Mizohata–Takeuchi estimates in the plane

Abstract

Suppose that (Formula presented.) is a smooth compact hypersurface in (Formula presented.) and (Formula presented.) is an appropriate measure on (Formula presented.). If (Formula presented.) is the extension operator associated with (Formula presented.), then the Mizohata–Takeuchi conjecture asserts that (Formula presented.) for all functions (Formula presented.) and weights (Formula presented.), where the (Formula presented.) is taken over all tubes (Formula presented.) in (Formula presented.) of cross-section 1, and (Formula presented.). This paper investigates how far we can go in proving the Mizohata–Takeuchi conjecture in (Formula presented.) if we only take the decay properties of (Formula presented.) into consideration. As a consequence of our results, we obtain new estimates for a class of convex curves that include exponentially flat ones such as (Formula presented.), (Formula presented.), (Formula presented.). © 2023 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.