Sharp restricted weak-type estimates for sparse operators
Irina Holmes Fay, Guillermo Rey, Kristina Ana Škreb
该研究确定了与作用于特征函数的稀疏算子水平集对应的精确Bellman函数，推进了稀疏算子限制弱型估计相关研究。
我们找到了与作用于特征函数的稀疏算子水平集相关的精确Bellman函数。
We find the exact Bellman function associated to the level-sets of sparse operators acting on characteristic functions.
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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.
We prove the sharp weighted-$L^2$ bounds for the strong-sparse operators introduced in \cite{KaragulyanM}. The main contribution of the paper is the construction of a weight that is a lacunary mixture of dual power weights. This weights helps to prove the sharpness of the trivial upper bound of the operator norm.
Sparse domination of sharp variational truncations
我们提供了一个通用的莱西最近稀疏逐点控制技术的制定与当地的弱型估计非切极大函数作为唯一的假设。我们验证了这一假设尖锐的变分截断奇异积分的情况下，未加权$L^2$估计。这扩展了以前已知的尖锐的加权估计光滑变分截断的情况下，尖锐的变分截断。我们还包括一个稀疏控制结果的多重线性算子与BMO函数的迭代算子。
For a class of sparse operators including majorants of singular integral, square function, and fractional integral operators in a uniform manner, we prove off-diagonal two-weight estimates of mixed type in the two-weight and $A_{\infty}$-characteristics. These bounds are known to be sharp in many cases; as a new result, we prove their sharpness for fractional square functions.
Sharp weighted non-tangential maximal estimates via Carleson-sparse domination
本文利用伴随奇异积分的逐点稀疏控制，证明了将函数从$\mathbf{R}^n$映射到上半空间的奇异积分非切向极大函数的尖锐加权估计，并用Carleson平均值替代通常平均值建立了非标准稀疏控制。
Sharp Sobolev regularity of restricted X-ray transforms
本文利用归纳策略，建立了由非退化曲线生成的限制X射线变换在$\mathbb R^{d+1}$（$d\ge 3$）中的精确$L^p$-Sobolev正则性估计，将已有三维结果推广至任意维度。
Restricted averaging operators to cones over finite fields
研究了q元有限域F_q上d维向量空间F_q^d的锥C上的限制平均算子A_C的L^p \to L^r估计.锥C的限制平均算子A_C由关系式A_Cf=f \ast σ定义|_C，其中σ表示锥C上的归一化表面测度，并且f是具有归一化计数测度dx的空间F_q^d上的复值函数。在以前的工作中，我们在奇数维d \ge 3中得到了A_C的有界性，但在偶数维d \ge 4中只得到了部分结果.本文证明了当锥C \subset F_q^d包含d/2维子空间时，在偶数维d中\ge 6的最优估计.
In this paper, we continue some recent work on two weight boundedness of sparse operators to the "off-diagonal" setting. We use the new "entropy bumps" introduced in by Treil-Volberg ([21]) and improved by Lacey-Spencer ([9]) and the "direct comparison bumps" introduced by Rahm-Spencer ([19]) and improved by Lerner ([10]). Our results are "sharp" in the sense that they are sharp in various particular cases. A feature is that given the current machinery, the proofs are almost trivial.
We prove sharp weak type weighted estimates for a class of sparse operators that includes majorants of standard $α$-fractional singular integrals, fractional integral operators, Marcinkiewicz integral operators, and square functions. These bounds are knows to be sharp in many cases, and our main new result is the optimal bound $$[w]_{A_{p,q}}^{\frac{1}{q}}[w^{q}]_{A_{\infty}}^{\frac{1}{2}-\frac{1}{p}}\lesssim[w]_{A_{p,q}}^{\frac{1}{2}-\fracα{d}}$$ for proper conditions which satisfy that three index $p$, $q$ and $α$ ensure weak type norm of fractional square functions on $L^{q}(w^{q})$ with $p…
In this note we give a sharp weighted estimate for square function from $L^2(w)$ to $L^2(w)$, $w\in A_2$. This has been known. But we also give a sharpening of this weighted estimate in the spirit of $T1$-type testing conditions. Finally we show that for any weight $w\in A^d_2$ and any characteristic function of a measurable set $\|S_wχ_E\|_{L^{2, \infty}(w^{-1})} \le C \sqrt{[w]_{A^d_2}}\, \|χ_E\|_w$, and this estimate is sharp. So on characteristic functions of measurable sets at least, no logarithmic correction is needed for the weak type of the dyadic square function.The sharp estimate for…
$L^p$-improving bounds for spherical maximal operators over restricted dilation sets: radial improvement
本文研究受限伸缩集上径向函数的球面极大算子$M_E$的$L^p$改进界，高维仅依赖上Minkowski维数，二维涉及分形几何其他概念，且两种情形的界对应区域几何形状差异显著。
We provide an estimate from below for the lower Hausdorff dimension of measures on the unit circle based on the arithmetic properties of their spectra. We obtain our bounds via application of a general result for abstract $q$-regular martingales to the Gundy--Varopoulos backwards martingale. To show the sharpness of our method, we improve the best known numerical lower bound for the Hausdorff dimension of certain Riesz products.
