Sergey Sadov
It is shown that the Laplace transform of an Lp (1<p<=2) function defined on the positive semiaxis satisfies the Hausdorff-Young type inequality with a positive weight in the right complex half-plane if and only if the weight is a Carleson measure.
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Weak Carleson conditions in uniformly rectifiable metric spaces: the WCD and alpha numbers
该研究通过弱Carleson条件刻画一致可求长度度量空间，证明UR空间满足弱常数密度条件，引入度量空间版alpha数并证明其弱Carleson条件可刻画度量一致可求长度性，且两结果均为更一般抽象结论的推论。
We obtain Paley-type and Hausdorff-Young-Paley-type inequalities for Jacobi expansions.
Using well-known techniques, we establish Hardy-Littlewood-type and Hausdorff-Young-type inequalities for generalized Gegenbauer expansions and their unification.
We establish Paley-type and Hausdorff-Young-Paley-type inequalities for generalized Gegenbauer expansions.
The nonlinear Hausdorff-Young inequality follows from the work of Christ and Kiselev. Later Muscalu, Tao, and Thiele asked if the constants can be chosen independently of the exponent. We show that the nonlinear Hausdorff-Young quotient admits an even better upper bound than the linear one, provided that the function is sufficiently small in the $L^1$ norm. The proof combines perturbative techniques with the sharpened version of the linear Hausdorff-Young inequality due to Christ.
Following the works of Lyons and Oberlin, Seeger, Tao, Thiele and Wright, we relate the variation of certain discrete curves on the Lie group $\text{SU}(1,1)$ to the corresponding variation of their linearized versions on the Lie algebra. Combining this with a discrete variational Menshov-Paley-Zygmund theorem, we establish a variational Hausdorff-Young inequality for a discrete version of the nonlinear Fourier transform on $\text{SU}(1,1)$.
Let $Ω$ be a domain in $\mathbb{R}^{d+1}$, $d \geq 1$. In the paper's references [HMM2] and [GMT] it was proved that if $Ω$ satisfies a corkscrew condition and if $\partial Ω$ is $d$-Ahlfors regular, i.e. Hausdorff measure $\mathcal{H}^d(B(x,r) \cap \partial Ω) \sim r^d$ for all $x \in \partial Ω$ and $0 < r < {\rm diam}(\partial Ω)$, then $\partial Ω$ is uniformly rectifiable if and only if (a) a square function Carleson measure estimate holds for every bounded harmonic function on $Ω$ or (b) an $\varepsilon$-approximation property for all $0 < \varepsilon <1$ for every such function. Here we…
Analogues of Hausdorff-Young inequalities for the Dirac scattering transform (a.k.a. SU(1,1) nonlinear Fourier transform) were first established by Christ and Kiselev [1],[2]. Later Muscalu, Tao, and Thiele [5] raised a question if the constants can be chosen uniformly in $1\leq p\leq 2$. Here we give a positive answer to that question when the Euclidean real line is replaced by its Cantor group model.
It is known that if $q$ is an even integer then the $L^q(\mathbb{R}^d)$ norm of the Fourier transform of a superposition of translates of a fixed gaussian is monotone increasing as their centres "simultaneously slide" to the origin. We provide explicit examples to show that this monotonicity property fails dramatically if $q > 2$ is not an even integer. These results are equivalent, upon rescaling, to similar statements involving solutions to heat equations. Such considerations are natural given the celebrated theorem of Beckner concerning the gaussian extremisability of the Hausdorff--Young i…
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。
The Hausdorff-Young inequality for Euclidean space, in its sharp form due to Beckner, gives an upper bound for the Fourier transform in terms of Lebesgue space norms, with an optimal constant. The extremizers have been identified by Lieb to be the Gaussians. We establish an improved upper bound, for functions that nearly extremize the inequality, with a negative second term roughly proportional to the square of the distance to the set of extremizers. One formulation of this term comes with its own sharp constant. The main step is to show that any extremizing sequence is precompact, modulo the …
Let p and q be conjugate exponents, with p in [1,2]. It is shown that the Laplace transform acts boundedly between the Lp space with unit weight on the positive real semiaxis and the Lq space weighted by a well-projected measure (a term defined in the paper) in the right complex half-plane. The operator norm is uniformly bounded over classes of measures with the same "projection constants". Particular cases are arclength measures on some classes of rectifiable curves. In addition, an analog of the Hausdorff-Young inequality in Lorentz spaces is obtained in the case of "wrong" exponents p>2.
