ESTIMATION OF A QUADRATIC FUNCTION AND THE p-BANACH–SAKS PROPERTY

Let E be a rearrangement-invariant Banach function space on [0, 1], and let Γ(E) denote the set of all p ≥ 1 such that any sequence {xn} in E converging weakly to 0 has a subsequence {yn} with supm m−1/p‖ ∑ 1≤k≤m yn‖ < ∞. The set Γi(E) is defined similarly, but only sequences {xn} of independent random variables are taken into account. It is proved (under the assumption Γ(E) = {1}) that if Γi(E) \ Γ(E) = ∅, then Γi(E) \ Γ(E) = {2}. §1 A classical Banach–Saks theorem (see [B, Chapter 12, Theorem 2]) states that if a sequence xn ∈ Lp[0, 1], 1 < p < ∞, converges weakly to zero, then there exists a sequence nk ∈ N and a number C > 0 such that ∥∥∥ m ∑ k=1 xnk ∥∥∥ Lp ≤ Cm for all m ∈ N (N denotes the set of positive integers). For p ∈ (2,∞) this estimate follows also from [KP]. The exponent max(1/2, 1/p) is sharp. It suffices to consider the Rademacher system xn(t) = sign sin 2πt, n ∈ N, for p ≥ 2 and any sequence of normalized Lp-functions with disjoint supports for p ≤ 2. This theorem leads to the following definitions (see [J, Be]). Let E be a Banach space, and let p ≥ 1. A bounded sequence {xn} in E is called a p-BS-sequence if there exists a subsequence {yn} ⊂ {xn}, such that sup m∈N m− 1 p ∥∥∥ m ∑ k=1 yk ∥∥∥ E < ∞. We shall say that E possesses the p-BS-property and write E ∈ BS(p) if any sequence converging weakly to zero contains a p-BS-subsequence. Obviously, every Banach space possesses the 1-BS-property. The set Γ(E) = {p : p ≥ 1, E ∈ BS(p)} is either [1, α] or [1, α), for some α ∈ [1,∞]. This set will be called the index set of the space E, and α is the Banach–Saks index of E; we write γ(E) = α if Γ(E) = [1, α] and γ(E) = α − 0 if Γ(E) = [1, α). A related notion was introduced in [R], where the coordinate Orlicz spaces with the p-BS-property were described. The Banach–Saks theorem says that γ(Lp) = min(p, 2) for 1 < p < ∞. It is known also that γ(L1) = γ(L∞) = 1. In lp, the sequences that converge weakly to zero have a 2000 Mathematics Subject Classification. Primary 46E30.