Ciprian Demeter
We construct $δ$-regular sets with $δ\ge \frac12$ for which the analog of the Bourgain--Dyatlov Fractal Uncertainty Principle fails for the Walsh--Fourier transform.
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We prove that, for functions in the Orlicz class LloglogLloglogloglogL, lacunary subsequences of the Fourier and the Walsh-Fourier series converge almost everywhere. Our integrability condition is less stringent than the homologous assumption in the almost everywhere convergence theorems of Lie (Fourier case) and Do-Lacey (Walsh-Fourier case), where the quadruple logarithmic term is replaced by a triple logarithm. Our proof of the Walsh-Fourier case is self-contained and, in antithesis to Do and Lacey's argument, avoids the use of Antonov's lemma, arguing directly via novel weak-L^p bounds for…
In this paper we investigate strong summability of the two-dimensional Walsh-Fourier series obtained in Weisz \cite{We} (see Theorem W) and prove sharpness of this result.
In this paper we investigate some convergence and divergence of some specific subsequences of partial sums with respect to Walsh system on the martingale Hardy spaces. By using these results we obtain relationship of the ratio of convergence of the partial sums of the Walsh series with the modulus of continuity of martingale. These conditions are in a sense necessary and sufficient.
Almost everywhere strong exponential summability of Fourier series in Walsh and trigonometric systems established by Rodin in 1990. We prove, that if the growth of a function $Φ(t):[0,\infty)\to[0,\infty)$ is bigger than the exponent, then the strong $Φ$-summability of a Walsh-Fourier series can fail everywhere. The analogous theorem for trigonometric system was proved before by one of the author of this paper.
We prove a version of Carleson's Theorem in the Walsh model for vector-valued functions: For $1<p< \infty$, and a UMD space $Y$, the Walsh-Fourier series of $f \in L ^{p}(0,1;Y)$ converges pointwise, provided that $Y$ is a complex interpolation space $Y=[X,H]_θ$ between another UMD space $X$ and a Hilbert space $H$, for some $θ\in(0,1)$. Apparently, all known examples of UMD spaces satisfy this condition.
In 1987 Harris proved (Proc. Amer. Math. Soc., 101) - among others- that for each $1\le p<2$ there exists a two-dimensional function $f\in L^p$ such that its triangular Walsh-Fourier series diverges almost everywhere. In this paper we investigate the Fejér (or $(C,1)$) means of the triangle two variable Walsh-Fourier series of $L^1$ functions. Namely, we prove the a.e. convergence $σ_n^{\bigtriangleup}f = \frac{1}{n}\sum_{k=0}^{n-1}S_{k, n-k}f\to f$ ($n\to\infty$) for each integrable two-variable function $f$.
Approximation by a special de la Vallée Poussin type matrix transform mean of Walsh-Fourier series
该文研究Walsh系统下Fourier级数的特殊类de la Vallée Poussin矩阵平均的范数收敛，给出了用函数连续模表示的逼近误差上界。
We prove endpoint-type sparse bounds for Walsh-Fourier Marcinkiewicz multipliers and Littlewood-Paley square functions. These results are motivated by conjectures of Lerner in the Fourier setting. As a corollary, we obtain novel quantitative weighted norm inequalities for these operators. Among these, we establish the sharp growth rate of the $L^p$ weighted operator norm in terms of the $A_p$ characteristic in the full range $1<p<\infty$ for Walsh-Littlewood-Paley square functions, and a restricted range for Marcinkiewicz multipliers. Zygmund's $L{(\log L)^{\frac12}}$ inequality is the core of…
In this paper we derive the restricted weighted maximal operator, defined by ${\sup }_{k\in \mathbb{N}}\left(\left\vert σ_{k}F\right\vert/A^2_k\right)$ of Fejér means of Walsh-Fourier series and prove that the it is bounded from the martingale Hardy space $H_{1/2}(G)$ to the Lebesgue space $L_{1/2}(G).$ The sharpness of this result is also proved. As a consequence we obtain some new and and well-know results.
In this paper we derive the maximal subspace of natural numbers $\left\{n_{k}:k\geq 0\right\}$, such that the restricted maximal operator, defined by $\sup_{k\in \mathbb{N}}\left\vert σ_{n_{k}}F \right\vert$ on this subspace of Fejér means of Walsh-Fourier series is bounded from the martingale Hardy space $H_{1/2}$ to the Lebesgue space $L_{1/2}$. The sharpness of this result is also proved.
