Upper bounds for Fourier decay rates of fractal measures

$L^p$ Fourier asymptotics, Hardy type inequality and fractal measures

Suppose $μ$ is an $α$-dimensional fractal measure for some $0<α<n$. Inspired by the results proved by R. Strichartz in 1990, we discuss the $L^p$-asymptotics of the Fourier transform of $fdμ$ by estimating bounds of $$\underset{L\rightarrow\infty}{\liminf}\ \frac{1}{L^k} \int_{|ξ|\leq L}\ |\widehat{fdμ}(ξ)|^pdξ,$$ for $f\in L^p(dμ)$ and $2<p<2n/α$. In a different direction, we prove a Hardy type inequality, that is, $$\int\frac{|f(x)|^p}{(μ(E_x))^{2-p}}dμ(x)\leq C\ \underset{L\rightarrow\infty}{\liminf} \frac{1}{L^{n-α}} \int_{B_L(0)} |\widehat{fdμ}(ξ)|^pdξ$$ where $1\leq p\leq 2$ and $E_x=E\c…

Restricting Fourier transforms of measures to curves in R^2

We establish estimates for restrictions to certain curves in R^2 of the Fourier transforms of some fractal measures.

Quantitative uniqueness properties for $L^2$ functions with fast decaying, or sparsely supported, Fourier transform

This paper builds upon two key principles behind the Bourgain-Dyatlov quantitative uniqueness theorem for functions with Fourier transform supported in an Ahlfors regular set. We first provide a characterization of when a quantitative uniqueness theorem holds for functions with very quickly decaying Fourier transform, thereby providing an extension of the classical Paneah-Logvinenko-Sereda theorem. Secondly, we derive a transference result which converts a quantitative uniqueness theorem for functions with fast decaying Fourier transform to one for functions with Fourier transform supported on…

分形测度及其前推的多项式傅里叶衰减

Polynomial Fourier decay for fractal measures and their pushforwards

本文证明一大类分形测度在非线性映射前推下具有多项式傅里叶衰减，并由此推出非仿射解析压缩迭代函数系统的非原子自共形测度也具有该衰减性质，进而给出在傅里叶唯一性、分形不确定性原理、限制估计和定量均匀分布等方面的应用。

Application of a Fourier restriction theorem to certain families of projections in R^3

We use a restriction theorem for Fourier transforms of fractal measures to study projections onto families of planes in R^3 whose normal directions form nondegenerate curves.

多卷积与多线性分形傅里叶扩展估计

Knapp-type obstructions in multilinear fractal Fourier extension

本文研究了多线性分形傅里叶扩展估计，提出了一种适用于卷积属于$ L^p $空间的测度的估计方法，并获得了Knapp型例子及两个必要条件。

$L^p$-Asymptotics of Fourier transform of fractal measures

One of the basic questions in harmonic analysis is to study the decay properties of the Fourier transform of measures or distributions supported on thin sets in $\mathbb{R}^n$. When the support is a smooth enough manifold, an almost complete picture is available. One of the early results in this direction is the following: Let $f\in C_c^{\infty}(\mathbb{R}^n)$ and $dσ$ be the surface measure on the sphere $S^{n-1}\subset\mathbb{R}^n$. Then $$|\widehat{fdσ}(ξ)|\leq\ C\ (1+|ξ|)^{-\frac{n-1}{2}}.$$ It follows that $\widehat{fdσ}\in L^p(\mathbb{R}^n)$ for all $p>\frac{2n}{n-1}$. This result can be…

低分形维数中的傅里叶限制

Fourier restriction in low fractal dimensions

本文研究低分形维数集合上的傅里叶限制估计，通过替换权函数并分三个区间处理，改进Du-Zhang定理在0<α<n/2范围的结果。

Average decay of the Fourier transform of measures with applications

We consider spherical averages of the Fourier transform of fractal measures and improve both the upper and lower bounds on the rate of decay. Maximal estimates with respect to fractal measures are deduced for the Schrödinger and wave equations. This refines the almost everywhere convergence of the solution to its initial datum as time tends to zero. A consequence is that the solution to the wave equation cannot diverge on a $(d-1)$-dimensional manifold if the data belongs to the energy space $\dot{H}^1(\mathbb{R}^d)\times L^2(\mathbb{R}^d)$.

