A proof of boundedness of the Carleson operator

Abstract

Abstract. We give a simplified proof that the Carleson operator is of weak type (2, 2). This estimate is the main ingredient in the proof of Carleson's theorem on almost everywhere convergence of Fourier series of functions in L 2 ([0, 1]).

Pointwise convergence of vector-valued Fourier series

“…Hence, the A T is bounded on L p (R; X), with constant depending only upon X and p. In the reverse direction, the argument in [7,Section 2] shows that the operators Π ξ with (Π ξ f ) = 1 (−∞,ξ) f , are the appropriate averages of the uptree operators A T . Hence, the Hilbert transform is bounded on L p (R; X), showing that X must be UMD.…”

“…Remark. Proposition 6.1 is essentially contained in the proof of Proposition 3.2 in [7], but not explicitly formulated as here, so we reproduce the proof for completeness.…”

“…The proof of Carleson's theorem à la Lacey-Thiele [7] is based on controlling two key quantities associated to any collection of tiles P: density and energy. (The terminology is slightly variable over different papers on the subject.)…”

“…We can then establish this property for all the intermediate UMD spaces as in Theorem 1.1. Once these critical quasi-orthogonality bounds are available, it is possible to once again adopt the general proof scheme of Lacey-Thiele [7]; we also frequently borrow from [11]. While the Fourier tiletype property is applied exactly once in the proof, UMD is used more frequently to control various Calderón-Zygmund type operators produced in the analysis.…”

“…The proof is based on an elaboration of our method from [6] where, adapting the phase plane analysis à la Lacey-Thiele [7], we treated the model case of WalshFourier series. The key point of this strategy is that a critical quasi-orthogonality estimate, which we call tile-type, can be obtained in the intermediate UMD spaces by interpolating between strong orthogonality estimates in a Hilbert space and a weaker version available in general UMD spaces.…”