Lower regularity well-posedness for a higher-order Schrödinger equation with cubic nonlinearities on the half-line

In this paper, we continue the study of Himonas and Yan\cite{himonas2024schrodinger,himonas2026higher,himonas2024higher} on the nonlinear Schrödinger equation with a dispersion of order $2m$ and cubic nonlinearities, $$i{u}_t+{(-1)}^{m+1}{\mathrm{\partial }}_x^{2m}u=N_k(u,u,u),$$ where, $m\ge 1$ being an integer, $$N_0(u,u,u)=uuu,N_1(u,u,u)=\bar{u}uu,N_2(u,u,u)=\bar{u}\bar{u}u,N_3(u,u,u)=\bar{u}\bar{u}\bar{u},$$ We derive trilinear estimates at lower regularity and thereby prove that the cNLS-2m on the half-line is well-posed at the optimal regularity $s=-\text{fr}m-12$. This improves the previous result, $s>-\text{fr}m-12$, and answers an open question left in \cite{himonas2026higher}. In addition, for $k=0,2$, we establish well-posedness at $s=-\text{fr}m-12$ as well. Moreover, for $k=3$, the nonlinearity exhibits a stronger resonance relation, which implies well-posedness for $s>-\text{fr}2m-13$. Our derivation of the trilinear estimates relies on varieties of Strichartz estimates, which differs from the $[k;Z]$-multiplier norm method employed in \cite{himonas2026higher}.