arXiv subjects

Ana E. de Orellana

Publications and source records attributed to Ana E. de Orellana.

6 recordsLinked to original sources

arXiv2026

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.

math.CA↗

arXiv2026

Fourier restriction estimates based on $L^q$-dimensions: beyond Stein--Tomas

The well-known Stein--Tomas restriction theorem gives the sharp range of $p$ for which $L^p\to L^2$ restriction estimates hold for the surface measure on the sphere. This was generalised to arbitrary measures satisfying certain Fourier decay and Frostman conditions by Mockenhaupt, Mitsis, and Bak--Seeger, with the most general version now a fundamental result in harmonic analysis. The Frostman condition essentially asks for uniform control on the measure of small balls and is the endpoint of a continuum of more nuanced conditions which describe the local fluctuations of the measure. This analysis gives rise to the $L^q$-dimensions of a measure and these are a central concept in fractal geometry and a crucial tool in multifractal analysis and the theory of large deviations. In this paper we prove a new Fourier restriction theorem which uses the $L^q$-dimensions instead of the Frostman condition, thus providing a continuum of estimates which recover Stein--Tomas at the endpoint. Our proof gives the endpoint estimate for all values of $q\in(1,\infty]$ via Stein's complex interpolation. In particular, in the case $q=\infty$ this partially resolves a question raised by Bak and Seeger. We explore when our theorem improves on Stein--Tomas, that is, when the range is not optimised at $q=\infty$, and show that this is the case quite generally, including for certain Mandelbrot cascade measures and measures with multifractal behaviour. On the way to proving our main theorem we obtain a novel description of the $L^q$-dimensions based on certain convolution norms, which may be of interest in its own right.

math.CA↗

arXiv2026

Fourier restriction for the additive Brownian sheet

The Fourier restriction problem asks when it is meaningful to restrict the Fourier transform of a function to a given set. Many of the key examples are smooth co-dimension 1 manifolds, although there is increasing interest in fractal sets. Here we propose a natural intermediary problem where one considers the fractal surface generated by the graph of the additive Brownian sheet in $\mathbb{R}^k$. We obtain the first non-trivial estimates in this direction, giving both a sufficient condition on the range of $q\in[1,2]$ for the Fourier transform to be $L^{q}(\mathbb{R}^{k+1})\to L^2(G(W))$ bounded and a necessary condition for it to be $L^{q}(\mathbb{R}^{k+1})\to L^p(G(W))$ bounded. The sufficient condition is obtained via the Fourier spectrum, which is a family of dimensions that interpolate between the Fourier and Hausdorff dimensions. Our main technical result, which is of interest in its own right, gives a precise formula for the Fourier spectrum of the natural measure on the graph of the additive Brownian sheet, and we apply this result to the Fourier restriction problem. Our restriction estimate is stronger than the estimate obtained from the well-known Stein--Tomas restriction theorem for all $k\geq3$. We obtain the necessary condition in two different ways, one via the Fourier spectrum and one via an appropriate Knapp example.

math.PR↗

arXiv2025

$L^2$ restriction estimates from the Fourier spectrum

The Stein--Tomas restriction theorem is an important result in Fourier restriction theory. It gives a range of $q$ for which $L^q\to L^2$ restriction estimates hold for a given measure, in terms of the Fourier and Frostman dimensions of the measure. We generalise this result by using the Fourier spectrum; a family of dimensions that interpolate between the Fourier and Sobolev dimensions for measures. This gives us a continuum of Stein--Tomas type estimates, and optimising over this continuum gives a new $L^q\to L^2$ restriction theorem which often outperforms the Stein--Tomas result. We also provide results in the other direction by giving a range of $q$ in terms of the Fourier spectrum for which $L^q\to L^2$ restriction estimates fail, generalising an observation of Hambrook and Łaba. We illustrate our results with several examples, including the surface measure on the cone, the moment curve, and several fractal measures.

math.CA↗

arXiv2024

A Fourier analytic approach to exceptional set estimates for orthogonal projections

Marstrand's celebrated projection theorem gives the Hausdorff dimension of the orthogonal projection of a Borel set in Euclidean space for almost all orthogonal projections. It is straightforward to see that sets for which the Fourier and Hausdorff dimension coincide have no exceptional projections, that is, \emph{all} orthogonal projections satisfy the conclusion of Marstrand's theorem. With this in mind, one might believe that the Fourier dimension (or at least, Fourier decay) could be used to give better estimates for the Hausdorff dimension of the exceptional set in general. We obtain projection theorems and exceptional set estimates based on the Fourier spectrum; a family of dimensions that interpolates between the Fourier and Hausdorff dimensions. We apply these results to show that the Fourier spectrum can be used to improve several results for the Hausdorff dimension in certain cases, such as Ren--Wang's sharp bound for the exceptional set in the plane, Peres--Schlag's exceptional set bound and Bourgain--Oberlin's sharp $0$-dimensional exceptional set estimate.

math.CA↗

arXiv2024

Obtaining the Fourier spectrum via Fourier coefficients

The Fourier spectrum is a family of dimensions that interpolates between the Fourier and Hausdorff dimensions and are defined in terms of certain energies which capture Fourier decay. In this paper we obtain a convenient discrete representation of those energies using the Fourier coefficients. As an example application, we use this representation to establish sharp bounds for the Fourier spectrum of a general measure with bounded support, improving previous estimates of the second-named author

math.CA↗