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Multiple convolutions and multilinear fractal Fourier extension estimates

Itamar Oliveira, Ana E. de Orellana

TL;DR

This work studies multilinear Fourier extension estimates for fractal measures, aiming to surpass what linear fractal restriction would predict by exploiting transversality in the fractal setting. The authors prove a general sufficiency: if the $k$-fold convolution $\mu_1*\cdots*\mu_k$ lies in a Lebesgue space $L^{\frac{q(p-1)}{q(p-1)-p}}$, then the multilinear restriction bound $\|\widehat{f_1 d\mu_1}\cdots \widehat{f_k d\mu_k}\|_{L^q} \lesssim \prod_{m=1}^k \|f_m\|_{L^p(d\mu_m)}$ holds for $q\ge 2$, $p\ge 1$, generalizing Chen–Trainor’s framework. The paper then derives concrete corollaries for bilinear settings and self-similar Cantor-type measures, illustrating both when convolution densities exist and when they do not. A complementary set of results constructs multilinear Knapp-type examples using random Cantor measures to establish necessary conditions in dimension one, showing that fractal transversality can yield multilinear gains beyond linear theory even when the convolution is singular. Altogether, the work broadens the landscape of fractal restriction theory by identifying convolution-regularity and arithmetic-transversality mechanisms that enable new multilinear estimates.

Abstract

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.

Multiple convolutions and multilinear fractal Fourier extension estimates

TL;DR

This work studies multilinear Fourier extension estimates for fractal measures, aiming to surpass what linear fractal restriction would predict by exploiting transversality in the fractal setting. The authors prove a general sufficiency: if the -fold convolution lies in a Lebesgue space , then the multilinear restriction bound holds for , , generalizing Chen–Trainor’s framework. The paper then derives concrete corollaries for bilinear settings and self-similar Cantor-type measures, illustrating both when convolution densities exist and when they do not. A complementary set of results constructs multilinear Knapp-type examples using random Cantor measures to establish necessary conditions in dimension one, showing that fractal transversality can yield multilinear gains beyond linear theory even when the convolution is singular. Altogether, the work broadens the landscape of fractal restriction theory by identifying convolution-regularity and arithmetic-transversality mechanisms that enable new multilinear estimates.

Abstract

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 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.
Paper Structure (14 sections, 11 theorems, 135 equations, 1 figure)

This paper contains 14 sections, 11 theorems, 135 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Linear theory: smooth setting versus fractal setting
  3. Multilinear theory: smooth setting versus fractal setting
  4. Preliminaries
  5. Main Results
  6. Proof of Theorem \ref{['thm:mainthm']}
  7. A multilinear Knapp example for random Cantor measures
  8. Setup, arithmetic structure and linear independence
  9. Choosing $\tau_{N,m}$ and $t_{N,m}$
  10. Linear independence
  11. Choosing $d_{N,m}$
  12. Construction of Cantor sets
  13. Proof of Theorem \ref{['nec-sing']}
  14. Second construction of Cantor sets

Key Result

Theorem 1.1

Let $\mu$ be a finite, compactly supported, Borel measure on $\mathbb{R}^d$ such that for some $\alpha,\beta>0$, for all $x\in\mathbb{R}^d$ and $r>0$, and for all $\xi\in\mathbb{R}^d$. Then for all $f\in L^2(\mu)$ and $q\geq 2+ 4\frac{d-\alpha}{\beta}$,

Figures (1)

  • Figure 1: If $\sum_{m=1}^{k}\alpha_{m}<1$, the dark grey region represents the necessary conditions of Theorem \ref{['nec-sing']} intersected with $q\geq\frac{2}{\sum_{m=1}^{k}\alpha_{m}}$ from Proposition \ref{['prop:topLid']}, since \ref{['hyp:H1']} implies $\overline{\dim}_{\textup{B}}{\rm supp}(\mu_m) = \alpha_m$. Notice that the latter are more restrictive than the conditions from Proposition 5.3 of Trainor (represented by the union of the dark and light grey regions) in a setting with $k$ measures $\mu_{1},\ldots,\mu_{k}$ satisfying \ref{['hyp:H1']} and \ref{['hyp:H2']}.

Theorems & Definitions (25)

  • Conjecture 1
  • Theorem 1.1: Stein--Tomas
  • Conjecture 2: Benn1
  • Remark 1.1
  • Theorem 3.1
  • Corollary 3.2
  • Corollary 3.3
  • Example 3.4
  • Example 3.5
  • Theorem 3.6
  • ...and 15 more