The height function of a sparse collection: a Bellman function approach
本研究聚焦调和分析中稀疏算子的弱(1,1)型不等式问题，采用Bellman函数方法，得到了与二元Carleson序列相关的局部化稀疏算子作用于常值函数的水平集的精确Bellman函数，为提取不等式精确常数提供了新结果。
Sparse domination of Hilbert transforms along curves
我们得到了$\mathbb{R}^n$中沿曲线的希尔伯特变换的清晰的稀疏界，并作为推论导出了这些算子的加权范数不等式。我们考虑的曲线包括单项曲线和任意具有非零挠率的$C^n$曲线。
In this paper we refine the recent sparse domination of the integrated $p = 2$ matrix weighted dyadic square function by T. Hytonen, S. Petermichl, and A. Volberg to prove a pointwise sparse domination of general matrix weighted dyadic square functions. We then use this to prove sharp two matrix weighted strong type inequalities for matrix weighted dyadic square functions when $1 < p \leq 2$.
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.
Sharp $L^p$-$L^q$ estimates for generalized $k$-plane transforms
本文推广了$k$ -平面变换，得到了多项式子流形上平均函数的算子的最优$L^p-L^q$估计.一个重要的进步，以前的工作是，充分的$L^p-L^q$估计传统上只产生有限的弱型估计的方法。在这个过程中，我们得到了对$p < 1$的$L^p$上的某些泛函的连续性估计.
To prove Fourier restriction estimate using polynomial partitioning, Guth introduced the concept of $k$-broad part of regular $L^p$ norm and obtained sharp $k$-broad restriction estimates. To go from $k$-broad estimates to regular $L^p$ estimates, Guth employed $l^2$ decoupling result. In this article, similar to the technique introduced by Bourgain-Guth, we establish an analogue to go from regular $L^p$ norm to its $(m+1)$-broad part, as the error terms we have the restricted $k$-broad parts ($k=2,\cdots,m$). To analyze the restricted $k$-broadness, we prove an $l^p$ decoupling result, which …
We consider the averages of a function $ f$ on $ \mathbb R ^{n}$ over spheres of radius $ 0< r< \infty $ given by $ A_{r} f (x) = \int_{\mathbb S ^{n-1}} f (x-r y) \; d σ(y)$, where $ σ$ is the normalized rotation invariant measure on $ \mathbb S ^{n-1}$. We prove a sharp range of sparse bounds for two maximal functions, the first the lacunary spherical maximal function, and the second the full maximal function. $$ M_{lac} f = \sup_{j\in \mathbb Z } A_{2^j} f , \qquad M_{full} f = \sup_{ r>0 } A_{r} f . $$ The sparse bounds are very precise variants of the known $L^p$ bounds for these maximal …
In this article, we address pointwise sparse domination for multilinear Calderón-Zygmund operators on upper doubling, geometrically doubling metric measure spaces. As a consequence, we have obtained sharp quantitative weighted estimates for multilinear Calderón-Zygmund operators.
We prove sparse bounds for pseudodifferential operators associated to Hörmander symbol classes. Our sparse bounds are sharp up to the endpoint and rely on a single scale analysis. As a consequence, we deduce a range of weighted estimates for pseudodifferential operators. The results naturally apply to the context of oscillatory Fourier multipliers, with applications to dispersive equations and oscillatory convolution kernels.
Burq-Gérard-Tzvetkov and Hu established $L^p$ estimates ($2\le p\le \infty$) for the restriction of eigenfunctions to submanifolds. The estimates are sharp, except for the log loss at the endpoint $L^2$ estimates for submanifolds of codimension 2. It has long been believed that the log loss at the endpoint can be removed in general, while the problem is still open. So this paper is devoted to the study of sharp endpoint restriction estimates for eigenfunctions in this case. Chen and Sogge removed the log loss for the geodesics on 3-dimensional manifolds. In this paper, we generalize their resu…
Weighted Estimation by Discrete-time Sparse Domination on Martingale Spaces
本研究针对离散时间鞅变换建立稀疏控制，引入条件稀疏性概念，推导了优于现有结果的精确加权估计与混合范数估计，还将稀疏控制应用于Doob极大算子及定量双权估计。
Sharp mixed $A_p$-$A_\infty$ estimates for sparse operators on filtered and nonhomogeneous measure spaces
本文在两类非倍测度背景下证明稀疏算子的混合$A_p$-$A_\infty$估计，获得与经典情形相同幂次的强、弱型界，并解决Rubio de Francia平方函数及Haar平移的开放问题。
A sharp boundedness result concerning some maximal operators of Vilenkin-Fejér means
本文给出了一个极大的正数子空间，使得在这个子空间上的限制的费耶平均极大算子对所有的$0<p\leq 1/2.$从Hardy空间$H_{p}$到$L_{p}$是有界的，并证明了这个结果在某种意义上是尖锐的.
