James Barker, Christoph Thiele, Pavel Zorin-Kranich
Among the class of functions on the circle with Fourier modes up to degree 120, constant functions are the unique real-valued maximizers for the endpoint Tomas-Stein inequality.
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We give a necessary and sufficient condition for the precompactness of all optimizing sequences for the Stein-Tomas inequality. In particular, if a well-known conjecture about the optimal constant in the Strichartz inequality is true, we obtain the existence of an optimizer in the Stein-Tomas inequality. Our result is valid in any dimension.
We show that constant functions are global maximizers for the adjoint Fourier restriction inequality for the sphere.
Band-limited maximizers for a Fourier extension inequality on the circle
研究在傅里叶模数不超过30的函数类中，常数函数是圆上端点Tomas-Stein不等式的唯一实值最大值
An inequality of Brascamp-Lieb-Luttinger and of Rogers states that among subsets of Euclidean space $\mathbb{R}^d$ of specified Lebesgue measures, balls centered at the origin are maximizers of certain functionals defined by multidimensional integrals. For $d>1$, this inequality only applies to functionals invariant under a diagonal action of $\text{Sl}(d)$. We investigate functionals of this type, and their maximizers, in perhaps the simplest situation in which $\text{Sl}(d)$ invariance does not hold. Assuming a more limited symmetry involving dilations but not rotations, we show under natura…
A symmetrization inequality of Rogers and of Brascamp-Lieb-Luttinger states that for a certain class of multilinear integral expressions, among tuples of sets of prescribed Lebesgue measures, tuples of balls centered at the origin are among the maximizers. Under natural hypotheses, we characterize all maximizing tuples for these inequalities for dimensions strictly greater than 1. We establish a sharpened form of the inequality.
In this paper, we will establish the best constants for certain classes of weighted Moser-Trudinger inequalities on the entire Euclidean spaces $\mathbb{R}^N$. We will also prove the existence of maximizers of these sharp weighted inequalities. The class of functions considered here are not necessarily spherically symmetric. Our inequality in Theorem 1.1 improves the earlier one where such type of inequality was only considered for spherically symmetric functions by M. Ishiwata, M. Nakamura, H. Wadade in \cite{INW} (except in the case $s\not=0$). Since $\int_{\mathbb{R}^N}Φ_N(α|u|^{N/(N-1)})\f…
Conditional existence of maximizers for the Tomas-Stein inequality for the sphere
在球面与薛定谔方程Strichartz不等式最佳常数严格比较的条件下，利用精细Tomas-Stein不等式和剖面分解，证明了存在函数极大化该不等式，且任何极大化序列有子序列收敛到极大元。
We show that, possibly after a compactification of spacetime, constant functions are local maximizers of the Tomas-Stein adjoint Fourier restriction inequality for the cone and paraboloid in every dimension, and for the sphere in dimension up to 60. For the cone and paraboloid we work from the PDE framework, which enables the use of the Penrose and the Lens transformations, which map the conjectured optimal functions into constants.
The Stein-Tomas inequality under the effect of symmetries
该研究在O(d-k)×O(k)对称函数类上得到了单位球面的新傅里叶限制估计，证明了端点Stein-Tomas不等式在此类中极大值元的存在性，并通过构造例子验证了指数范围的最优性。
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。
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.
