L. Brandolini, S. Hofmann, A. Iosevich
We prove that the spherical mean of the Fourier transform of the characteristic function of a bounded convex set (without any additional assumptions) or a bounded set with a C^{3/2} boundary decays at infinity at the same rate as the Fourier transform of the characteristic function of the ball.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We present a restriction theorem for the Fourier transform to a 2-dimensional conical surface of finite type, obtaining a sharp result, which improves previous work by Barcelo.
On a Sharp Fourier Extension Inequality on the Circle with Lacunary Spectrum
本文证明了圆上对满足谱间隔大于三倍的函数，在Tomas-Stein指数下成立尖锐的傅里叶延拓不等式。
We construct Salem sets on the real line with endpoint Fourier decay and near-endpoint regularity properties. This complements a result of Łaba and Pramanik, who obtained near-endpoint Fourier decay and endpoint regularity properties. We then modify the construction to extend a theorem of Hambrook and Łaba to show sharpness of the $L^2$-Fourier restriction estimate by Mockenhaupt and Bak-Seeger, including the case where the Hausdorff and Fourier dimension do not coincide.
Sharp Fourier decay estimates for measures supported on the well-approximable numbers
本文在良逼近数上构造了傅里叶变换近乎最优衰减的测度，从而对Kaufman的先前构造实现了对数级别的改进。
We prove qualitatively sharp heat kernel bounds in the setting of Fourier-Bessel expansions when the associated type parameter $ν$ is half-integer. Moreover, still for half-integer $ν$, we also obtain sharp estimates of all kernels subordinated to the heat kernel. Analogous estimates for general $ν> -1$ are conjectured. Some consequences concerning the related heat semigroup maximal operator are discussed.
The heat kernel in the setting of classical Fourier-Bessel expansions is defined by an oscillatory series which cannot be computed explicitly. We prove qualitatively sharp estimates of this kernel. Our method relies on establishing a connection with a situation of expansions based on Jacobi polynomials and then transferring known sharp bounds for the related Jacobi heat kernel.
On sharp heat kernel estimates in the context of Fourier-Dini expansions
本文研究了配备 Lebesgue 测度且右端点施加 Neumann 条件的区间 (0,1) 上 Fourier-Dini 展开相关热核的精确估计，并给出了其在 Poisson 核与位势核的精确界、极大热半群映射性质及边界收敛性等方面的应用。
The purpose of this note is to discuss several results that have been obtained in the last decade in the context of sharp adjoint Fourier restriction/Strichartz inequalities. Rather than aiming at full generality, we focus on several concrete examples of underlying manifolds with large groups of symmetries, which sometimes allows for simple geometric proofs. We mention several open problems along the way, and include an appendix on integration on manifolds using delta calculus.
A sharp Hörmander condition for bilinear Fourier multipliers with Lipschitz singularities
该研究针对带有限Lipschitz曲线奇异集的双线性Fourier乘子，提出关于Sobolev指数精确的Hörmander条件，证明其局部L²范围内的L^p有界性，推广了非退化双线性Hilbert变换并规避退化附近一致有界性难题。
The restricted maximal operators of partial sums with respect to bounded Vilenkin systems are investigated. We derive the maximal subspace of positive numbers, for which this operator is bounded from the Hardy space $%H_{p}$ to the Lebesgue space $L_{p}$ for all $0<p\leq 1.$ We also prove that the result is sharp in a particular sense.
A sharp Hörmander estimate for multi-parameter and multi-linear Fourier multiplier operators
本文研究多线性多参数Fourier乘子的Hörmander型定理，给出了基于L^u的Sobolev范数刻画下的光滑性假设结果，在多参数双线性情形下精确、多线性情形下几乎精确，且改进了单参数多线性情形的已有结论。
In this paper we study direct and inverse approximation inequalities in $L^{p}(\mathbb{R}^{d})$, $1<p<\infty$, with the Dunkl weight. We obtain these estimates in their sharp form substantially improving previous results. We also establish new estimates of the modulus of smoothness of a function $f$ via the fractional powers of the Dunkl Laplacian of approximants of $f$. Moreover, we obtain new Lebesgue type estimates for moduli of smoothness in terms of Dunkl transforms. Needed Pitt-type and Kellogg-type Fourier--Dunkl inequalities are derived.
