Pointwise convergence of Fourier series (I). On a conjecture of Konyagin

  • Victor Lie

    Purdue University, West Lafayette, USA

Abstract

We provide a near-complete classification of the Lorentz spaces for which the sequence of partial Fourier sums is almost everywhere convergent along lacunary subsequences. Moreover, under mild assumptions on the fundamental function , we identify log log log log log log as the largest Lorentz space on which the lacunary Carleson operator is bounded as a map to . As a consequence, we

  • disprove a conjecture stated by Konyagin in his 2006 ICM address;

  • provide a negative answer to an open question related to the Halo conjecture.

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 , being responsible for the unboundedness of the weak- norm of a "grand maximal counting function“ associated with the mass levels.

Cite this article

Victor Lie, Pointwise convergence of Fourier series (I). On a conjecture of Konyagin. J. Eur. Math. Soc. 19 (2017), no. 6, pp. 1655–1728

DOI 10.4171/JEMS/703