Maximizers of the $L^2\to L^4$ Fourier extension inequality for cones in finite fields
本文研究了有限域中$L^2\to L^4$傅里叶扩展不等式在不同锥体上的极值问题，建立了适用于所有剩余锥体的尖锐不等式并分类了极值函数。
The more then hundred years old Bernstein inequality states that the supremum norm of the derivative of a trigonometric polynomial of fixed degree can be bounded from above by supremum norm of the polynomial itself. The reversed Bernstein inequality, that we prove in this note, says that the reverse inequality holds for functions in the orthogonal complement of the space of polynomials of fixed degree. In fact, we derived a more general result for the lower bounds on higher derivatives. These bounds are better then those obtained by applying bound for the first derivative successively several …
We prove the following sharp Sobolev inequality on the circle $$\int_{\mathbb{S}^1} [4(v')^2 - v^2] \mathrm{d} θ\geq - \frac{4π^2}{\int_{\mathbb{S}^1} v^{-2} \mathrm{d} θ},$$ with the equality being achieved when $v^{-2} (θ) = \frac{k\sqrt{1-α^2}}{1+ α\cos(θ- θ_o)}$where$k>0$, $α\in (-1,1)$, $θ_0 \in \mathbb{R}$. If $v$ vanishes somewhere on the circle, then $$4 \int_{\mathbb{S}^1} (v')^2 \mathrm{d} θ\geq\int_{\mathbb{S}^1} v^2 \mathrm{d} θ.$$ The basic tools to prove the inequality are the rearrangement inequality on the circle and the variational method. We investigate the variational proble…
A Sharpened Rearrangement Inequality for Convolution on the Sphere
本文针对球面上涉及两个函数和一个单调函数的三线性卷积形式，在指示函数情形下建立了强化重排不等式，证明对称重排是唯一最大化子。
Sharp $L^2 \to L^4$ extension inequality for quadratic surfaces in finite fields
本文针对有限域中一类二次曲面（不含原点的锥面和特定球面）建立了尖锐的 $L^2 \to L^4$ 扩展不等式，并证明极大化子具有常数模。
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。
We prove that in dimensions $d \geq 3$, the non-endpoint, Lorentz-invariant $L^2 \to L^p$ adjoint Fourier restriction inequality on the $d$-dimensional hyperboloid $\mathbb{H}^d \subseteq \mathbb{R}^{d+1}$ possesses maximizers. The analogous result had been previously established in dimensions $d=1,2$ using the convolution structure of the inequality at the lower endpoint (an even integer); we obtain the generalization by using tools from bilinear restriction theory.
Exponentials rarely maximize Fourier extension inequalities for cones
本文研究锥面上傅里叶延拓不等式的极大化问题，证明了尺度不变情形下极大化元的存在性与极大化序列的预紧性，并刻画了低维 $L^2$ 全局极大化元，进一步证明这些函数仅在 $p=2$ 时才是 $L^p$ 到 $L^q$ 不等式的临界点。
We give an extension of Poincaré's type capacitary inequality for Dirichlet spaces and provide an application to study the uniqueness sets on the unit circle for these spaces.
Borwein and Erdélyi proved a Bernstein type inequality for rational functions on the unit circle and on the real line. Here we establish asymptotically sharp extensions of their inequalities for rational functions on analytic Jordan arcs and curves. In the proofs key roles are played by Borwein-Erdélyi inequality on the unit circle, Gonchar-Grigorjan type estimate of the norm of holomorphic part of meromorphic functions and Totik's construction of fast decreasing polynomials.
The Best Constant For Inequality Involving Sum Of The Reciprocals And Product Of Positive Numbers With Unit Sum
本文通过新的优化论证确定了涉及倒数和与和为1的正数乘积的代数不等式的最佳参数值，并在三数情形下推广了欧拉不等式，应用于三角形外接圆与内切圆半径之比。
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。
Sharp endpoint extension inequalities for the moment curve on finite fields
本文研究了有限域中矩曲线的尖锐端点延拓不等式，通过结合分析、代数与组合学方法，确定了最优常数，并在低维与大基域基数两种互补情形下刻画了最大化子。
For a polynomial $P_n$ of degree $n$, Bernstein's inequality states that $\|P_n'\| \le n \|P_n\|$ for all $L^p$ norms on the unit circle, $0<p\le\infty,$ with equality for $P_n(z)= c z^n.$ We study this inequality for random polynomials, and show that the expected (average) and almost sure value of $\Vert P_n' \Vert/\Vert P_n\Vert$ is often different from the classical deterministic upper bound $n$. In particular, for circles of radii less than one, the ratio $\Vert P_n' \Vert/\Vert P_n\Vert$ is almost surely bounded as $n$ tends to infinity, and its expected value is uniformly bounded for all…
Sharp uniform stability of the logarithmic Sobolev inequality on finite cycles and stability of the cubic Sobolev inequality
本文确定有限环上对数Sobolev不等式四次稳定性估计的尖锐一致常数，并给出三次Sobolev不等式的定量稳定性，关键新工具包括非负函数的傅里叶系数界和Gross两点不等式的尖锐稳定性。
In this paper we prove an inequality inspired by a conjecture of Brezis, which asks for a bound for the topological degree of a map from the circle to itself in terms of a nonlocal integral.
