AN IMPROVED MENSHOV-RADEMACHER THEOREM

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 3, March 1996 AN IMPROVED MENSHOV-RADEMACHER THEOREM FERENC MÓRICZ AND KÁROLY TANDORI (Communicated by J. Marshall Ash) Abstract. We study the a.e. convergence of orthogonal series defined over a general measure space. We give sufficient conditions which contain the Menshov-Rademacher theorem as an endpoint case. These conditions turn outtobenecessaryintheparticularcasewherethemeasurespaceistheunit interval [0, 1] and the moduli of the coefficients form a nonincreasing sequence. We also prove a new version of the Menshov-Rademacher inequality. 1. Introduction In this note we consider an arbitrary measure space X with measure µ. We denote by Ω the class of orthonormal systems ϕ := {ϕ (x): 1,x X} of functions. That is, we always assume that ϕ (x)ϕ l (x) dµ(x) =0 ( l), ϕ 2 (x)dµ(x) =1 (, l =1,2,...). Here and in the sequel, the integrals are extended over the whole space X. Given a sequence a = {a : 1} of real numbers we introduce the quantity ( ) p 2 a(m, n) := sup max a ϕ (x) dµ(x), m p n where m, n are integers, 1 m n, and the supremum is taen over all systems ϕ Ω. Furthermore, set a := lim a(1,n), n which may be infinite. The following theorem proved by the second named author characterizes those sequences a for which the orthogonal series (1.1) a ϕ (x) converges a.e. for all ϕ Ω. Received by the editors November 1, 1993 and, in revised form, September 26, Mathematics Subject Classification. Primary 42C05. Key words and phrases. Orthonormal system, a.e. convergence, Menshov-Rademacher inequality and theorem. This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant #234. =m c 1996 American Mathematical Society 877

2 878 FERENC MÓRICZ AND KÁROLY TANDORI Theorem A ([8, Theorem 1]). The orthogonal series (1.1) converges a.e. for all ϕ Ω if and only if a <. In general, it is a difficult tas to chec whether the norm a is finite or not. Therefore it maes sense to loo for necessary or sufficient conditions, which are easier to apply in concrete situations. 2. Sufficient conditions The well-nown Menshov-Rademacher theorem (see [2], [5], and also [1, p. 80]) states that if (2.1) (log ) 2 <, then the orthogonal series (1.1) converges a.e. for all ϕ Ω. Hereandinthesequel, the logarithms are to the base 2. We note that condition (2.1) is the best possible in the sense that for any nondecreasing sequence {w()} of positive numbers such that w() = o(log ) ( ) there exist a sequence {a } of real numbers and a particular system ϕ Ωsuch that w 2 () <, but the orthogonal series (1.1) diverges a.e. (see, e.g., [1, p. 88]). The second named author improved the above result by showing (see [9]) that if a 2 1 (2.2) (log )log + a 2 <, then the orthogonal series (1.1) converges a.e. for all ϕ Ω. Here we agree to put { log + (1/a 2 log(1/ )= ) if 0 <a2, 1 if > or a =0. In [4] we proved that condition (2.2) can be replaced by the somewhat better condition ( ) a 2 2 (2.3) (log )log a 2 a 2 l <, r=0 J r l J r where (2.4) J r := {n r +1,n r +2,...,n r+1 } and n r := 2 2r (r 0). It is easy to see that condition (2.3) follows from (2.2), and condition (2.2) follows from (2.1), but not conversely. Before stating the main result of this paper, we introduce the following notations: (2.5) I ν := {2 ν +1,2 ν +2,...,2 ν+1 }, A 2 ν := I ν (ν 0), and we agree to put log(2a2 ν /a2 )=0incasea =0.