Hausdorff-Young inequalities for $\mathrm{SU}(1,1)$ Fourier products indexed by binary cubes
本文证明了二进制立方体索引的 $\mathrm{SU}(1,1)$ 傅里叶乘积的最优对数 Hausdorff-Young 不等式，推广至 $1/p+1/q<1$，并导出链不等式及交替链生成函数的非线性界。
We consider Carleson's problem regarding pointwise convergence for the Schrödinger equation. Bourgain recently proved that there is initial data, in $H^s(\mathbb{R}^n)$ with $s<\frac{n}{2(n+1)}$, for which the solution diverges on a set of nonzero Lebesgue measure. We provide a different example enabling the generalisation to fractional Hausdorff measure.
We prove several results about the best constants in the Hausdorff-Young inequality for noncommutative groups. In particular, we establish a sharp local central version for compact Lie groups, and extend known results for the Heisenberg group. In addition, we prove a universal lower bound to the best constant for general Lie groups.
In this paper we study the vector-valued analogues of several inequalities for the Fourier transform. In particular, we consider the inequalities of Hausdorff--Young, Hardy--Littlewood, Paley, Pitt, Bochkarev and Zygmund. The Pitt inequalities include the Hausdorff--Young and Hardy--Littlewood inequalities and state that the Fourier transform is bounded from $L^p(\mathbb{R}^d,|\cdot|^{βp})$ into $L^q(\mathbb{R}^d,|\cdot|^{-γq})$ under certain condition on $p,q,β$ and $γ$. Vector-valued analogues are derived under geometric conditions on the underlying Banach space such as Fourier type and rela…
Those functions which nearly extremize Young's convolution inequality are characterized for discrete groups which have no nontrivial finite subgroups. Near-extremizers of the Hausdorff-Young inequality are characterized for Z^d.
Analog version of Hausdorff--Young's theorem for quadratic Fourier transforms and boundedness of oscillatory integral operator
本文基于 Riesz--Thorin 插值定理证明了二次 Fourier 变换和线性正则变换的 Hausdorff--Young 型不等式，并研究了多项式相位振荡积分算子的有界性。
The nonlinear Hausdorff-Young inequality
该研究证明了常数1的离散非线性豪斯多夫-杨不等式，通过极限论证得到连续空间的对应不等式，解决了Muscalu-Tao-Thiele一致性问题，其证明利用隐藏的希尔伯特空间结构将非线性问题转化为经典插值问题。
Representation of quasi-periodic functions and Hausdorff-Young inequalities for Besicovitch almost periodic functions
本文研究了准周期函数的表示及其与贝西科维奇几乎周期函数的豪斯多夫-扬不等式的关系，通过建立Banach代数的同构关系，给出了简化的不等式证明并探讨了母函数的正则性。
$L^p$-$L^q$ Fourier multipliers and Hausdorff-Young-Paley inequalities on Riemannian symmetric spaces of noncompact type
本文在任意秩非紧型黎曼对称空间上建立Hörmander型Lp-Lq Fourier乘子定理，推导谱乘子定理，证明Laplace-Beltrami算子函数的有界性，并给出Hausdorff-Young和Paley不等式的混合范数版本，结果适用于Damek-Ricci空间。
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.
Multiversion of the Hausdorff--Young inequality
本文研究相关联合高斯向量上经Mehler变换或噪声算子作用后的多函数不等式，统一推广实复超压缩性，并得到Hausdorff–Young等不等式的多函数版本及含噪Brascamp–Lieb刻画。
We obtain Hausdorff-Young inequalities for the one-dimensional Cherednik--Opdam transform and its inverse, and we establish a real Paley-Wiener theorem for its inverse that generalizes an analogous result by N. B. Andersen for the Jacobi transform.
