Endpoint Christ–Kiselev estimate for the derivative nonlinear Schrödinger equation

Let $S(t)$ be the linear propagator, let $\mathscr{A}f(t)=\int_{\mathbb{R}}S(t-\tau)f(\tau)\,d\tau$ be the associated full-time integral operator, and let $\Box_k$ be the frequency-localization operator. For a function $f$ on $\mathbb{R}^{1+n}$, write $D^{1/2}_{x_2}$ and $D^{-1/2}_{x_1}$ for the corresponding fractional derivatives. Endpoint estimate conjecture.

$$\left\|\Box_k\partial_{x_2}\mathscr{A}f\right\|_{L^\infty_{x_1}L^2_{x_2,\ldots,x_n}L^2_t(\mathbb{R}^{1+n})} \lesssim \left\|D^{1/2}_{x_2}D^{-1/2}_{x_1}\Box_kf\right\|_{L^1_{x_2}L^2_{x_1,x_3,\ldots,x_n}L^2_t(\mathbb{R}^{1+n})}.$$

If this endpoint estimate holds, the authors state that the related estimate holds for all $\sigma\geqslant 1/2$, improving the regularity assumptions in their derivative nonlinear Schrödinger equation results. They do not know whether the Christ–Kiselev lemma applies at the endpoint, so the conjecture remains open.