Endpoint bounds for the bilinear Hilbert transform (Q2790613)

The classical Hilbert transform is bounded in \(L_p\) for \(1 < p <\infty\) and weak type \(L_1\). Similarly the Coifman-Meyer bilinear singular integrals with Calderón-Zygmund kernel in \({\mathbb R}^2\), obey the weak endpoint bound \(T : L_1({\mathbb R}) \times L_1({\mathbb R}) \rightarrow L^{1/2,\infty}({\mathbb R})\). The article concerns endpoint bounds for the more singular family of bilinear operators known as bilinear Hilbert transforms NEWLINE\[NEWLINE BHT_{b}(f_1, f_2)(x) = p.v. \int_{\mathbb R}f_1(x -b_1t)f_2(x -b_2t) \frac{dt}{t},\quad x\in {\mathbb R}, NEWLINE\]NEWLINE and the corresponding trilinear form NEWLINE\[NEWLINE \Lambda_\beta(f_1, f_2, f_3) = <BHT_{b}(f_1, f_2), f_3> = \int_{\mathbb R} p.v. \int_{\mathbb R} f_1(x - \beta_1t)f_2(x - \beta_2t)f_3(x - \beta_3 t) \frac{dt}{t}, NEWLINE\]NEWLINE with \(\beta_1 - \beta_3 = b_1\), \(\beta_2 - \beta_3 = b_2\). The authors study the behaviour of the bilinear Hilbert transform (BHT) at the boundary of the known boundedness region. For example it is proved that if \(f_2 \in L_2\) and sets \(F_1, F_3 \subset {\mathbb R}\) of finite measure are given. Then, there exists a major subset \(F'_3\) of \(F_3\), depending on \(f_2\), \(F_1\), \(F_3\), such that NEWLINE\[NEWLINE |\Lambda_\beta(f_1, f_2, f_3)| \leq C_\beta |F_1| \|f_2\|_2 |F_3|^{-1/2} \log ( e + \frac{|F3|}{|F1|}) \quad \forall |f_1| \leq {1 }_{F_2} , |f_3| \leq 1_{F'_3} NEWLINE\]NEWLINE Similarly NEWLINE\[NEWLINE |\Lambda_\beta(f_1, f_2, f_3)| \leq C_\beta |F_1|^{3/4} |F_2|^{3/4} |F_3|^{-1/2} \log \log ( e^e + \frac{|F3|}{\min\{|F1|,|F_2|}), \quad \forall |f_j| \leq {1 }_{F_j} NEWLINE\]NEWLINE and \(f_3\) be supported on the major subset \(F'_3\) of \(F_3\). This improves the earlier results by \textit{D. Bilyk} and \textit{L. Grafakos} [J. Geom. Anal. 16, No. 4, 563--584 (2006; Zbl 1112.42005); Collect. Math. 2006, 141--169 (2006; Zbl 1112.42006)]. The bounds in Lorentz-Orlicz spaces near \(L^{2/3}\) are also discussed. To prove the results the authors elaborate an enhanced version of the multi-frequency Calderón-Zygmund decomposition.