Fourier restriction in low fractal dimensions

Abstract

Let be a smooth compact hypersurface with a strictly positive second fundamental form, be the Fourier extension operator on, and be a Lebesgue measurable subset of. If contains a ball of each radius, then the problem of determining the range of exponents for which the estimate holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set: there is a number <![CDATA[0 such that for all balls in of radius. On the left-hand side of this estimate, we are integrating the function against the measure. Our approach consists of replacing the characteristic function of by an appropriate weight function, and studying the resulting estimate in three different regimes: small values of, intermediate values of, and large values of. In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted Hölder-type inequality that holds for general non-negative Lebesgue measurable functions on and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du-Zhang theorem in the range <![CDATA[0 < \alpha. Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.