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A Littlewood-Paley type inequality for the quotient norms of an analytic measure is obtained; one consequence of this inequality is the classical theorem of F. and M. Riesz. In this paper T is the circle group, Z the additive group of integers and M(T) the customary space of Borel measures on T; for ,u E M(T) and n E Z define Af(n)= e-ino d1(0). If ,u E M(T), put ,u = ,ua + ass where ta, is absolutely continuous with respect to Lebesgue measure on T and i,u is singular with respect to Lebesgue measure on T; let Ma(T) denote the space of all absolutely continuous measures. A measure ,u E M(T) is said to be of analytic type if ,t(n) = 0 for all n 2 for all k and Dk = [n2k, n2k+1) = {m E Z: n2k 2 for all k and Dk = [n2k,fn2k+1)U (-n2k+1,-n2k] C Z. Our quantitative generalization of the F. and M. Riesz Theorem is then THEOREM V. There is a constant C > 0 such that for any sequence (Dn)1 of symmetric dyadic intervals in Z and any ,u E H'(T)

In this paper G is a nondiscrete compact abelian group with character group Г and M(G) the usual convolution algebra of Borel measures on G. We designate the following subspaces of M(G) employing the customary notations: Ma(G) those measures which are absolutely continuous with respect to Haar measure; MS(G) the space of measures concentrated on sets of Haar measure zero and Md(G) the discrete measures.

The purpose of this note is to show how the characterization of an absolutely continuous or singular measure of R. Doss can be used to prove the Helson-Lowdenslager Theorem for measures of analytic type. Our methods were inspired by the work of P. Cohen and H. Davenport on the L' norm of exponential sums. Let G be an infinite LCA group with character group r and M(G) the customary convolution algebra of bounded Borel measures on G; let ^ denote the FourierStieltjes transform. We say r is totally ordered if r is discrete and there exists a semigroup P such that (P) n (-P) = (0) and (P) U (-P) = r. A measure IL E M(G) is said to be of analytic type if i(-y) = 0 for all y E r \ P = {y E r: y 0 and M E Z+ there is a polynomial of the form TM(z) = alz +*i + a,z (#) Received by the editors May 6, 1980. 1980 Mathematics Subject Classification Primary 43A17. ? 1981 American Mathematical Society 0002-9939/81/0000-0357/$02.75

Let G be a compact abelian group with dual group Γ. For 1 ≦ p < ∞, 1 ≦ q < ∞, let denote the Banach space of complex-valued functions on Γ which are multipliers of type (p, q) and the subspace of compact multipliers.Grothendieck [10; 11] has proven that a function in LP(G), 1 ≦ p < 2, has an unconditionally converging Fourier series in LP(G) if and only if it is in L2(G), and Helgason [12] has proven that the derived algebra of LP(G), 1 ≦ p < 2, is L2(G). Using these results we show in § 2 that a multiplier of type (p, g), 1 ≦ p ≦ 2, 1 ≦ q ≦ 2, has an unconditionally converging Fourier series in if and only if it is in (Theorem 2.1), and that, for 1 ≦ p ≦ q ≦ 2, the derived algebra of is (Theorem 2.2). Statements equivalent to the above are also given.

In this paper G is a locally compact Abelian group, φ a complex-valued function defined on the dual Γ, Lp(G) (1 ≤ p ≤ ∞) the usual Lebesgue space of index p formed with respect to Haar measure, C(G) the set of all bounded continuous complex-valued functions on G, and C0(G) the set of all f ∈ C(G) which vanish at infinity.

