Some sharp inequalities of Mizohata–Takeuchi-type

  • Anthony Carbery

    University of Edinburgh, Edinburgh, Scotland, UK
  • Marina Iliopoulou

    National and Kapodistrian University of Athens, Zografou, Greece
  • Hong Wang

    New York University, New York, USA
Some sharp inequalities of Mizohata–Takeuchi-type cover
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Abstract

Let be a strictly convex, compact patch of a hypersurface in , with non-vanishing Gaussian curvature and surface measure induced by the Lebesgue measure in . The Mizohata–Takeuchi conjecture states that

for all and all weights , where denotes the -ray transform. As partial progress towards the conjecture, we show, as a straightforward consequence of recently-established decoupling inequalities, that for every , there exists a positive constant , which depends only on and , such that for all and all weights , we have

where ranges over the family of tubes in of dimensions . From this we deduce the Mizohata–Takeuchi conjecture with an -loss; i.e., that

for any ball of radius and any . The power here cannot be replaced by anything smaller unless properties of beyond ‘decoupling axioms’ are exploited. We also provide estimates which improve this inequality under various conditions on the weight, and discuss some new cases where the conjecture holds.

Cite this article

Anthony Carbery, Marina Iliopoulou, Hong Wang, Some sharp inequalities of Mizohata–Takeuchi-type. Rev. Mat. Iberoam. 40 (2024), no. 4, pp. 1387–1418

DOI 10.4171/RMI/1463