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

Abstract: 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.

• The paper introduces new Fourier restriction estimates based on Lq-dimensions to sharpen the classical Stein–Tomas bounds.
• It leverages convolution-based characterizations and complex interpolation to account for multifractal fluctuations in singular measures.
• Implications extend to geometric problems, offering tighter bounds for distance sets and related harmonic analysis challenges.

Fourier Restriction Estimates Based on $L^q$-Dimensions: Beyond Stein–Tomas

Context and Motivation

The restriction problem in harmonic analysis deals with determining for which exponents $p$ and $r$ the Fourier transform of functions, or more generally, measures, can be meaningfully restricted to lower-dimensional sets or measures. The classical Stein–Tomas theorem characterizes $L^p \rightarrow L^2$ restriction for the Lebesgue surface measure on the sphere and was subsequently generalized to singular measures by Mockenhaupt, Mitsis, and Bak–Seeger via the introduction of Frostman and Fourier decay conditions. The Frostman condition controls the measure of small balls and becomes the endpoint of a family of finer geometric conditions: the $L^q$-dimensions, which capture multifractal and local-mass fluctuation effects that are undetectable to Frostman-type analysis.

This work establishes new Fourier restriction theorems in terms of the $L^q$-dimensions of $\mu$, strictly refining the Stein–Tomas paradigm. The results interpolate between the (Frostman-based) endpoint and a multifractal continuum, fundamentally connecting harmonic analysis and fractal geometry (2606.07143). The technical core leverages convolution-based characterizations of $L^q$-dimensions and applies Stein’s complex interpolation, enabling endpoint results in this more nuanced setting.

Theoretical Framework

Restricting Fourier transforms to singular measures $\mu$ is only nontrivial in the range $1 < p' < 2$ (i.e., $p$0), since for $p$1 the transform is continuous, and for $p$2 restriction to a null set is always impossible. The classical Stein–Tomas theorem for singular measures provides an $p$3-based extension estimate,

$p$4

provided that, for all $p$5 and small $p$6, $p$7 (Frostman exponent) and $p$8 (Fourier decay), with

$p$9

Sharpness is achieved on very regular Salem-type sets or those with arithmetic structure, but such Frostman-based bounds generally ignore substantial local variation.

The $r$0-dimension,

$r$1

with varying $r$2, encodes $r$3-moment fluctuations. As $r$4, one recovers the Frostman exponent; as $r$5, the $r$6 (correlation) dimension. The main innovation lies in replacing the Frostman hypothesis with precise $r$7-dimensional conditions, yielding a family of restriction theorems that adapt to multifractality.

Figure 1: A multifractal measure supported on the Sierpiński carpet; dark regions denote areas of high local mass—the $r$8-dimensional approach is sensitive to such variance.

Main Results

General Restriction Theorem

Let $r$9 be a compactly supported Borel measure. Suppose for some $L^p \rightarrow L^2$0 and $L^p \rightarrow L^2$1

$L^p \rightarrow L^2$2

with $L^p \rightarrow L^2$3 and the Fourier spectrum parameter $L^p \rightarrow L^2$4, for suitable $L^p \rightarrow L^2$5. Then

$L^p \rightarrow L^2$6

implies

$L^p \rightarrow L^2$7

where $L^p \rightarrow L^2$8 can be optimized in $L^p \rightarrow L^2$9 and $L^q$0, recovering Stein–Tomas for $L^q$1, $L^q$2.

In explicit $L^q$3-dimension terms (see Corollary 1.2): $L^q$4 where $L^q$5 is the Fourier spectrum parameter. For multifractal measures, one may strictly improve $L^q$6 compared to the Frostman-only bound, whenever

$L^q$7

for some $L^q$8.

This provides a direct path to finer extension estimates whenever the measure is multifractal (e.g., non-uniformly distributed self-similar measures, random cascades).

Convolution Characterization of $L^q$9-dimensions

A notable technical contribution is the equivalence (for $L^q$0)

$L^q$1

with the extreme $L^q$2 case corresponding to the Frostman potential condition. This description is highly compatible with convolution-based and harmonic analysis arguments.

Numerical and Probabilistic Examples

The framework is validated by application to Mandelbrot multiplicative cascades—a canonical family of multifractal random measures arising in turbulence theory and probability:

• The $L^q$3-dimension and (recently computed) Fourier dimension of the cascade are known, allowing explicit evaluation of the restriction threshold as a function of $L^q$4.
• In subcritical or suitably tuned cases, the optimal value of $L^q$5 for the restriction exponent may be finite (i.e., not the Frostman endpoint), demonstrating substantial gains over Stein–Tomas.

Figure 2: Two realizations of a Mandelbrot cascade; each realization exhibits a hierarchy of local mass concentrations absent in uniform-Salem sets.

Figure 3: The threshold lower bound for $L^q$6 as a function of $L^q$7 for a Mandelbrot cascade; the value is optimized for an intermediate $L^q$8, with the dashed line marking the Stein–Tomas bound.

The authors also show that for any $L^q$9 one may construct cascades where the restriction estimate is optimized at that precise $\mu$0, underscoring the necessity of a full $\mu$1-dimensional continuum.

Figure 4: The bound for $\mu$2 can be tuned to reach its minimal value at a prescribed $\mu$3, depending on the choice of multiplicative weights in the cascade.

Implications and Future Directions

This work provides a direct, sharp conduit between harmonic analysis and geometric measure theory: harmonic analysts must now reckon with multifractal $\mu$4-dimensions, rather than relying wholly on extreme-case (Frostman) control. From a geometric perspective, it yields new constraints and insights into the possible behavior of singular measures with respect to Fourier analytic quantities.

Practical consequences include potentially tighter bounds for geometric problems relating to sumsets, distance sets, and projections, as these restriction estimates are critical input to those applications. The convolution-based framework also unifies the landscape, enabling endpoint results via complex interpolation for multifractal measures, partially resolving an open question of Bak and Seeger.

Possible future directions include:

• Extending these techniques to multilinear restriction, decoupling, or more exotic ambient groups;
• Investigating the role of $\mu$5-dimensions in distance set and Falconer-type problems;
• Exploring stochastic models (random cascades, random fractals) where the multifractal properties are pronounced.

Conclusion

This paper extends Fourier restriction theory by incorporating the full spectrum of $\mu$6-dimensions into the analysis, thus moving "beyond Stein–Tomas" and capturing the local and multifractal nature of singular measures. The results generalize and improve classical bounds in many cases, especially for multifractal supports. The convolution-norm perspective supplies a robust analytic handle for future work in restriction-type problems, and the methodology is broadly transferable across analysis and geometric measure theory (2606.07143).