Restricted weak-type endpoint estimates for discrete k-spherical maximal functions
本文利用k-球面上格点解集的傅里叶变换的近似公式和Bourgain及Ionescu方法，将离散k-球面极大函数的l^p（Z^d）-有界性结果改进为端点处的限制弱型结果.此外，我们引入了一个新的近似公式为一个单一的平均值，这使我们能够提高我们的边界离散k-球面极大函数沿稀疏的连续性的半径，利用最近的进展Wooley的Vinogradov平均值矩阵。特别地，我们改进了缺项离散k-球面极大函数的界，当k>2时。我们引入了一个密度参数，它可以被看作是Minkowski维数的离散版本中使用的相关作品的连续现象，证明我们的结果稀疏平均。
Following the ideas of A. Lerner, F. Nazarov, S. Ombrosi from [12] we prove that there is a sequence of weights $w\in A^d_1$ such that $[w]^d_{A_1}\to \infty$, and martingale transforms $T$ such that with an absolute positive $c$ $\|T: L^1(w) \to L^{1, \infty}(w)\| \ge c [w]^d_{A_1}\log [w]^d_{A_1}$. We also show the existence of the sequence of weights (now in $A_2$) such that $[w]^d_{A_2}\to \infty$, and such that the following holds: $[w]_{A_2^d}\asymp \|M^d\|_{w^{-1}}^2$; $\|S_{w}: L^{2} (w) \to L^2(w^{-1})\| \ge c\, \|M^d\|_{w^{-1}}\sqrt{\log \|M^d\|_{w^{-1}}}$; $\|S_{w}: L^{2,1} (w) \to …
Sparse domination implies vector-valued sparse domination
本文证明多线性算子的标量值稀疏控制蕴含向量值稀疏控制，引入UMD条件的多线性类比，从而直接获得精确的向量值加权界，并应用于改进Calderón-Zygmund算子与双线性Hilbert变换的界。
We derive bilateral estimates for the constants appearing in the Fourier transform restricted theorems on the Euclidean sphere for the ordinary and especially radial functions belonging to the Lebesgue-Riesz spaces as well as belonging to the Grand Lebesgue Spaces. We obtain an exact estimate for the norm of the restriction Fourier transform operator acting on the radial functions.
Sharp weighted bounds without testing or extrapolation
我们给出了一个简短的证明的尖锐的加权范围的稀疏运营商，持有所有$p$，$1<p<\infty$。根据最近的发展，这意味着任何Calderón-Zygmund算子的界都成立。我们的方法的新颖之处在于，我们避免了其他证明中存在的两种技术：两个权重不等式和外推。我们的技巧也适用于分数次积分算子。
The multilinear fractional sparse operator theory II: refining weighted estimates via multilinear fractional sparse forms
本文通过引入新的多线性分数阶稀疏形式和权类，改进多线性分数阶奇异积分算子广义交换子的加权估计，获得尖锐定量界并应用于分数阶Laplacian方程解的正则性。
Random Tessellations, Restricted Isometric Embeddings, and One Bit Sensing
我们得到了改进的边界为一位传感器。例如，令$ K_s$表示具有稀疏性参数$ 0 < s < n+1$的维度$ n+1$中的球体$ \mathbb S ^{n}$中的$ s$ -稀疏单位向量的集合，并且假设$ 0 < δ< 1$。我们表明，对于$ m \gtrsim δ^{-2} s \log \frac ns$，一位映射$$ x \mapsto \bigl[ {sgn} \langle x,g_j \rangle \bigr] _{j=1} ^{m}, $$，其中$ g_j$是$ \mathbb R ^{n+1}$上的iid高斯向量，具有很高的概率具有从$ K_s$到$ m$维汉明立方体的$ δ$ -RIP。这些界限与由$ x \mapsto \frac 1m[\langle x,g_j \rangle ] _{j=1} ^{m} $给出的{线性} $ δ$-RIP的界限相匹配，从$ \mathbb R ^{n}$中的稀疏向量到$ \ell ^{1}$中。换句话说，一位RIP和线性RIP同样有效。对于其他一位属性，如符号积RIP属性，也有相应的改进。
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