3 AN IMPROVED MENSHOV-RADEMACHER THEOREM 879 Theorem 1. If for some 0 <ε 2we have (log ) ε log 2A2 ν (2.6) a 2 <, ν=0 I ν then the orthogonal series (1.1) converges a.e. for all ϕ Ω. It turns out that even more is true. Namely, under (2.6), the rearranged series (2.7) a j ϕ j (x) j=1 converges a.e. for all ϕ Ω and for all (so-called wea) permutations { 1, 2,...} of the positive integers {1, 2,...} such that { j : j I ν } is a permutation of I ν in the case of ν large enough. It is instructive to consider the special case of (2.6) corresponding to ε =1: (log )log 2A2 ν (2.8) a 2 <. ν=0 I ν We claim that even this particular condition is better than (2.3). Indeed, a simple estimation gives ν=1 I ν (log )log 2A2 ν 2 r+1 1 = (log )log 2A2 ν a 2 r=0 ν=2 r I ν ( ) a 2 2 (log )log a 2 a 2 l r=0 J r l J r ( ) = a 2 2 (log )log a 2 a 2 l, r=0 J r l J r which justifies our claim. In order to prove Theorem 1, first we prove a new version of the famous Menshov- Rademacher inequality (see, e.g., [1, p. 79] and [3, Theorem 3]), which says ( ) n 2 N (2.9) max a ϕ (x) dµ(x) (log 2N) 2 (N 1). 1 n N Lemma 1. For every ε, 0 <ε 2, there exists a constant C depending only on ε such that for all sequences a, allϕ Ω, and all integers N 1, we have ( ) n 2 max a ϕ (x) dµ(x) 1 n N (2.10) N ) 2 ε N C(log 2N) (log ε 2A2, A 2 :=.

4 880 FERENC MÓRICZ AND KÁROLY TANDORI Proof of Lemma 1. Inequality (2.10) is obvious in case N = 1 and 2, provided C 2. In case N 3, we may assume that (2.11) N = n r := 2 2r for some r 1. Otherwise, we may supplement the given coefficients a 1,a 2,...,a N by an appropriate number of zeros, which does not affect the left-hand side in (2.10), while the right-hand side increases at most by 2 ε. We rearrange the a in a descending order of magnitude: (2.12) a d(1) a d(2) a d(nr). For each p =0,1,...,r 1, we consider the integers of the bloc {d(): J p } in their original (increasing) order, where J p is defined in (2.4). More precisely, let {e(): J p } be the permutation of J p such that d(e(n p +1))<d(e(n p +2))< <d(e(n p+1 )). Our ey estimate is the following: n M(x) := max a ϕ (x) 1 n N r 1 a d(1) ϕ d(1) (x) + a d(2) ϕ d(2) (x) + max n n J p p=0 =n p+1 Applying Minowsi s inequality and (2.9) gives { r 1 (2.13) M 2 (x) dµ(x)} a d(1) + a d(2) + (log 2n p+1 ) By the Cauchy-Schwarz inequality, we obtain r 1 (log 2)2 (2.14) p=0 J p a 2 d() r 1 4 ε/2 2pε a 2 d() J p p=0 { r 1 } r 1 4 ε/2 2 pε p=0 4 ε/2 (2 ε 1) { 2 rε n r p=0 (log 2)2 ε a 2 d() (log 2)2 ε p=0 J p } a 2 d() (log 2)2 ε. =3 By (2.12), we have a 2 d() A2 for all. Thus, (2.15) n r =3 a 2 d() (log n r 2)2 ε =3 a 2 d() a d(e()) ϕ d(e()) (x). log 2A2 a 2. d() J p a 2 d().

