Multiple convolutions and multilinear fractal Fourier extension estimates

The classical Stein–Tomas theorem extends the theory of linear Fourier restriction estimates from smooth manifolds to fractal measures exhibiting Fourier decay. In the multilinear setting, transversality allows for Fourier extension estimates that go beyond those implied by the linear theory to hold. We establish a multilinear Fourier extension estimate for measures whose convolution belongs to an $L^p$ space, applicable to known results by Shmerkin and Solomyak that exploit `transversality’ between self-similar measures. Moreover, we generalise work by Hambrook–Łaba and Chen from the linear setting to obtain Knapp-type examples for multilinear estimates; we obtain two necessary conditions: one in terms of the upper box dimension of the measures’ supports, and another one in terms of their Fourier decay and a ball condition. In particular, these conditions give a more restrictive range compared with previously known results whenever the convolution of the measures at play is singular.

💡 Research Summary

The paper investigates Fourier extension estimates in the setting of fractal measures, focusing on the multilinear regime where several measures interact simultaneously. It begins by recalling the classical Stein–Tomas theorem, which links curvature of a smooth hypersurface to decay of its surface measure’s Fourier transform and yields $L^{2}(\mu)\to L^{q}(\mathbb{R}^{d})$ restriction estimates for measures satisfying a Frostman condition and a Fourier decay bound. The authors point out that, beyond the $L^{2}$ range, known results for fractal measures are limited, typically relying on the assumption that the measure’s convolution power belongs to $L^{\infty}$.

In the multilinear context, transversality—quantified by a uniform lower bound on the $k$‑dimensional volume spanned by normal vectors—allows one to obtain estimates that are stronger than what would follow from applying Hölder’s inequality to the linear theory. The paper’s central contribution is Theorem 3.1, which provides a sufficient condition for multilinear Fourier extension estimates: if $\mu_{1},\dots,\mu_{k}$ are compactly supported finite Borel measures on $\mathbb{R}^{d}$ and their $k$‑fold convolution $\nu=\mu_{1}\cdots\mu_{k}$ is absolutely continuous with respect to Lebesgue measure and belongs to $L^{r}(\mathbb{R}^{d})$ for some $1\le r\le\infty$, then for any exponents $p_{j}\ge2$ and $q$ satisfying
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