on the left half of I. By L, [, ), we denote the sace of Radon random variables f : [0,) such that ) / f L ft) <. 0 2
Haar tye. Given,2], a Banach sace is of Haar tye rovided that there exists a constant c > 0 such that h I x I c I D I / L for all finitely suorted families ) I D. We let HT ) be the infimum of all ossible c > 0 as above. The central result concerning Haar tye is due to G. Pisier [6] and asserts that Haar tye is equivalent to the fact that can be equivalently renormed such that the new norm has a modulus of smoothness of ower tye. For additional information see [, 5] and the references therein. Carleson Constants. Let E D be a non-emty collection of dyadic intervals. The Carleson constant of E is given by equation 2). Next we define consecutive generations of E and, using [E], describe a dichotomy for E known as the almost disjointification and condensation roerties. We define G 0 E) to be the maximal dyadic intervals of E where maximal refers to inclusion. Suose, we have already defined G 0 E),..., G E), then we form G + E) : G 0 E \G 0 E) G E))). I D Given I D, let I E {J E, J I} and ut G k I,E) : G k I E) for k. Assume that [E] < anhat M is thelargest integer smaller than4[e]+. Then E i : G Mk+i E), 0 i M, 3) k0 consists of almost disjoint dyadic intervals, in the sense that for I E i, J G I,E i ) J I 2 and K I E i 2 I. Conversely, if [E], then forall n and ε 0,)there exists a K 0 E that is densely acked in the sense that J ε) K 0. J G nk 0,E) 3
Based on this we show in Lemma 3.4 that for any n the san h I ) contains asystem offunctions, withthesamejoint distributionasthefirst 2 n elements of the Haar basis. The roof of this basic dichotomy and some of its alications can be found in [4, Chater 3]. 3 Haar Tye and Carleson Constants The main results of this note are Theorems 3. and 3.2. Combined they give an answer to the question as to which families of dyadic intervals E will detect whether a Banach sace has Haar tye. The answer is a dichotomy: either [E] <, then E does not detect any Haar tye of any Banach sace, or [E], then E determines the Haar tye constant exactly, for any Banach sace and each < 2. Theorem 3.. Let,2] and E D be a non-emty collection of dyadic intervals. Then the following statements are equivalent: ) A Banach sace is Haar tye if there exists c > 0 such that for all finitely suorted families ) I D one has h I I / L c ; the infimum over all such c > 0 coincides with HT ). 2) [E]. Theorem 3.2. Let,2], E D be a non-emty collection of dyadic intervals, and be a Banach sace which is not of Haar tye. Then the following statements are equivalent: ) There exists a constant c > 0 such that for all finitely suorted families ) I D one has h I I / L c. 4
2) [E] <. Theorem 3. and Theorem 3.2 follow immediately from the following two lemmas and the obvious fact that there are Banach saces without Haar tye if,2]). Lemma 3.3. Let, ), [E] <, and be a Banach sace. Then there is a constant c > 0, deending at most on, such that one has h I c [E] I / L for all finitely suorted ). Proof. Using 3), we write E E 0 E M with M < 4[E]+ such that the collections {A I : I E i } of airwise disjoint and measurable sets defined by A I : I \ J, I E i, satisfy Since h I I / it is sufficient to rove h I i But here we get that h I i I L I / L L J G I,E i ) 2 I A I I. M h I I / i0 i L M M h I i0 i c i 5 I / L for i M. /
K Ei K Ei K Ei K Ei K Ei K E i h I t) x I i I i i K i hi t) I hi t) I hi t) I nk) x G l K,E i ) G l K,E i ) l0 x G l K,E i )χ {l nk)} G l K,E i ) l0 hg lk,e i )t) hg lk,e i )t) where G l K,E i ) form the maximal sequence of dyadic intervals from E i such that K G 0 K,E i ) G K,E i ) G nk) K,E i ) and G nk) K,E i ) is the unique maximal interval in E i containing K. Next we obtain an uer estimate for the last exression as follows: l0 K E i l0 K E i x G l K,E i )χ {l nk)} G l K,E i ) xg l K,E i )χ {l nk)} G l K,E i ) hg lk,e i )t) 6
