Global Mizohata-Takeuchi for Indicator Weights of Bounded Complexity in ${\mathbb{R}}^2$

We prove the global Mizohata-Takeuchi estimate for bounded complexity weights in ${\mathbb{R}}^2$ and an arbitrary $C^2$ uniformly convex curve on the Fourier side, with an implied constant depending only on the complexity and the curve. A version with a logarithmic loss follows from a short argument using the $T^{\ast }T$ method and the main estimate of Basu, Guo, Zhang, and Zorin-Kranich (2026). The main effort of this paper is devoted to removing the logarithmic factor. To achieve this, we use tools from semialgebraic geometry and adapt and develop ideas of Ricci and Stein (1987). As a by-product, we obtain a uniform geometric inequality for planar regions of bounded complexity that may be of independent interest.