The one-dimensional Mizohata–Takeuchi conjecture for the truncated parabola

Let $E$ be the extension operator for the truncated parabola, corresponding to $\Gamma(\xi)=\xi^2$, and let $\omega:B_R\to[0,\infty)$ be a one-dimensional weight, so that

$$\omega(B(y,r))\le r$$

for every ball of radius $r$ centered at $y$, with $1\le r\le R$. Parabolic one-dimensional Mizohata–Takeuchi conjecture. Then

$$\int_{B_R}|Eg|^2\omega\lesssim_\epsilon R^\epsilon\sup_T\omega(T)^{\frac12}\|g\|_2^2,$$

where $T$ ranges over $R\times R^{\frac12}$ tubes. This is the special quadratic case retained after the corresponding assertion for general convex curves is shown to fail; the source gives no resolution.