MAXIMAL RESTRICTION ESTIMATES AND THE MAXIMAL FUNCTION OF THE FOURIER TRANSFORM

We prove inequalities concerning the restriction of the strong maximal function of the Fourier transform to the circle, providing an answer to a question left open by Muller, Ricci, and Wright. We employ methods similar in spirit to the classical proofs of the two-dimensional restriction theorem, with the addition of a suitable trick to help us linearise our maximal function. In the end, we comment on how to use the same linearisation trick in combination with Vitturi's duality argument to obtain sharper high-dimensional results for the Hardy-Littlewood maximal function.