Knapp-type obstructions in multilinear fractal Fourier extension

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--Łaba 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.