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

Abstract

In his 2006 ICM invited address, Konyagin mentioned the following conjecture: if S n f stands for the n-th partial Fourier sum of f and {n j } j ⊂ N is a lacunary sequence, then S n j f is a.e. pointwise convergent for any f ∈ L log log L. In this paper we will show that sup j |S n j (f )| 1,∞ C f 1 log log(10 + f ∞ f 1 ).As a direct consequence we obtain that S n j f → f a.e. for f ∈ L log log L log log log L. 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 sup j |S n j (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 L 1 " apply.

Endpoint bounds for the bilinear Hilbert transform

“…This motivates the study of endpoint estimates for the bilinear Hilbert transform at the boundary of the hexagon. Similar endpoint questions for Carleson's operator, the other stalwart of time-frequency analysis, have enjoyed some popularity as well in recent years [8,9,10,22,23].…”

The pointwise convergence of Fourier Series (II). Strong L1 case for the lacunary Carleson operator

“…• in Section 8 we present the proof of Theorem 19 that can be reduced to the proofs of the L 2 -boundedness of the Carleson operator and of the Main Lemma. • in Section 9 we prove Theorem 20 by closely following the spirit of our approach in [25]. • Section 10 is the most technical component of our paper.…”

“…non-convex) trees of 0-frequency tiles relative to the standard order relation " ≤ ". 25 24 Throughout the paper we adopt the following convention: when setting the parameter r ∈ N we allow the value r = 0, and, in this case, we simply set…”

“…In this section we will perform a final decomposition of our tiles depending on the deep geometric and additive combinatoric properties of the set F encapsulated in the time-frequency regularization of the set F developed in Section 4.3. The procedure described here is a major improvement over the initial approach in [25] that was designed to treat the L 1,∞ behavior of the lacunary Carleson operator. 33 • Fix 1 < k < k F and I ∈ I k .…”

“…3 For a more detailed account of this, please see the statement of the Main Theorem B at the end of this section. 4 This should be understood as the adjoint of the linearized Carleson operator -see (25), where the linearized function N (·) is regarded as an arbitrary measurable function independent of the input function. 5 Throughout the paper, all the sets consisting of real numbers are assumed to be measurable.…”