On sharp Fourier extension from spheres in arbitrary dimensions
本文研究单位球面上的 Fourier 延拓估计，证明了适用于任意维数 $d\geq 3$ 的一族新的尖锐 $L^2\to L^4$ 不等式。
In this paper we prove a sharp trilinear inequality which is motivated by a program to obtain the sharp form of the $L^2 - L^6$ Tomas-Stein adjoint restriction inequality on the circle. Our method uses intricate estimates for integrals of sixfold products of Bessel functions developed in a companion paper. We also establish that constants are local extremizers of the Tomas-Stein adjoint restriction inequality as well as of another inequality appearing in the program.
Sharp Fourier restriction to monomial curves
本文建立了带有仿射弧长测度的单项曲线的傅里叶限制/延拓算子范数的下界，并证明了极值化序列预紧性与范数严格大于阈值的等价关系，同时引入了适用于更一般流形的新方法。
Sharp Fourier extension for functions with localized support on the circle
本文证明了支撑在$S^1$一对对偶点的$\sqrt{6}/80$-邻域内的函数满足圆上$L^2 \to L^6$ Tomas-Stein延拓不等式的锐利猜想，并推进了证明所有函数该不等式的计划。
Sharp estimates for the Fourier transform of surface-carried measures and maximal operators associated with hypersurfaces in $\mathbb{R}^4$ with vanishing Gaussian curvature
针对 $\mathbb{R}^4$ 中高斯曲率为零且为多项式函数图像的超曲面，推导了测度傅里叶变换关于频率方向的尖锐一致估计，并确定了相关极大算子 $L^p$ 有界性的精确指数。
We study the boundedness on the Wiener amalgam spaces $W^{p,q}_s$ of Fourier multipliers with symbols of the type $e^{iμ(ξ)}$, for some real-valued functions $μ(ξ)$ whose prototype is $|ξ|^β$ with $β\in (0,2]$. Under some suitable assumptions on $μ$, we give the characterization of $W^{p,q}_s\rightarrow W^{p,q}$ boundedness of $e^{iμ(D)}$, for arbitrary pairs of $0< p,q\leq \infty$. Our results are an essential improvement of the previous known results, for both sides of sufficiency and necessity, even for the special case $μ(ξ)=|ξ|^β$ with $1<β<2$.
Bilinear Rough Singular Integrals near the Critical Integrability via Sharp Fourier Multiplier Criteria
本文研究了双线性粗糙奇异积分算子在临界可积性附近的有界性问题，提出了一种新的傅里叶乘子算子有界性准则，改进了已有条件并接近线性理论推测的端点估计。
Asymptotically sharp discrete nonlinear Hausdorff-Young inequalities for the SU(1,1)-valued Fourier products
针对单位圆正交多项式中SU(1,1)值非线性Fourier变换的离散Hausdorff-Young不等式，本文研究其精确常数，并对小序列或远离线性极值化子的序列给出两个结果。
We consider Fourier transforms of densities supported on curves in R^d. We obtain sharp lower and close to sharp upper bounds for the L^q decay rates.
A sharp weighted Fourier extension estimate for the cone in $\mathbb{R}^3$ based on circle tangencies
本文应用圆切触估计证明了 $\mathbb{R}^3$ 中锥和 $1$ 维权重的精确加权 Fourier 延拓估计，并作为推论改进了相应的最佳 Mizohata--Takeuchi 型估计。
Sharp Fourier Extension on the Circle Under Arithmetic Constraints
本文在 Fourier 系数支撑集的算术约束下，建立了圆周上端点 Tomas-Stein 限制定理的精确伴随 Fourier 限制不等式，该约束推广了 $B_3$-集。
We derive bilateral estimates for the constants appearing in the Fourier transform restricted theorems on the Euclidean sphere for the ordinary and especially radial functions belonging to the Lebesgue-Riesz spaces as well as belonging to the Grand Lebesgue Spaces. We obtain an exact estimate for the norm of the restriction Fourier transform operator acting on the radial functions.
Sharpness of the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions
本文证明了Mockenhaupt-Mitsis-Bak-Seeger傅里叶限制定理在所有维度和完整参数范围内的指数最优性，构造是确定性的且能产生Salem集。
We prove sharp endpoint results for the Fourier restriction operator associated to nondegenerate curves in $\Bbb R^d$, $d\ge 3$, and related estimates for oscillatory integral operators. Moreover, for some larger classes of curves in $\Bbb R^d$ we obtain sharp uniform $L^p\to L^q$ bounds with respect to affine arclength measure, thereby resolving a problem of Drury and Marshall.
