Knapp-type obstructions in multilinear fractal Fourier extension

Abstract

For curved, smooth hypersurfaces, the classical Knapp example shows that the Stein--Tomas theorem, which gives linear Fourier restriction estimates, is sharp. Variants of this example combined with the geometric notion of \textit{transversality} motivate the $L^{2}$-based multilinear Fourier extension conjecture. In the fractal setting, work by Mockenhaupt, Mitsis, and Bak-Seeger extended the linear Fourier restriction estimate beyond the smooth setting, and subsequent work showed this extension to be sharp. In this article, we construct multilinear Knapp-type examples for fractal measures inspired by the works of Hambrook--{\L}aba and Chen. This yields two necessary conditions for a fractal multilinear Fourier extension estimate to hold: one in terms of the upper box dimension of the measures' supports, and another in terms of their Fourier decay and a ball condition. These conditions give a more restrictive range compared with previously known results whenever the convolution of the underlying measures is singular. In contrast, we complement this with a result in the positive direction by establishing a multilinear Fourier extension estimate for measures whose convolution lies in an $L^p$ space. This provides a rich class of examples of `transversal' self-similar measures through the work of Shmerkin and Solomyak.