Multipliers of the Haar series

Abstract

We calculate the norms of multipliers for the Haar system in some rearrangement invariant spaces for which the Haar system is not an absolute basis. 1.The system of functions χ 0 0 (t) = 1,where 1 ≤ k ≤ 2 n with n = 0, 1, . . . , is called the Haar system. Denote the set of indices (n, k) corresponding to the Haar functions by Ω. The formula m = 2 n + k establishes a one-to-one correspondence between Ω and the set of positive integers N and enables us to use the Haar system {χ m } with one index. Each sequence s = (s 1 , s 2 , . . .) generates a multiplier S, which on the Haar polynomials is defined asAccording to the classical Paley-Marcinkiewicz theorem [1, Theorem 3, Corollary 2; 2, Section 2, p. 5], the multiplier S is bounded in L p with 1 < p < ∞ provided that sup m |s m | < ∞. Moreover, forwhere the sign ≈ stands for two-sided estimates with constants independent of s. This article, as well as [3], is devoted to studying the Fourier-Haar multipliers in r.i. spaces and strengthens the result that is announced in [4]. Multipliers for the Haar system were studied in [5-8] and many other articles. 2.Let us give needed definitions. A Banach function space E on [0, 1] equipped with the Lebesgue measure is called the rearrangement invariant (r.i.) or symmetric space whenever:(1) from |x(t)| ≤ |y(t)| and y ∈ E it follows that x ∈ E and x E ≤ y E ;(2) the equimeasurability of two functions x(t) and y(t), where y ∈ E, implies that x ∈ E and x E = y E .In accordance with [2], assume henceforth that E is a separable space or the dual of a separable space. Denote by κ e (t) the characteristic function of a measurable set e ⊂ [0, 1].Denote by Φ the set of increasing concave functions on [0, 1] with ϕ(0) = 0 and ϕ(1) = 1. Every function ϕ ∈ Φ generates the Lorentz space Λ(ϕ) equipped with the normwhere x * (t) is the rearrangement of x(t) [9, p. 81].

The space $L_1(L_p)$ is primary for $1

“…We denote the space of bounded Haar multipliers D : L 1 → L 1 by L HM (L 1 ). In this subsection we recall a formula for the norm of a Haar multiplier that was observed by Semenov and Uksusov in [29]. We then use Haar multipliers to sketch a proof of the fact that every bounded linear operator on L 1 is an approximate 1-projectional factor of a scalar operator.…”

“…Specifically we build on compact families of L 1 -operators, extracted from span{T I,J }, and large deviation estimates for empirical processes: (a) (Compactness.) We utilize the Semenov-Uksusov characterization [29] of Haar multipliers on L 1 and uncover compactness properties of the operators T I,J : L 1 → L 1 . See Theorem 3.2 and Theorem 3.4.…”

“…Theorem 2.6 (Semenov-Uksusov, [29]). Let (a I ) I∈D + be a collection of scalars, and D be the associated Haar multiplier.…”

Strategically reproducible bases and the factorization property

“…It is common to call diagonal operators on the Haar system Haar multipliers. Loosely following [25] we use the following notation. Notation 6.8.…”

The space is primary for 1 < p < ∞

“…Specifically, we build on compact families of -operators, extracted from , and large deviation estimates for empirical processes: (Compactness.) We utilise the Semenov-Uksusov characterisation [34, 35] of Haar multipliers on and uncover compactness properties of the operators . See Theorem 3.2 and Theorem 3.4. (Stabilisation.)…”