Bilinear operators along variable curves

Abstract

Motivated by Zygmund's differentiation conjecture and Stein's conjecture for Lipschitz vector fields, we study bilinear singular integral and maximal operators along variable curves given by $$ \begin{aligned} BH_a^α(f,g)(x) &= \operatorname{p.v.}\int_{\mathbb R} f(x-t) g\left(x + a(x)t^α\right)\,\frac{dt}{t},\\ BM_a^α(f,g)(x) &= \sup_{r>0}\frac{1}{2r}\int_{-r}^{r} \left|f(x-t)g\left(x+a(x)t^α\right)\right|\,dt,\\ BSH_c^α(f,g)(x) &= \operatorname{p.v.} \int_{\mathbb R} f(x-t) g(x + t^α)\, \frac{dt}{t-c(x)}, \end{aligned} $$ where $α> 0$ and $a,c:\mathbb R\to\mathbb R$ are measurable. For every $α>0$ and $0 1}L^s(\mathbb R)$, one of which is compactly supported, whose outputs under $BH_a^α$ and $BM_a^α$ do not belong to $L^1(\mathbb R)$. For any measurable $a:\mathbb R\to\mathbb R$, $BM_a^α$ maps $L^p(\mathbb R) \times L^q(\mathbb R)$ boundedly into $L^r(\mathbb R)$ whenever $1 1$, and maps $L^p(\mathbb R) \times L^q(\mathbb R)$ into weak $L^1(\mathbb R)$ when $1/p+1/q=1$, with constants independent of $a$. Finally, we investigate the shifted operator $BSH_c^α$, including dyadic-integer-valued $c$, and prove that there is a measurable choice of $c$ such that the operator admits no bounded $L^p(\mathbb R) \times L^q(\mathbb R) \to L^{r,\infty}(\mathbb R)$ estimates in the range $1 \leq p,q \leq \infty$ and $\frac{1}{2}<r<\infty$, resolving an open problem posed by Gaitan and Lie.