For 1<p<infty, and weight w in A_p, and function f in L^p(w), we show that the r-variation of the Walsh-Fourier sums are finite, for r sufficiently large as function of w. (That r is a function of w is necessary.) This strengthens a result of Hunt-Young and is a weighted extension of a variation norm Carleson theorem of Oberlin-Seeger-Tao-Thiele-Wright. The proof uses phase plane analysis and a weighted extension of a variational inequality of Lepingle.
We study the Walsh-Fourier series of S_{n_j}f, along a lacunary subsequence of integers {n_j}. Under a suitable integrability condition, we show that the sequence converges to f a.e. Integral condition is only slightly larger than what the sharp integrability condition would be, by a result of Konyagin. The condition is: f is in L loglog L (logloglog L). The method of proof uses four ingredients, (1) analysis on the Walsh Phase Plane, (2) the new multi-frequency Calderon-Zygmund Decomposition of Nazarov-Oberlin-Thiele, (3) a classical inequality of Zygmund, giving an improvement in the Hausdor…
We prove a weak-$L^p$ bound for the Walsh-Carleson operator for $p $ near 1, improving on a theorem of Sjolin. We relate our result to the conjectures that the Walsh-Fourier and Fourier series of a function $f\in L\log L(\mathbb T)$ converge for almost every $x \in \mathbb T$.
The properties of the maximal operator of the $(C,α)$-means ($α=(α_1,\ldots,α_d)$) of the multi-dimensional Walsh-Kaczmarz-Fourier series are discussed, where the set of indices is inside a cone-like set. We prove that the maximal operator is bounded from $H_p^γ$ to $L_p$ for $p_0< p $ ($p_0=\max{1/(1+α_k): k=1,\ldots, d}$) and is of weak type $(1,1)$. As a corollary we get the theorem of Simon \cite{S1} on the a.e. convergence of cone-restricted two-dimensional Fejér means of integrable functions. At the endpoint case $p=p_0$, we show that the maximal operator $σ^{κ,α,*}_L$ is not bounded fro…
The main aim of this paper is to investigate the quadratical partial sums of the two-dimensional Walsh-Fourier series.
In this PhD thesis we are dealing with convergence and summability of partial sums, Fejér and Marcinkiewicz means with respect to one- and two-dimensional Walsh-Fourier series on the martingale Hardy spaces. This thesis is focus to achieve the following main results: To find estimation of convergence and divergence of the subsequences of partial sums of the one-dimensional Walsh-Fourier series on the martingale Hardy spaces $H_p(G)$, when $0<p\leq1$. To find necessary and sufficient conditions in terms of modulus of continuity of martingale Hardy spaces, for which subsequences of partial sums …
We prove that certain mean of the quadratical partial sums of the two-dimensional Walsh-Fourier series are uniformly bounded operators from the Hardy space $H_{p}$ to the space $L_{p}$ for $0<p<1.$
We prove a variation norm Carleson theorem for Walsh-Fourier series of functions with values in a UMD Banach space. Our only hypothesis on the Banach space is that it has finite tile-type, a notion introduced by Hytönen and Lacey. Given q \geq 2 we show that, if the space X has tile-type t for all t>q, then the r-variation of the Walsh-Fourier sums of any function f \in L^p ([0,1) ; X) belongs to L^p, whenever q<r \leq \infty and (r/(q-1))' < p < \infty. We also show that if this conclusion is true for a variant of the variational Carleson operator then the space X necessarily has tile-type t …
A class of increasing sequences of natural numbers $(n_k)$ is found for which there exists a function $f\in L[0,1)$ such that the subsequence of partial Walsh-Fourier sums $(S_{n_k}(f))$ diverge everywhere. A condition for the growth order of a function $φ:[0,\infty)\rightarrow[0,\infty)$ is given fulfilment of which implies an existence of above type function $f$ in the class $φ(L)[0,1)$.
Almost everywhere divergence of Cesaro means of subsequences of Walsh--Paley Fourier partial sums
证明存在严格递增正整数序列和L^1函数，使得Walsh-Fourier部分和子序列的Cesàro均值几乎处处发散，揭示了二进结构下仍存在发散现象。
We prove that Cesàro means of one-dimensional Walsh-Fourier series are uniformly bounded operators in the martingale Hardy space $H_{p}$ for $% 0<p<1/\left( 1+α\right).$
In this paper we study the a. e. strong convergence of the quadratical partial sums of the two-dimensional Walsh-Fourier series. Namely, we prove the a.e. relation $(\frac{1}{n}\sum\limits_{m=0}^{n-1}\left\vert S_{mm}f - f \right\vert^{p})^{1/p}\rightarrow 0$ for every two-dimensional functions belonging to $L\log L$ and $0<p\le 2$. From the theorem of Getsadze \cite{Gets} it follows that the space $L\log L$ can not be enlarged with preserving this strong summability property.
