Sparse domination and an extended $L^p$ range for the trilinear Hilbert transform

We establish sparse bounds for the trilinear Hilbert transform $T(f_1,f_2,f_3)(x):=\textrm{p.v.}\int_{\mathbb R} f_1(x-t)f_2(x-2t)f_3(x-3t)\,\frac{dt}{t},$ with averaging exponents arbitrarily close to $2$ from above. As a consequence, we extend OpenAI's $L^3\times L^3\times L^3\to L^1$ bound (entry 086 in the OpenAI Research Catalog) to $L^{p_1}\times L^{p_2}\times L^{p_3}\to L^r$ throughout the range $2<p_1,p_2,p_3<\infty, \qquad \frac1r=\frac1{p_1}+\frac1{p_2}+\frac1{p_3}, \qquad \frac23<r<2,$ and obtain corresponding weighted estimates for weights in multilinear Muckenhoupt classes. Our proof relies on the normalized local estimate and stopping time argument developed in OpenAI's work.