Let G be a locally compact abelian group with dual Γ \Gamma , E a subset of Γ \Gamma , and ϕ \phi a complex-valued function defined on Γ \Gamma . Assume ϕ \phi has σ \sigma -compact support. In this paper we prove that ϕ \phi is a multiplier of type ( L 1 , L p 1 ∩ L p 2 , E ) ( 1 ≦ p 1 ≦ 2 , 1 > p 2 ≦ ∞ ) ({L^1},{L^{{p_1}}} \cap {L^{{p_2}}},E)\;(1 \leqq {p_1} \leqq 2,1 > {p_2} \leqq \infty ) if and only if ϕ = f ^ \phi = \hat f a.e. on E for some f ∈ L p 1 ( G ) ∩ L p 2 ( G ) f \in {L^{{p_1}}}(G) \cap {L^{{p_2}}}(G) . We give applications of this result to the problems of restrictions, uniqueness, inversion and characterization of L p {L^p} transforms.

Let G G be a compact abelian group, and Γ \Gamma its character group. Given E ⊂ Γ , E a E \subset \Gamma ,{E^a} denotes the set of all accumulation points of E E in Γ ¯ \overline \Gamma , the Bohr compactification of Γ \Gamma . In this paper it is shown that the inclusion ( L 1 ( G ) ) ∧ | E ⊂ ( l 1 ( G ) ) ∧ | E {({L^1}(G))^ \wedge }{|_E} \subset {({l^1}(G))^ \wedge }{|_E} obtains if and only if E ∩ E a = ∅ E \cap {E^a} = \emptyset and there exists a measure μ ϵ M ( G ) \mu \epsilon M(G) such that μ ^ = 1 \widehat \mu = 1 on E E and μ ^ = 0 \widehat \mu = 0 on Γ ∩ E a \Gamma \cap {E^a} .

Let ,u be a measure of analytic type on the unit circle. We give a short direct proof that L2 is absolutely continuous with respect to Lebesgue measure; our method also gives a convolution product version of some related several variable results. Let T be the circle group, Z the integers and M(T) the complex-valued regular Borel measures on T. For , E M(T) and n E Z define ji(n) = fTe-in djI(O). Denote by Ma(T) those , E M(T) which are absolutely continuous with respect to Lebesgue measure on T. We now cite the theorem of F. and M. Riesz. THEOREM 1. Let p E M(T) such that p(n) = O for all n 0, N>O,~~~~~~~~~~~~~O Let dT = td,L where t is a trigonometric polynomial on T. An easy consequence of (1) is that limN, . IfITB(. -N = 0; this implies that if co E M(T) and c 0 be given. We gather from (2) that there is a measure /Ae and a positive integer N(e) satisfying II tLeII N(e). (3) Notice that because of (1) and the second part of (3), ,u * (p) is a trigonometric polynomial. Inasmuch as Ma(T) is closed, we conclude from (3) that lim{, * 0,-ye)} = U2 E Ma(T). Received by the editors August 3, 1979. 1980 Mathematics Subject Classification. Primary 43A75. i 1980 American Mathematical Society 0002-9939/80/0000-0367/$0 1.50

In this paper we generalize Hardy's inequality [3] for measures of analytic type and obtain a proof of the Littlewood conjecture [4] for the L' norm of exponential sums as a simple consequence. Let T be the circle group, Z the additive group of integers and M(T) the customary convolution algebra of Borel measures on T; for pe e M(T) and n G Z put M(%)= e-ino dp(O) Denote by H'(T) the classical space of all measures e e M(T) such that j(n) = 0 for all n < 0 and let C denote the complex numbers. We now state Hardy's inequality for measures of analytic type: THEOREM 1. If M C H'(T) then

Let G be a compact abelian group and T a complex-valued function defined on the dual r. The main result of this paper is that p is a compact multiplier of type (p, q), 1 0 there corresponds a finite set KC P such that ffE a,b,rp(y) I 0 there corresponds a finite set Kcii such that 1_ byp(y)I 0. (See Gaudry [8] or Bachelis and Gilbert [1] for details.) Our main result is the following characterization of mp(P): THEOREM. Let T be a complex-valued function defined on F, 1 ?p 0, there corresponds a finite subset Kc F such that a,b,cp(y) I 0, there corresponds a finite subset Kc F such that Ebyg(y)I (ii). Let E > 0 be given. There corresponds by the preceding lemma a trigonometric polynomial L such that 11L-Tepj1(p,q)