We prove that there exists a martingale $f\in H_p $ such that the subsequence $\{L_{2^n}f \}$ of Nörlund logarithmic means with respect to the Walsh system are not bounded in the Lebesgue space $weak-L_p $ for $0<p<1 $. Moreover, we prove that for any $f\in L_p(G),$ $p\geq 1, $ $L_{2^n}f$ converge to $f$ at any Lebesgue point $x$. Some new related inequalities are derived.
Dyadic Martingale Transforms and Weighted Walsh-Carleson Operators
本文研究了由双侧马丁格尔变换产生的加权瓦尔什-卡拉森极大算子，证明了在满足均匀二进制变化条件和顶部二进制尺度上均匀界的情况下，这些算子在子序列上的弱型(1,1)估计，并给出了与权重在顶部二进制尺度附近行为相关的发散准则。
In this paper we derive a new strong convergence theorem of Riesz logarithmic means of the one-dimensional Vilenkin-Fourier (Walsh-Fourier) series. The corresponding inequality is pointed out and it is also proved that the inequality is in a sense sharp, at least for the case with Walsh-Fourier series.
We prove $L^p$ estimates for the Walsh model of the maximal bi-Carleson operator (which is a hybrid of the bilinear Hilbert transform and the Carleson maximal operator which appears naturally in the eigenfunction problem for one-dimensional Schrodinger operators). The corresponding estimates for the Fourier model will be obtained in the sequel of this paper.
We prove L^p estimates for the Walsh model of the "biest", a trilinear multiplier with singular symbol which arises naturally in the expansion of eigenfunctions of a Schrodinger operator, and which is also related to the bilinear Hilbert transform. The corresponding estimates for the Fourier model will be obtained in the sequel of this paper.
As main result we prove that Fejér means of Walsh-Fourier series are uniformly bounded operators from $\ H_{p}$ to $H_{p}$ $\left( 0<p\leq 1/2\right) .
For spherical and parabolic averages of the Fourier transform of fractal measures, we obtain new upper bounds on rates of decay by an "intermediate dimension" trick.
By considering a Moran-type construction of fractals on $[0,1]$, we show that for any $0\le s\le 1$, there exists some Moran fractal set, which is perfect, with Hausdorff dimension $s$ whose Fourier dimension is zero and it contains arbitrarily long arithmetic progressions.
In a recent paper [Ann. of Math. 189 (2019), 837--861], Du and Zhang proved a fractal Fourier restriction estimate and used it to establish the sharp $L^2$ estimate on the Schrödinger maximal function in $\Bbb R^n$, $n \geq 2$. In this paper, we show that the Du-Zhang estimate is the endpoint of a family of fractal restriction estimates such that each member of the family (other than the original) implies a sharp Kakeya result in $\Bbb R^n$ that is closely related to the polynomial Wolff axioms. We also prove that all the estimates of our family are true in $\Bbb R^2$.
Fourier decay of fractal measures on surfaces of co-dimension two in $\mathbb{R}^5$
本文研究了高余维二次曲面上分形测度的 Fourier 衰减问题，采用 Du 和 Zhang 的策略及宽窄分析，获得了 $\mathbb{R}^5$ 中余维为 2 的二次曲面上 Fourier 衰减的若干下界。
We prove L^p estimates for the "biest", a trilinear multiplier with singular symbol which arises naturally in the expansion of eigenfunctions of a Schrodinger operator, and which is also related to the bilinear Hilbert transform. In a previous paper these estimates were obtained for a simpler Walsh model for this operator, but in the Fourier case additional complications arise due to the inability to perfectly localize in both space and frequency.