$\mathbb{R}^2$ 中分形测度的圆周平均的 $L^p$ 衰减估计

$L^p$ Decay Estimates for Circular Means of Fractal Measures in $\mathbb{R}^2$

本文针对平面曲线上的傅里叶延拓算子建立加权 $L^p\rightarrow L^q$ 限制估计，并由此得到 $\mathbb{R}^2$ 中分形测度圆周平均的 $L^p$ 衰减估计，同时构造例子给出 $p\in[1,2]$ 时衰减指数的上界。

Large sets without Fourier restriction theorems

We construct a function that lies in $L^p(\mathbb{R}^d)$ for every $p \in (1,\infty]$ and whose Fourier transform has no Lebesgue points in a Cantor set of full Hausdorff dimension. We apply Kovač's maximal restriction principle to show that the same full-dimensional set is avoided by any Borel measure satisfying a nontrivial Fourier restriction theorem. As a consequence of a near-optimal fractal restriction theorem of Łaba and Wang, we hence prove a lack of valid relations between the Hausdorff dimension of a set and the range of possible Fourier restriction exponents for measures supported i…

傅里叶支撑在矩曲线上分形集上的函数的$L^{p}$可积性

$L^{p}$-integrability of functions with Fourier supports on fractal sets on the moment curve

本研究针对矩曲线上满足特定覆盖数条件的分形紧集$E$，证明了傅里叶支撑在$E$上的$L^p$函数在特定$p$范围内恒为零，验证了$p$范围的最优性，推广了已有结果并给出相关应用。

Fourier integral operators, fractal sets and the regular value theorem

We prove that if ${\mathcal E} \subset {\Bbb R}^{2d}$, $d \ge 2$, is an Ahlfors-David regular product set of sufficiently large Hausdorff dimension, denoted by $dim_{\mathcal H}({\mathcal E})$, and $ϕ$ is a sufficiently regular function, then the upper Minkowski dimension of the set $$ \{w \in {\mathcal E}: ϕ_l(w)=t_l; 1 \leq l \leq m \}$$ does not exceed $dim_{\mathcal H}({\mathcal E})-m$, in line with the regular value theorem from the elementary differential geometry. Our arguments are based on the mapping properties of the underlying Fourier Integral Operators and are intimately connected …

带有分数集缩放的球形最大算子

Spherical maximal operators with fractal sets of dilations on radial functions

研究了在径向函数上，具有分数集缩放的球形最大算子在勒贝格空间中的映射性质，确定了高维情况下类型集仅依赖于E的上闵可夫斯基维度，并在二维情况下通过与E的上阿索夫谱相关的维度谱确定了L^p→L^q类型集的闭包。

A class of random Cantor measures, with applications

We survey some of our recent results on the geometry of spatially independent martingales, in a more concrete setting that allows for shorter, direct proofs, yet is general enough for several applications and contains the well-known fractal percolation measure. We study self-convolutions and Fourier decay of measures in our class, and present applications of these results to the restriction problem for fractal measures, and the connection between arithmetic structure and Fourier decay.

Upper estimates for stable dimensions of fractal sets with variable number of foldings

For a hyperbolic map f on a saddle type fractal Lambda with self-intersections, the number of f- preimages of a point x in Lambda may depend on x. This makes estimates of the stable dimensions more difficult than for diffeomorphisms or for maps which are constant-to-one. We employ the thermodynamic formalism in order to derive estimates for the stable Hausdorff dimension function delta^s on Lambda, in the case when f is conformal on local stable manifolds. These estimates are in terms of a continuous function on Lambda which bounds the preimage counting function from below. As a corollary we o…