The theory of Fourier integral operators is surveyed, with an emphasis on local smoothing estimates and their applications. After reviewing the classical background, we describe some recent work of the authors which established sharp local smoothing estimates for a natural class of Fourier integral operators. We also show how local smoothing estimates imply oscillatory integral estimates and obtain a maximal variant of an oscillatory integral estimate of Stein. Together with an oscillatory integral counterexample of Bourgain, this shows that our local smoothing estimates are sharp in odd spati…
Let $S$ be a hypersurface in $\Bbb R^3$ which is the graph of a smooth, finite type function $ϕ,$ and let $μ=ρ\, d\si$ be a surface carried measure on $S,$ where $d\si$ denotes the surface element on $S$ and $ρ$ a smooth density with suffiently small support. We derive uniform estimates for the Fourier transform $\hat μ$ of $μ,$ which are sharp except for the case where the principal face of the Newton polyhedron of $ϕ,$ when expressed in adapted coordinates, is unbounded. As an application, we prove a sharp $L^p$-$L^2$ Fourier restriction theorem for $S$ in the case where the original coordin…
We study potential operators (Riesz and Bessel potentials) associated with classical Jacobi and Fourier-Bessel expansions. We prove sharp estimates for the corresponding potential kernels. Then we characterize those $1 \le p,q \le \infty$, for which the potential operators are of strong type $(p,q)$, of weak type $(p,q)$ and of restricted weak type $(p,q)$. These results may be thought of as analogues of the celebrated Hardy-Littlewood-Sobolev fractional integration theorem in the Jacobi and Fourier-Bessel settings. As an ingredient of our line of reasoning, we also obtain sharp estimates of t…
We prove sharp L^p-L^q endpoint bounds for singular fractional integral operators and related Fourier integral operators, under the nonvanishing rotational curvature assumption.
It is considered Fourier transform of convex analytic hypersufaces on $R^{4} $. We prove that the Fourier restriction operator associated to convex analytic hypersufaces is \textit{$(L_{p}, L_{2})$} bounded whenever $1\le p\le \frac{2h+2}{h+2}$. The result is sharp.
We prove Fourier restriction estimates to arbitrary compact $C^{N}$ curves for any $N > d$ in the (sharp) Drury range, using a power of the affine arclength measure as a mitigating factor. In particular, we make no nondegeneracy assumption on the curve.
The quaternion Fourier transform (QFT) satisfies some uncertainty principles similar to the Euclidean Fourier transform. In this paper, we establish Miyachi's theorem for this transform.
The two-sided quaternion Fourier transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization of Beurling's theorem, Hardy, Cowling-Price and Gelfand-Shilov theorems, is obtained for the two-sided quaternion Fourier transform.
In this paper we generalize the continuous quaternion windowed Fourier transform called the multivariate two sided continuous quaternion windowed Fourier transform. Using the two sided quaternion Fourier transform we derive several important properties.
We provide sharp two-sided estimates of the Fourier-Bessel heat kernel and we give sharp two-sided estimates of the transition probability density for the Bessel process in (0,1) killed at 1 and killed or reflected at 0.
We prove a Calderón-Zygmund type estimate which can be applied to sharpen known regularity results on spherical means, Fourier integral operators and generalized Radon transforms.
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 $\Bbb R^2$.
Certain relations between the Fourier transform of a function of bounded variation and the Hilbert transform of its derivative are revealed. The widest subspaces of the space of functions of bounded variation are indicated in which the cosine and sine Fourier transforms are integrable.
In this note we propose a generalization of the Laplace and Fourier transforms which we call symmetric Laplace transform. It combines both the advantages of the Fourier and Laplace transforms. We give the definition of this generalization, some examples and basic properties. We also give the form of its inverse by using the theory of the Fourier transform. Finally, we apply the symmetric Laplace transform to a parabolic problem and to an ordinary differential equation.
In this paper, we estabish an analogue of Hardy's theorem and Miyachi's theorem for the Clifford-Fourier transform.
A fundamental theme in classical Fourier analysis relates smoothness properties of functions to the growth and/or integrability of their Fourier transform. By using a suitable class of $L^{p}-$multipliers, a rather general inequality controlling the size of Fourier transforms for large and small argument is proved. As consequences, quantitative Riemann-Lebesgue estimates are obtained and an integrability result for the Fourier transform is developed extending ideas used by Titchmarsh in the one dimensional setting.
We provide asymptotically sharp bounds for the $L_p$ norms of the Fourier multipliers with the symbols $e^{iλφ(ξ/|ξ|)}$, where $λ\in \mathbb{R}$ is a large parameter.
