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

2014/08/20 by Victor Lie, Lie, Victor
Mathematics · #42A20 #42A55 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.1408.4783

openalex publication_date 2014/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide a near-complete classification of the Lorentz spaces Λ_φ for which the sequence \S_n\_{n∈ ℕ} of partial Fourier sums is almost everywhere convergent along lacunary subsequences. Moreover, under mild assumptions on the fundamental function φ, we identify Λ_φ:= Lloglog Lloglogloglog L as the largest Lorentz space on which the lacunary Carleson operator is bounded as a map to L^1,∞. 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 L¹, being responsible for the unboundedness of the weak-L¹ norm of a "grand maximal counting function" associated with the mass levels.

Citations

• Une série de Fourier-Lebesgue divergente presque partout
• The Polynomial Carleson Operator
• Cancellation for the multilinear Hilbert transform

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