l0 I,K E i G l K,E i )I i l0 2 l l0 i ) l0 2 l I K E i G l K,E i )I i I. Next we turn to the case [E] for which it is known that the Gamlen- Gaudet construction yields an aroximation of the Haar system by aroriate blocks of h I ). The next lemma demonstrates that this construction fits erfectly with our Haar tye inequalities. We carefully avoid using the unconditionality of the Haar system and therefore the UMD roerty of Banach saces) and exhibit a system of functions with exactly the same joint distribution as the ususal Haar basis, rather than to allow that the measures of the suort of a Haar function and its aroximation are related by uniformly bounded multilicative constants. Lemma 3.4. Let be a Banach sace,,2], and E be a collection of dyadic intervals such that [E]. If there is a constant c > 0 such that h I I / L c for all finitely suorted families ), then is of Haar tye with HT ) c. Remark 3.5. In Lemma 3.4 the range 2, ) does not make sense since already R does not have Rademacher tye 2, ) and henceforth 7 4)
Haar tye 2, ). In other words, for [E] and 2, ) the inequality 4) fails to be true. Proof of Lemma 3.4. Let n, δ 0,), and ε 2 n δ. Since [E] the condensation roerty cf. [4, Lemma 3..4]) yields the existence of a K 0 E such that J ε) K 0. J G nk 0,E) Examining the Gamlen-Gaudet construction [2] as for examle) resented in [4, Proosition 3..6], we obtain a family B I ) I Dn of collections of dyadic intervals such that i) B I K 0 E, ii) the elements of B I are airwise disjoint, iii) for B I : K B I K one has that B I B J if and only if I J, and B I B J if and only if I J, iv) for k I : K B I h K and I,I,I + D n such that I is the right half of I and I + the left half of I, one has B I {k I } and B I + {k I }, v) for 0 k n and I 2 k one has As a consequence and K 0 2 k 2ε K 0 B I K 0 2 k. δ 2 n K 0 B I K 0 2 n for I 2 n δ) K 0 I 2 n B I K 0. Choose measurable subsets A I B I for I 2 n such that a) A I δ)2 n K 0, 8
b) the k I restricted to A I are symmetric. Let S : I 2 n A I, so that S δ) K 0, and A I : B I S for all remaining) I D n. When restricted to the robability sace S, ) the system k S I) I Dn has the same joint distribution as the usual Haar system h I ) I Dn on the unit interval. Hence, as a consequence of the Gamlen-Gaudet construction, we obtain that h I I / L k I I / S S L S, S ) k I I / K B I L S,) h K / / I / ). L Recall that we selected the collection B I as a sub-collection of E. Using our hyothesis concerning E and, we obtain an uer estimate for the last term as follows, c S I K B I c B I I S B I c I δ) K 0 c x δ) I. Letting δ 0 yields our statement. 9
References [] R. Deville, G. Godefroy, V. Zizler. Smoothness and renorming in Banach saces. Longman 993. [2] J.L.B. Gamlen, R.J. Gaudet. On subsequences of the Haar system in L [0, ] ), Israel J. Math., 5 973), 404 43. [3] S. Geiss, P.F.. Müller. Extraolation of vector valued rearrangement oerators, rerint 2006). [4] Müller, P. F.., Isomorhisms between H saces, Birkhäuser Verlag, Basel, 2005. [5] A. Pietsch, J. Wenzel. Orthonormal systems and Banach sace geometry. Cambridge University Press, 998. [6] G. Pisier. Martingales with values in uniformly convex saces. Israel J. Math. 20 975), 326 350. [7] E.M. Semenov. Equivalence in L of ermutations of the Haar system. Dokl. Akad. Nauk SSSR, 242 978), 258 260. Addresses Deartment of Mathematics and Statistics P.O. Box 35 MaD) FIN-4004 University of Jyväskylä Finland Deartment of Analysis J. Keler University A-4040 Linz Austria 0