Transactions of the American Mathematical Society

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ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Endpoint bounds for the bilinear Hilbert transformHTML articles powered by AMS MathViewer

by Francesco Di Plinio and Christoph Thiele
Trans. Amer. Math. Soc. 368 (2016), 3931-3972 Request permission

Abstract:

We study the behavior of the bilinear Hilbert transform $(\mathrm {BHT})$ at the boundary of the known boundedness region $\mathcal H$. A sample of our results is the estimate \begin{equation*} |\langle \mathrm {BHT}(f_1,f_2),f_3 \rangle | \leq \textstyle C |F_1|^{\frac 34}|F_2| ^{\frac 34} |F_3|^{-\frac 12} \log \log \Big (\mathrm {e}^{\mathrm {e}} +\textstyle \frac {|F_3|}{\min \{|F_1|,|F_2|\}} \Big ),\end{equation*} valid for all tuples of sets $F_j\subset \mathbb {R}$ of finite measure and functions $f_j$ such that $|f_j| \leq \boldsymbol {1}_{F_j}$, $j=1,2,3$, with the additional restriction that $f_3$ be supported on a major subset $F_3’$ of $F_3$ that depends on $\{F_j:j=1,2,3\}$. The use of subindicator functions in this fashion is standard in the given context. The double logarithmic term improves over the single logarithmic term obtained by D. Bilyk and L. Grafakos. Whether the double logarithmic term can be removed entirely, as is the case for the quartile operator, remains open.

We employ our endpoint results to describe the blow-up rate of weak-type and strong-type estimates for $\mathrm {BHT}$ as the tuple $\vec \alpha$ approaches the boundary of $\mathcal H$. We also discuss bounds on Lorentz-Orlicz spaces near $L^{\frac 23}$, improving on results of M. Carro et al. The main technical novelty in our article is an enhanced version of the multi-frequency Calderón-Zygmund decomposition.

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• Francesco Di Plinio
• Affiliation: Department of Mathematics, Brown University, Box 1917, Providence, Rhode Island 02912