Endpoint bounds for the bilinear Hilbert transform

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 $| \langle\mathrm{BHT}(f_1,f_2),f_3 \rangle | \leq C |F_1|^{\frac34}|F_2| ^{\frac34} |F_3|^{-\frac12} \log\log \Big(\mathrm{e}^{\mathrm{e}} + \frac{|F_3|}{\min\{|F_1|,|F_2|\}} \Big)$ valid for all tuples of sets $F_j \subset \mathbb R$ of finite measure and functions $f_j$ such that $|f_j| \leq \mathbf{1}_{F_j}$, $j=1,2,3$, 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 double logarithmic term improves over the single logarithmic term obtained by Bilyk and Grafakos. Whether the double logarithmic term can be removed entirely, as is the case for the quartile operator discussed by Demeter and the first author, 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 $L^{\frac23}$, improving on results of Carro, Grafakos, Martell and Soria. The main technical novelty in our article is an enhanced version of the multi-frequency Calder\'on-Zygmund decomposition by Nazarov, Oberlin and the second author.