COLLOQUIUM MATHEMATICUM VOL. 106 2006 NO. 2 A MULTIPLIER THEOREM FOR FOURIER SERIES IN SEVERAL VARIABLES BY NAKHLE ASMAR (Columbia, MO), FLORENCE NEWBERGER (Long Beach, CA) and SALEEM WATSON (Long Beach, CA) Abstract. We define a new type of multiplier operators on L p (T N ), where T N is the N -dimensional torus, and use tangent sequences from probability theory to prove that the operator norms of these multipliers are independent of the dimension N . Our construction is motivated by the conjugate function operator on L p (T N ), to which the theorem applies as a particular example. 1. Introduction. On the one-dimensional torus T, we can define the conjugate function  f of f ∈ L p (T) by the Fourier multiplier operator   f (n)= −i sgn(n)  f (n) (n ∈ Z), where sgn(n)=1, −1, or 0, according as n is positive, negative or 0. Parse- val’s theorem implies that the operator f →  f is bounded from L 2 (T) into L 2 (T) with norm equal to 1. The celebrated theorem of M. Riesz establishes the boundedness of this operator from L p (T) into L p (T), where 1 <p< ∞. M. Riesz’s theorem plays an important role in harmonic analysis. It has been generalized in many directions (for a brief history of this theorem, including the extensions cited below, see [1] or [3]). One version of the M. Riesz theorem on the N -dimensional torus, due to S. Bochner (1939), was extended by H. Helson to any compact (connected) abelian group G whose dual group Γ contains an order P . Recall that a subset P of Γ is called an order if it satisfies the following three axioms: P ∩ (−P )= {0}, P ∪ (−P )= Γ, P + P = P. Helson’s definition of the conjugate function is as follows. Given an order P ⊂ Γ , we define a signum function with respect to P by sgn P (χ)= −1, 0, or 1, according as χ ∈ (−P ) \{0}, χ = 0, or χ ∈ P \{0}. For f ∈ L 2 (G), define  f by the Fourier multiplier 2000 Mathematics Subject Classification : Primary 42B15; Secondary 60G42. Key words and phrases : Fourier multiplier operator, martingale difference decompo- sition, conjugate function, tangent sequence. [221]