Endpoint bounds for the bilinear Hilbert transform

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.