Fourier decay bound and differential images of self-similar measures

In this note, we investigate $C^2$ differential images of the homogeneous self-similar measure associated with an IFS $\mathcal{I}=\{ρx+a_j\}_{j=1}^m$ satisfying the strong separation condition and a positive probability vector $\vec{p}$. It is shown that the Fourier transforms of such image measures have power decay for any contractive ratio $ρ\in (0, 1/m)$, any translation vector $\vec{a}=(a_1, \ldots, a_m)$ and probability vector $\vec{p}$, which extends a result of Kaufman on Bernoulli convolutions. Our proof relies on a key combinatorial lemma originated from Erdős, which is important in …

关于 fractal 集合中 3APs 的数量

On the number of 3APs in fractal sets

本文研究 fractal 集合中三元等差数列的步长长度集合的大小问题，提出基于测度质量下界和傅里叶变换范数上界的新结果。

Lower S-Dimension of Fractal Sets

The interrelations between (upper and lower) Minkowski contents and (upper and lower) surface area based contents (S-contents) as well as between their associated dimensions have recently been investigated for general sets in R^d (cf. [3]). While the upper dimensions always coincide and the upper contents are bounded by each other, the bounds obtained in [3] suggest that there is much more flexibility for the lower contents and dimensions. We show that this is indeed the case. There are sets whose lower S-dimension is strictly smaller than their lower Minkowski dimension. More precisely, given…

关于抛物线上分形测度的傅里叶变换

On Fourier transforms of fractal measures on the parabola

本文研究抛物线上满足 Frostman 条件的分形测度的傅里叶变换衰减估计，证明其 $L^p$ 范数受 $R^{(2-t)/p}$ 控制，并由此导出抛物线上 Borel 集的 $n$ 重和集 Hausdorff 维数下界。

傅里叶比：关于紧支撑测度的不确定性、限制与近似

The Fourier Ratio: Uncertainty, Restriction, and Approximation for Compactly Supported Measures

本文提出傅里叶比概念，用于研究紧支撑测度的不确定性、限制与近似问题，揭示了信号恢复与分形测度之间的差异。

基于$L^2$-平坦化的傅里叶衰减

Fourier Decay from $L^2$-Flattening

本文提出一种利用$L^2$-平坦化定理和非集中性估计的统一方法，为动力学定义测度建立傅里叶变换衰减率，并应用于双曲流形谱隙与分形不确定性原理。

横截族的两端弗斯滕伯格不等式及其在傅里叶衰减中的应用

Two-ends Furstenberg inequality for transversal families and applications to Fourier decay

该研究将两端弗斯滕伯格不等式从直线推广到横截曲线并简化证明，还给出相关证明，应用于凸曲线上分形测度的傅里叶衰减研究，取得相应成果。

分形傅里叶级数的Hölder连续性与维数

Hölder continuity and dimensions of fractal Fourier series

该研究针对一类分形傅里叶级数，通过部分和一致界条件推导其Hölder连续性指数与盒计数维数上界，复现了经典函数的最优结果，并在广义黎曼假设下得到含莫比乌斯函数的级数的非平凡界。

Fourier decay in nonlinear dynamics

We study when Fourier transforms of Gibbs measures of sufficiently nonlinear expanding Markov maps decay at infinity at a polynomial rate. Assuming finite Lyapunov exponent, we reduce this to a nonlinearity assumption, which we verify for the Gauss map using Diophantine analysis. Our approach uses large deviations and additive combinatorics, which combines the earlier works on the Gibbs measures for Gauss map (Jordan-Sahlsten, 2013) and Fractal Uncertainty Principle (Bourgain-Dyatlov, 2017).

自相似集上自相似测度傅里叶变换衰减率的显式上界

Explicit Upper Bounds on Decay Rates of Fourier Transforms of Self-similar Measures on Self-similar Sets

本文研究自相似集上自相似概率测度的傅里叶变换，导出了衰减率的显式上界以改进先前研究，并应用于Lüroth表示数字受限的数集。