Harmonic Analysis on the Space of M-positive Vectors
本文在由伸缩矩阵M定义的M-正向量空间上建立调和分析框架，引入Walsh函数类似物和Fourier变换，证明Poisson求和公式、Plancherel定理等，并刻画了函数及其Fourier变换均具紧支撑的Schwartz类类似物。
Fractal Uncertainty and Quantitative Uniqueness for the Fourier Bessel Transform
针对傅里叶-贝塞尔变换，研究人员建立了其定量唯一性与分形不确定性原理，通过适配正则集合的阻尼函数及多尺度迭代等方法，克服了经典傅里叶论证无法直接应用的问题，得到了相关函数估计式。
Let $μ$ be an $α$-dimensional probability measure. We prove new upper and lower bounds on the decay rate of hyperbolic averages of the Fourier transform $\widehatμ$. More precisely, if $\mathbb{H}$ is a truncated hyperbolic paraboloid in $\mathbb{R}^d$ we study the optimal $β$ for which $$\int_{\mathbb{H}} |\hatμ(Rξ)|^2 \, d σ(ξ)\leq C(α, μ) R^{-β}$$ for all $R > 1$. Our estimates for $β$ depend on the minimum between the number of positive and negative principal curvatures of $\mathbb{H}$; if this number is as large as possible our estimates are sharp in all dimensions.
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 $E_x=E\c…
We establish estimates for restrictions to certain curves in R^2 of the Fourier transforms of some fractal measures.
This paper builds upon two key principles behind the Bourgain-Dyatlov quantitative uniqueness theorem for functions with Fourier transform supported in an Ahlfors regular set. We first provide a characterization of when a quantitative uniqueness theorem holds for functions with very quickly decaying Fourier transform, thereby providing an extension of the classical Paneah-Logvinenko-Sereda theorem. Secondly, we derive a transference result which converts a quantitative uniqueness theorem for functions with fast decaying Fourier transform to one for functions with Fourier transform supported on…
We use a restriction theorem for Fourier transforms of fractal measures to study projections onto families of planes in R^3 whose normal directions form nondegenerate curves.
Polynomial Fourier decay for fractal measures and their pushforwards
本文证明一大类分形测度在非线性映射前推下具有多项式傅里叶衰减，并由此推出非仿射解析压缩迭代函数系统的非原子自共形测度也具有该衰减性质，进而给出在傅里叶唯一性、分形不确定性原理、限制估计和定量均匀分布等方面的应用。
Knapp-type obstructions in multilinear fractal Fourier extension
本文研究了多线性分形傅里叶扩展估计，提出了一种适用于卷积属于$ 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。
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…
We show that subsets of $\mathbb{R}^n$ of large enough Hausdorff and Fourier dimension contain polynomial patterns of the form \begin{align*} ( x ,\, x + A_1 y ,\, \dots,\, x + A_{k-1} y ,\, x + A_k y + Q(y) e_n ), \quad x \in \mathbb{R}^n,\ y \in \mathbb{R}^m, \end{align*} where $A_i$ are real $n \times m$ matrices, $Q$ is a real polynomial in $m$ variables and $e_n = (0,\dots,0,1)$.
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。
$L^p$ 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。
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…
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 $\dot{H}^1(\mathbb{R}^d)\times L^2(\mathbb{R}^d)$.
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 …
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.
We introduce a duality for Affine Iterated Function Systems (AIFS) which is naturally motivated by group duality in the context of traditional harmonic analysis. Our affine systems yield fractals defined by iteration of contractive affine mappings. We build a duality for such systems by scaling in two directions: fractals in the small by contractive iterations, and fractals in the large by recursion involving iteration of an expansive matrix. By a fractal in the small we mean a compact attractor X supporting Hutchinson's canonical measure mu, and we ask when mu is a spectral measure, i.e., whe…
We establish a version of the fractal uncertainty principle, obtained by Bourgain and Dyatlov in 2016, in higher dimensions. The Fourier support is limited to sets $Y\subset \mathbb{R}^d$ which can be covered by finitely many products of $δ$-regular sets in one dimension, but relative to arbitrary axes. Our results remain true if $Y$ is distorted by diffeomorphisms. Our method combines the original approach by Bourgain and Dyatlov, in the more quantitative 2017 rendition by Jin and Zhang, with Cartan set techniques.
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。
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。
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。
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).
In this paper, we investigate the fractal uncertainty principle (FUP) for discrete Cantor sets, which are determined by an alphabet from a base of digits. Consider the base of M digits and the alphabets of cardinality A such that all the corresponding Cantor sets have a fixed dimension 0<log A/log M<2/3. We prove that the FUP with an improved exponent over Dyatlov-Jin (arXiv:1608.02238) holds for almost all alphabets, asymptotically as M tends to infinity. Our result provides the best possible exponent when the Cantor sets enjoy either the strongest Fourier decay assumption or strongest additi…
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