A Mizohata--Takeuchi-type estimate for the hyperbolic paraboloid

We prove Mizohata–Takeuchi-type estimates for the Fourier extension operator associated with the truncated hyperbolic paraboloid in ${\mathbb{R}}^3$. Our main result controls the weighted $L^2$ norm of the extension operator by the $L^2$ mass of the weight over a multiscale family of parallelepipeds arising from ${\mathrm{\ell }}^2$ decoupling for the hyperbolic paraboloid. We deduce this result from a weighted $L^2$ estimate for functions whose Fourier transforms are supported in a small neighborhood of the hyperbolic paraboloid, which we establish using a bilinear refined decoupling estimate.