On the pointwise convergence of the sequence of partial Fourier sums along lacunary subsequences

In his 2006 ICM invited address, Konyagin mentioned the following conjecture: if Snf stands for the n-th partial Fourier sum of f and {nj } j ⊂ N is a lacunary sequence, then Snj f is a.e. pointwise convergent for any f ∈ Llog logL. In this paper we will show that supj |Snj (f )| 1,∞ C f 1 log log(10 + f ∞ f 1 ). As a direct consequence we obtain that Snj f → f a.e. for f ∈ Llog logLlog log logL. The (discrete) Walsh model version of this last fact was proved by Do and Lacey but their methods do not (re)cover the (continuous) Fourier setting. The key ingredient for our proof is a tile decomposition of the operator supj |Snj (f )| which depends on both the function f and on the lacunary structure of the frequencies. This tile decomposition, called (f, λ)-lacunary, is directly adapted to the context of our problem, and, combined with a canonical mass decomposition of the tiles, provides the natural environment to which the methods developed by the author in “On the boundedness of the Carleson operator near L1” apply. © 2012 Elsevier Inc. All rights reserved.