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

TL;DR

This paper establishes the boundedness range for a bilinear operator along curves with non-zero curvature, extending harmonic analysis techniques to a new class of oscillatory integral operators.

Contribution

It introduces the Rank II LGC method to analyze the maximal boundedness of the Bilinear Hilbert-Carleson operator along curves with non-zero curvature.

Findings

Boundedness holds for 1/r between 1/2 and 1, with p1,p2 in (1,∞).

Operator norms are controlled by the Lp norms of the functions.

The method extends previous techniques to a broader class of oscillatory integrals.

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 $$ BHC_{[\vec{a},\vec{\alpha}]}(f_1,f_2)(x) := \sup_{\lambda\in\mathbb{R}} \left|\,p.v.\, \int_{\mathbb{R}} f_1(x - a_1 t^{\alpha_1}) \,f_2(x - a_2 t^{\alpha_2}) \,e^{i\,\lambda\,a_3 \,t^{\alpha_3}} \,\frac{dt}{t}\right|$$ obeys the bounds $$\|BHC_{[\vec{a},\vec{\alpha}]} (f_1,f_2)\|_{L^r} \lesssim_{\vec{a} \,\vec{\alpha},r,p_1,p_2} \|f_1\|_{L^{p_1}}\,\|f_2\|_{L^{p_2}}$$ 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 H\"older 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.