5 AN IMPROVED MENSHOV-RADEMACHER THEOREM 881 Taing into account (2.11), (2.13) (2.15), and the fact that log(2a 2 /a 2 d() ) 1, we conclude { } M 2 (x) dµ(x) ( 4 ε/2 ) n r 2 +2 (log (2 ε N)ε 1) a 2 d() ( ) 2 ε log 2A2 a 2 d() This proves (2.10), since the right-hand side is symmetric in the terms a 1,a 2,..., a nr. Denote by the nth partial sum of series (1.1). s n (x) := n a ϕ (x) Lemma 2 (see, e.g., [1, p. 83]). If (2.16) (log log )2 <, then the subsequence {s 2 ν(x): ϕ Ω. =2 Proof of Theorem 1. From (2.6) it follows that (log )ε <, ν 0} of the partial sums converges a.e. for all =2 which in turn implies (2.16). By Lemma 2, s 2 ν(x) converges a.e. as m. It remains to estimate the maximal fluctuation n M ν (x) :=max a ϕ (x) (ν 1). n I ν By Lemma 1, =2 ν +1 M 2 ν (x) dµ(x) C(ν +1)ε I ν log 2A2 ν a 2.. By (2.6), ν=1 Mν 2 (x) dµ(x) <, whence, by the dominated convergence theorem, lim max n a ϕ (x) ν n I ν =0 =2 ν +1 This completes the proof of Theorem 1. a.e.

6 882 FERENC MÓRICZ AND KÁROLY TANDORI 3. Necessary conditions From now on, let the measure space X be the unit interval [0, 1] with Borel measurable subsets and Lebesgue measure µ. Clearly, the condition < is necessary for the orthogonal series (1.1) to converge a.e. for all ϕ Ω. To see this, it is enough to tae the Rademacher system in the capacity of ϕ (see, e.g., [1, p. 54]). The second named author gave the following nontrivial necessary condition. Theorem B ([6, Theorem 1]). If the orthogonal series (1.1) converges a.e. for all ϕ Ω, then ( ) 2 a 2 1 (3.1) log + a 2 <. It is plain that (3.1) implies (3.2) ν=0 I ν ( ) 2 log 2A2 ν a 2 <. This latter condition coincides with (2.6) for ε = 0. We note that condition (3.1) is not sufficient in general for the orthogonal series (1.1) to converge a.e. for all ϕ Ω. This is shown by the next Example 1. The second named author proved in [7] that if a 1 a 2,then all rearrangements (2.7) of the orthogonal series (1.1) converge a.e. for all ϕ Ωif and only if { } (3.3) (log )2 < r=0 J r where the blocs J r are defined by (2.4). Now, for the sequence a := r 1 2 r (n r+1 n r ) if J r (r 0) condition (3.1) is satisfied, but (3.3) is not. Consequently, there exist a system {ϕ } Ω on the unit interval [0, 1] and a permutation { j } of the positive integers such that the rearranged series (2.7) diverges a.e. (see [7]). It remains to observe that condition (3.1) is invariant under a permutation of its terms. On the other hand, condition (2.6) for some ε > 0 is not necessary for the orthogonal series (1.1) to converge a.e. for all ϕ Ω. To see this, we present the following Example 2. Let a := { r 2 if = n r := 2 2r (r 1), 0 otherwise.

7 AN IMPROVED MENSHOV-RADEMACHER THEOREM 883 Then condition (2.6) is not satisfied for any ε > 0. Nevertheless, series (1.1) converges a.e. for all ϕ Ω, due to a = r 2 <. r=1 4. Analysis of condition (2.6) Given a sequence a := {a : 1} of real numbers, consider the series occurring in condition (2.6) and denote its sum by ε for some ε 0. That is, let ε := (log )ε log 2A2 ν a 2, ν=0 I ν which may be infinite. We note that condition (2.6) reduces to (2.1) if ε =2,to (2.8) if ε = 1, and to (3.2) if ε =0. We shall prove the following Theorem 2. Let 0 δ<ε 2. Then there exists an absolute constant C 1 such that for every sequence a = {a } (4.1) δ C 1 ε. If the sequence { a } is nonincreasing, then the converse inequality ε C (4.2) 2 δ also holds, with another absolute constant C 2. Combining Theorems 1 and 2 with Theorem B provides the following nown characterization (see, e.g., [8] and [9]). Corollary. If a sequence { a } is nonincreasing, then conditions (2.1), (2.8), and (3.2) are pairwise equivalent, and each of them is necessary and sufficient for the orthogonal series (1.1) to converge a.e. for all ϕ Ω. Before proving Theorem 2, we prove a simplified version for finite sequences. Lemma 3. There exists an absolute constant C 3 such that for all sequences a = {a } and all N 1 we have N ) 2 (log 2A2 N N (4.3) C 3 (log 2) 2, A 2 :=. If the sequence { a } is nonincreasing, then N N ) 2 (4.4) (log 2) 2 (log 2A2. Proof of Lemma 3. First, we assume that the sequence { a } is nonincreasing. Without loss of generality, we may assume that A = 1. By monotonicity, This proves (4.4). 1, whence 2 2/a2 (1 N).

