Weighted norm inequalities for the Fourier extension operator via the X-ray tomography

Abstract

This note is an announcement of forthcoming paper [J. Bennett and S, Nakamura, Tomography bounds for the Fourier extension operator and applications, Math. Ann. 380 (2021), 119–159.] which is a work with Professor Jonathan Bennett (University of Birmingham) and so the main purpose is to exhibit results in [J. Bennett and S, Nakamura, Tomography bounds for the Fourier extension operator and applications, Math. Ann. 380 (2021), 119–159.] especially related to the weighted norm estimate for the Fourier extension operator known as Stein and Mizohata-Takeuchi conjectures. To these open problems in [J. Bennett and S, Nakamura, Tomography bounds for the Fourier extension operator and applications, Math. Ann. 380 (2021), 119–159.] we apply the approach using the X-ray tomography principle which has its origin in work of Planchon and Vega [F. Planchon, L. Vega, Bilinear virial identities and applications, Ann. Scient. Ec. Norm. Sup., 42 (2009), 263–292.]. We will explain our results with motivations and how to apply the tomography principle to the weighted norm estimate. We will also provide the explicit and detailed proof of Theorem 4.1 in [J. A. Barceló, J. Bennett, A. Carbery, A note on localised weighted estimates for the extension operator, J. Aust. Math. Soc. 84 (2008), 289–179.] by Barceló-Bennett-Carbery.