In this work, we are interested by the $q$-Bessel Fourier transform with a new approach. Many important results of this $q$-integral transform are proved with a new constructive demonstrations and we establish in particular the associated $q$-Fourier-Neumen expansion which involves the $q$-little Jacobi polynomials.
In this paper, we define a new transform called the Gabor quaternionic Fourier transform (GQFT), which generalizes the classical windowed Fourier transform to quaternion valued-signals, we give several important properties such as the Plancherel formula and inversion formula. Finally, we establish the Heisenberg uncertainty principles for the GQFT.
We establish estimates for restrictions to certain curves in R^2 of the Fourier transforms of some fractal measures.
We obtain a characterisation of the Fourier transform on the space of Schwartz class functions on $\mathbb{R}^n.$ The result states that any appropriately additive bijection of the Schwartz space onto itself, which interchanges convolution and pointwise products is essentially the Fourier transform.
We give a generalization of Beurling's theorem for the Clifford-Fourier transform. Then, analogues of Hardy, Cowling-Price and Gelfand-Shilov theorems are obtained in Clifford analysis.
We prove $L^p \rightarrow L^q$ Fourier restriction estimates for 3-dimensional quadratic surfaces in $\mathbb{R}^5$. Our results are sharp, up to endpoints, for a few classes of surfaces.
The Fourier transform is considered as a Henstock--Kurzweil integral. Sufficient conditions are given for the existence of the Fourier transform and necessary and sufficient conditions are given for it to be continuous. The Riemann--Lebesgue lemma fails: Henstock--Kurzweil Fourier transforms can have arbitrarily large point-wise growth. Convolution and inversion theorems are established. An appendix gives sufficient conditions for interchanging repeated Henstock--Kurzweil integrals and gives an estimate on the integral of a product.
The purpose of this paper is to prove an essentially sharp L^2 Fourier restriction estimate for light cones, of the type which is called bilinear in the recent literature.
In this work, we obtain the Fourier transform of the first derivative of the generalized logistic growth curve in terms of Gamma functions and we discuss special cases.
This is the second of two articles in which we prove a sharp $L^p-L^2$ Fourier restriction theorem for a large class of smooth, finite type hypersurfaces in R^3, which includes in particular all real-analytic hypersurfaces.
A bilinear inequality of Geba, Greenleaf, Iosevich, Palsson, and Sawyer for the Fourier transform is shown to be equivalent to a simpler linear inequality, and the range of exponents is extended. Related mixed-norm inequalities are discussed.
We calculate the Fourier transform of a spherically symmetric exponential function. Our evaluation is much simpler than the known one. We use the polar coordinates and reduce the Fourier transform to the integral of a rational function of trigonometric functions. Its evaluation turns out to be much easier than expected because of homogeneity and a hidden symmetry. Relationship with a Fourier integral representation formula for harmonic functions is explained.
A Fourier approach to the sharp stability of Heisenberg Uncertainty Principles of higher and fractional orders: a new perspective
本文通过傅里叶方法将二阶海森堡不确定性原理的尖锐稳定性化为一阶Caffarelli--Kohn--Nirenberg不等式，并推广到分数阶，计算尖锐常数、刻画优化子并建立稳定性链。
We prove sharp inequalities for the symmetric-decreasing rearrangement in Fourier space of functions in $\mathbb{R}^d$. Our main result can be applied to a general class of (pseudo-)differential operators in $\mathbb{R}^d$ of arbitrary order with radial Fourier multipliers. For example, we can take any positive power of the Laplacian $(-Δ)^s$ with $s> 0$ and, in particular, any polyharmonic operator $(-Δ)^m$ with integer $m \geq 1$. As applications, we prove radial symmetry and real-valuedness (up to trivial symmetries) of optimizers for: i) Gagliardo-Nirenberg inequalities with derivatives of…
A mathematical survey on Fourier type integral transform and their offshoots: windowed Fourier transform, wavelet transform and Stockwell transform
本文综述了Fourier型积分变换（如FrFT、LCT、QPFT）及其衍生变换的卷积性质、不确定性原理和多分辨率分析，并详细给出了四元数情形下的对应不等式。
Sharp Fourier extension on fractional surfaces
本文研究分数曲面上的傅里叶延拓算子，利用缺失质量方法与双线性限制理论刻画了α-Strichartz不等式极值序列的预紧性，并在二维情形证明了α∈[2,α0)（α0>5）时极值的存在性。
The aim of this paper is to establish a few uncertainty principles for the Fourier and the short-time Fourier transforms. Also, we discuss an analogue of Donoho--Stark uncertainty principle and provide some estimates for the size of the essential support of the short-time Fourier transform.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。