Variable-coefficient Mizohata–Takeuchi conjecture

Let $\mathcal T^\lambda$ be the operator defined earlier in the paper, let $B_\lambda$ be its spatial domain, and let $\{\Gamma_{\theta,v}^\lambda\}_{\theta,v}$ be the associated collection of curves. For a weight function $w:B_\lambda\to[0,\infty)$, define

$$\|X^\lambda w\|_{L^\infty}:=\sup_{\theta,v}\int_{\Gamma_{\theta,v}^\lambda}w(x)\,dx.$$

The supremum is over the curves in the collection $\{\Gamma_{\theta,v}^\lambda\}_{\theta,v}$. Variable-coefficient Mizohata–Takeuchi conjecture. For every $\varepsilon>0$, there exists a constant $C_\varepsilon>0$, independent of $w$, such that

$$\int_{B_\lambda}|\mathcal T^\lambda f(x)|^2w(x)\,dx\leq C_\varepsilon\lambda^\varepsilon\|X^\lambda w\|_{L^\infty}\int_{\mathbb{R}}|f(\xi)|^2\,d\xi.$$

This is proposed as a variable-coefficient analogue of the planar Mizohata–Takeuchi conjecture and is intended to connect the weighted estimate with the paper’s preceding theorem. The supplied text gives no resolution evidence.