L Bounds for a Maximal Dyadic Sum Operator

The authors prove L bounds in the range 1 < p < ∞ for a maximal dyadic sum operator on R. This maximal operator provides a discrete multidimensional model of Carleson’s operator. Its boundedness is obtained by a simple twist of the proof of Carleson’s theorem given by Lacey and Thiele [6] adapted in higher dimensions [8]. In dimension one, the L boundedness of this maximal dyadic sum implies in particular an alternative proof of Hunt’s extension [3] of Carleson’s theorem on almost everywhere convergence of Fourier integrals. 1. The Carleson-Hunt theorem A celebrated theorem of Carleson [1] states that the Fourier series of a squareintegrable function on the circle converges almost everywhere to the function. Hunt [3] extended this theorem to L functions for 1 < p < ∞. Alternative proofs of Carleson’s theorem were provided by C. Fefferman [2] and by Lacey and Thiele [6]. The last authors proved the theorem on the line, i.e. they showed that for f in L(R) the sequence of functions SN(f)(x) = ∫ |ξ|≤N f̂(ξ)edξ converges to f(x) for almost all x ∈ R as N → ∞. This result was obtained as a consequence of the boundedness of the maximal operator C(f) = sup N>0 |SN(f)| from L(R) into L2,∞(R). In view of the transference theorem of Kenig and Tomas [4] the above result is equivalent to the analogous theorem for Fourier series on the circle. Lacey and Thiele [5] have also obtained a proof of Hunt’s theorem by adapting the techniques in [6] to the L case but this proof is rather complicated compared with the relatively short and elegant proof they gave for p = 2. Investigating higher dimensional analogues, Pramanik and Terwilleger [8] recently adapted the proof of Carleson’s theorem by Lacey and Thiele [6] to prove weak type (2, 2) bounds for a discrete maximal operator on R similar to the one which arises in the aforementioned proof. After a certain averaging procedure, this result provides an alternative proof of Sjölin’s [10] theorem on the weak L boundedness of Date: December 11, 2002. 1991 Mathematics Subject Classification. Primary 42A20. Secondary 42A24.