8 884 FERENC MÓRICZ AND KÁROLY TANDORI From now on, we do not use monotonicity of { a }. We shall distinguish between two cases accordingly as (i) 4 a 2 2 1, whence log 4log2; a ( 2 ) (ii) 4 < 1, whence a2 log 2 C4, 2 where ( C 4 := max t log 2 ) 2 (4.5) 0<t 1 t 2. To sum up, N ( log 2 ) 2 N N a 2 16 (log 2) C 4 2. Hence (4.3) follows with C 3 := 16 + C 4 π 2 6. Proof of Theorem 2. It is plain that if ε < for some ε>0, then A := <. Without loss of generality, we may assume that A =1. Again, we distinguish between two cases accordingly as (i) 4 A2 ν, whence log 2A2 ν 4log2; (ii) 4 >A2 ν, whence a2 (log 2A2 ν ) ε δ A C ν a 2 4, 2 where C 4 is defined in (4.5). Hence δ 4ε δ ν=0 I ν + C 4 ν=0 I ν (log 2)ε (log 2)δ 2 A ν. Since A ν A=1,weget δ 16 ε + C 4 =2 log 2A2 ν (log 2) 2 2. Due to the fact that ε A2 = 1, this gives (4.1) with (log 2) 2 C 1 := 16 + C 4 2. =2 Relying on (4.1) just proved, it is enough to chec (4.2) in the special case when δ =0andε= 2. By Lemma 3, for ν 6wehave (4.6) ν+1 =2 ν +2 ν 1 (log 2) 2 2 ν+1 (log 2( 2 ν )) 2 =2 ν +2 ν 1 ( ) 2 log 2A2 ν a 2. I ν

9 AN IMPROVED MENSHOV-RADEMACHER THEOREM 885 By monotonicity, for ν 6wehave 1 2 ν+1 (4.7) a 2 8 (log 2) 2 (log 2) 2. I ν+1 =2 ν +2 ν 1 Combining (4.6) and (4.7) yields (4.2). Acnowledgment The authors are grateful to the referee for the valuable suggestions which improved the presentation of this paper. References 1. G. Alexits, Convergence problems of orthogonal series, Hungarian Acad. Sci., Budapest, MR 36: D. E. Menchoff, Sur les séries des fonctions orthogonales (Première partie), Fund. Math. 4 (1923), F. Móricz, Moment inequalities and the strong laws of large numbers, Z. Wahrsch. Verw. Gebiete 35 (1976), MR 53: F. Móricz and K. Tandori, Almost everywhere convergence of orthogonal series revisited, J. Math. Anal. Appl. 182 (1994), MR 95a: H. Rademacher, Einige Sätze über Reihen von allgemeinen Orthogonalfuntionen, Math. Ann. 87 (1922), K. Tandori, Über die Divergenz der Orthogonalreihen, Publ. Math. Debrecen 8 (1961), MR 25: , Orthogonalen Funtionen X, Acta Sci. Math. (Szeged) 23 (1962), MR 26: , Über die Konvergenz der Orthogonalreihen II, Acta Sci. Math. (Szeged) 25 (1964), MR 30: , Bemerung zur Konvergenz der Orthogonalreihen, Acta Sci. Math. (Szeged) 26 (1965), MR 33:3041 Bolyai Institute, University of Szeged, Aradi Vértanú Tere 1, 6720 Szeged, Hungary address: moricz@math.u-szeged.hu