The pointwise convergence of Fourier Series (I). On a conjecture of Konyagin

Abstract

We provide a near-complete classification of the Lorentz spaces $\Lambda_{\varphi}$ for which the sequence $\{S_{n}\}_{n\in \mathbb{N}}$ of partial Fourier sums is almost everywhere convergent along lacunary subsequences. Moreover, under mild assumptions on the fundamental function $\varphi$, we identify $\Lambda_{\varphi}:= L\log\log L\log\log\log\log L$ as the \emph{largest} Lorentz space on which the lacunary Carleson operator is bounded as a map to $L^{1,\infty}$. In particular, we disprove a conjecture stated by Konyagin in his 2006 ICM address. Our proof relies on a newly introduced concept of a "Cantor Multi-tower Embedding," a special geometric configuration of tiles that can arise within the time-frequency tile decomposition of the Carleson operator. This geometric structure plays an important role in the behavior of Fourier series near $L^1$, being responsible for the unboundedness of the weak-$L^1$ norm of a "grand maximal counting function" associated with the mass levels.Comment: 82 pages, no figures. We have added the following items: 1) Section 5 presenting a suggestive example; 2) Section 6 explaining the fundamental role of the so called grand maximal counting function; 3) Section 12 presenting a careful analysis of the Lacey-Thiele discretized Carleson model and of the Walsh-Carleson operator. Accepted for publication in J. Eur. Math. Soc. (JEMS