We obtain several versions of the Hausdorff-Young and Hardy-Littlewood inequalities for the $(k,a)$-generalized Fourier transform recently investigated at length by Ben Saï d, Kobayashi, and Ø rsted. We also obtain a number of weighted inequalities - in particular Pitt's inequality - that have application to uncertainty principles. Specifically we obtain several analogs of the Heisenberg-Pauli-Weyl principle for $L^p$-functions, local Cowling-Price-type inequalities, Donoho-Stark-type inequalities and qualitative extensions. We finally use the Hausdorff-Young inequality as a means to obtain en…
It is shown in the present article that the Minkowski function does not preserve the Hausdorff-Besicovitch dimension. Results of this article were presented by the author of this article on the Second Interuniversity Scientific Conference on Mathematics and Physics for Young Scientists in April, 2011 (Links to the conference paper (in Ukrainian): www.researchgate.net/publication/301637057, www.researchgate.net/publication/311665828. A link to the presentation (in Ukrainian): www.researchgate.net/publication/303752636).
Note on Fourier inequalities
本文证明变指数勒贝格空间中豪斯多夫--杨不等式成立当且仅当指数为常数，并在函数单调性条件下给出经典与变指数情形下 Pitt 型加权傅里叶不等式的完整刻画。
Classical inequalities for all Fourier matrix coefficients of $\mathrm{SL}(2,\mathbb{R})$ and their applications
本文为 $\mathrm{SL}(2,\mathbb{R})$ 上的 $(l, n)$ 型函数建立了三个基础 Fourier 不等式，据此证明了 Fourier 乘子的 $L^p$-$L^q$ 有界性，并将其应用于热核估计与非线性方程的全局适定性研究。
Inequalities in Fourier analysis on binary cubes
本文刻画了二元立方体上离散函数的 Hausdorff-Young 不等式与等指数 Young 卷积不等式尖锐成立的 Lebesgue 指数精确范围，并由此导出广义加性能量上界、熵不确定性原理的二元变体及 Fourier 限制的无维数估计。
Let $\{S_i\}_{i\in Λ}$ be a finite contracting affine iterated function system (IFS) on ${\Bbb R}^d$. Let $(Σ,σ)$ denote the two-sided full shift over the alphabet $Λ$, and $π:Σ\to {\Bbb R}^d$ be the coding map associated with the IFS. We prove that the projection of an ergodic $σ$-invariant measure on $Σ$ under $π$ is always exact dimensional, and its Hausdorff dimension satisfies a Ledrappier-Young type formula. Furthermore, the result extends to average contracting affine IFSs. This completes several previous results and answers a folklore open question in the community of fractals. Some ap…
On Herz-Bochkarev limiting problem
本文利用新型 grand Lorentz 空间技术研究 $L_{p,q}$ 中 Hausdorff-Young 型不等式常数随 $p$ 的变化，得出 $p\ ightarrow 2$ 极限情形的最优界，解决 Herz-Bochkarev 问题并改进已有估计。
Smoothing inequalities for transport metrics in compact spaces
本文在紧致流形上建立了Wasserstein度量与傅里叶变换之间的上界估计，首次覆盖$p$全范围，并应用于证明球面设计的最优均匀性。
On the asymptotic behaviour of the Fourier transform of the Mittag-Leffler function
研究 Mittag-Leffler 函数的傅里叶变换在特定条件下的渐进行为，推导出上界估计，并应用得到 Lebesgue 指数 p 的值，扩展了传统方法的适用范围。
The $L^q$ spectrum of self-affine measures on sponges
本文研究了 $\mathbb{R}^d$ 中海绵上自仿测度的 $L^q$ 谱，通过引入新的压力函数并证明其变分原理，得出了 $L^q$ 谱的变分公式，确定了测度的 Frostman 维数和盒维数，并将现有结果推广至任意维度。
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