Szego Seminar: "Modulation Invariant Operators near $L^1$"

Speaker: Anastasios Fragkos, Washington University in Saint Louis

Abstract: We prove that the weak-$L^{p}$ norms, and in fact the sparse $(p,1)$-norms, of the Carleson maximal partial Fourier sum operator are $\lesssim (p-1)^{-1}$ as $p\to 1^+$. Furthermore, our sparse $(p,1)$-norms bound imply new and stronger results at the endpoint $p=1$. In particular, we obtain that the Fourier series of functions from the weighted Arias de Reyna space $ \mathrm{QA}_{\infty}(w) $, which contains the weighted Antonov space $L\log L\log\log\log L(\mathbb T; w)$, converge almost everywhere whenever $w\in A_1$. This is an extension of the results of Antonov and Arias De Reyna, where $w$ must be Lebesgue measure.

The center of our approach is a sharply quantified near-$L^1$ Carleson embedding theorem for the modulation-invariant wave packet transform. The proof of the Carleson embedding is based on a newly developed smooth multi-frequency decomposition which, near the endpoint $p=1$, outperforms the abstract Hilbert space approach of past works, including the seminal one by Nazarov, Oberlin and Thiele.

Host: Jeremy Cummings