New Fourier Restriction Theorem Utilizing L^q-Dimensions Beyond Stein-Tomas
This paper introduces a novel Fourier restriction theorem that replaces the Frostman condition with $L^q$-dimensions, thereby offering a continuous range of estimates that encompass the Stein-Tomas theorem at its endpoint. The proof employs Stein's complex interpolation and yields endpoint estimates for $q \in (1, \infty]$, partially addressing a question by Bak and Seeger for $q=\infty$. The developed theorem offers improvements over Stein-Tomas for various measures, including Mandelbrot cascades and multifractal measures, and introduces a new characterization of $L^q$-dimensions via convolution norms.