Around the Fejér-Jackson inequality: Tight bounds for certain oscillatory functions via Laplace transform representations
本文通过拉普拉斯变换表示法，为特定振荡函数提供了更紧的界，研究了锯齿函数的傅里叶多项式近似误差以及对数函数的泰勒级数余项的渐近紧致不等式。
In this paper, we establish the existence of extremals for two kinds of Stein-Weiss inequalities on the Heisenberg group. More precisely, we prove the existence of extremals for the Stein-Weiss inequalities with full weights in Theorem 1.1 and the Stein-Weiss inequalities with horizontal weights in Theorem 1.4. Different from the proof of the analogous inequality in Euclidean spaces given by Lieb [26] using Riesz rearrangement inequality which is not available on the Heisenberg group, we employ the concentration compactness principle to obtain the existence of the maximizers on the Heisenberg …
In this paper, we will use a suitable tranform to investigate the sharp constants and optimizers for the following Caffarelli-Kohn-Nirenberg inequalities for a wide range of parameters $(r,p,q,s,μ,σ)$ and $0\leq a\leq1$: \begin{equation} \left({\displaystyle\int} \left\vert u\right\vert ^{r}\frac{dx}{\left\vert x\right\vert ^{s}}\right)^{1/r}\leq C\left( {\displaystyle\int} \left\vert \nabla u\right\vert ^{p}\frac{dx}{\left\vert x\right\vert ^μ% }\right) ^{a/p}\left({\displaystyle\int} \left\vert u\right\vert ^{q}\frac{dx}{\left\vert x\right\vert ^σ% }\right) ^{\left( 1-a\right) /q}. \end{equa…
We establish good numerical estimates for a certain class of integrals involving sixfold products of Bessel functions. We use relatively elementary methods. The estimates will be used in the study of a sharp Fourier restriction inequality on the circle.
We find the best possible constant $C$ in the inequality $\|φ\|_{L^r}\leq C\|φ\|_{L^p}^{\frac{p}{r}}\|φ\|_{\mathrm{BMO}}^{1-\frac{p}{r}}$, where $2 \leq r$ and $p < r$. We employ the Bellman function technique to solve this problem in the case of an interval and then transfer our results to the circle and the line.
Sharp endpoint extension inequalities for the moment curve on finite fields II: an extremal property of the uniform distribution
本文确定了有限域上矩曲线傅里叶端点扩展不等式的最优常数及所有极大化子，证实了先前猜想，并证明其等价于均匀分布唯一最大化独立同分布样本不同重排期望数的概率极值命题。
We improve the constant $\fracπ{2}$ in $L^1$-Poincaré inequality on Hamming cube. For Gaussian space the sharp constant in $L^1$ inequality is known, and it is $\sqrt{\fracπ{2}}$. For Hamming cube the sharp constant is not known, and $\sqrt{\fracπ{2}}$ gives an estimate from below for this sharp constant. On the other hand, L. Ben Efraim and F. Lust-Piquard have shown an estimate from above: $C_1\le \fracπ{2}$. There are at least two other independent proofs of the same estimate from above (we write down them in this note). Since those proofs are very different from the proof of Ben Efraim and…
Stability of sharp Fourier restriction to spheres
本文证明了在维度 $d \in \{3,4,5,6,7\}$ 中，单位球面上的常数函数在受有界扰动时仍最大化加权伴随 Fourier 限制不等式，并完整分类了复值最大化子，同时在 $\mathbb S^7$ 上导出了新的 sharp 伴随限制不等式。
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