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 Haar system constitutes an unconditional basis for a separable rearrangement invariant (symmetric) space E if and only if the multiplier determined by the sequence λ nk = (−1) n , k = 0, 1, for n = 0 and k = 0, 1, . . . , 2 n for n ≥ 1, is bounded in E. If the Lorentz space Λ(ϕ) differs from L 1 and L ∞ then there is a multiplier with respect to the Haar system which is bounded in Λ(ϕ) and unbounded in L ∞ and L 1 .1. The Haar system is the following system of functions: χ 0 0 (t) = 1 andwhere 1 ≤ k ≤ 2 n , n = 0, 1, . . . . We denote by Ω the set of indices (n, k) corresponding to the Haar functions. The formula m = 2 n + k establishes a one-to-one correspondence between Ω and the set N of naturals and enables us to use the single index Haar system {χ m }. Every sequence λ = (λ 1 , λ 2 , . . . ) generates the multiplier Λ that is defined at the polynomials with respect to the Haar system as follows: Λ m c m χ m = m λ m c m χ m . By the classical Paley-Marcinkiewicz Theorem [1, Theorem 3 and Corollary 2] or [2, Section 2, p. 5] the multiplier Λ is bounded in L p (1 < p < ∞) if sup m |λ m | < ∞. Moreover, if 1 < p < ∞ then Λ Lp ≈ λ l∞ = sup m |λ m |,where the sign ≈ stands for two-sided estimates with constants independent of λ. This article studies the spaces of multipliers in some pairs of function spaces. The multipliers with respect to the Haar system were studied in [3-7] and many other articles. 2.We now give the relevant definitions. A Banach function space E on [0, 1] with the Lebesgue measure is called rearrangement invariant or symmetric if (1) |x(t)| ≤ |y(t)| and y ∈ E implies x ∈ E and x E ≤ y E ;(2) equimeasurability of x(t) and y(t) with y ∈ E implies x ∈ E and x E = y E . We suppose henceforth that E is separable or dual to a separable space. Denote the characteristic function of a measurable set e ⊂ [0, 1] by κ e (t). For every τ > 0 the operatoris bounded in a rearrangement invariant space E and σ τ E ≤ max(1, τ ). The numberslog σ τ E log τ , β E = lim τ →∞ log σ τ E log τ Tol yatti; Voronezh.

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