The Bilinear Hilbert-Carleson operator along curves. The purely non-zero curvature case

Abstract

In this paper, we provide the maximal boundedness range (up to end-points) for the Bilinear Hilbert-Carleson operator along curves in the (purely) non-zero curvature setting. More precisely, we show that the operator $$BH{C}_{[\vec{a},\vec{\alpha }]}(f_1,f_2)(x):=\underset{\lambda \in \mathbb{R}}{\sup }\left|p.v.{\int }_{\mathbb{R}}{f}_1(x-a_1{t}^{{\alpha }_1})f_2(x-a_2{t}^{{\alpha }_2})e^{i\lambda a_3{t}^{{\alpha }_3}}\frac{dt}{t}\right|$$ obeys the bounds $$\Vert BH{C}_{[\vec{a},\vec{\alpha }]}(f_1,f_2){\Vert }_{L^r}{\lesssim }_{\vec{a}\vec{\alpha },r,p_1,p_2}\Vert f_1{\Vert }_{L^{p_1}}\Vert f_2{\Vert }_{L^{p_2}}$$ whenever $\vec{a}=(a_1,a_2,a_3),\vec{\alpha }=({\alpha }_1,{\alpha }_2,{\alpha }_3)\in (\mathbb{R}\setminus \{0\}{)}^3$ with $\vec{\alpha }$ having pairwise distinct coordinates and for any Hölder range $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{r}$ with $1<p_1,p_2<\infty$ and $\frac{1}{2}<r<\infty$. This result is achieved via the Rank II LGC method introduced in arXiv:2308.10706.

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