Andreas Seeger, Terence Tao, James Wright
We show that if $f$ is locally in $L\log\log L$ then the lacunary spherical means converge almost everywhere. The argument given here is a model case for more general results on singular maximal functions and Radon transforms (see ref. 6).
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We prove that the lacunary Carleson operator is bounded from $L \log L$ to $L^{1}$. This result is sharp. The proof is based on two newly introduced concepts: 1) the \emph{time-frequency regularization of a measurable set} and 2) the \emph{set-resolution of the time-frequency plane at $0-$frequency}. These two concepts will play the central role in providing a special tile decomposition adapted to the interaction between the \emph{structure} of the lacunary Carleson operator and the corresponding \emph{structure} of a fix measurable set. Another key insight of our paper is that it provides for…
In his 2006 ICM invited address, Konyagin mentioned the following conjecture: if $S_n f$ stands for the $n$-th partial Fourier sum of $f$ and ${n_j}_j\subset \N$ is a lacunary sequence, then $S_{n_j} f$ is a.e. pointwise convergent for any $f\in L\log\log L$. In this paper we will show that $| \sup_{j} |S_{n_j}(f)| |_{1,\infty}\leq C |f|_{1} \log\log (10+\frac{|f|_{\infty}}{|f|_1})\:.$ As a direct consequence we obtain that $S_{n_j}f \rightarrow f $ a.e. for $f\in L\log\log L\log\log\log L$. The (discrete) Walsh model version of this last fact was proved by Do and Lacey but their methods do no…
Improved endpoint bounds for the lacunary spherical maximal operator
本文证明了缺项球面极大算子的改进端点界，并由此得出局部属于特定 $L$ 空间的函数的缺项球面均值几乎处处逐点收敛。
Lacunary Spherical Maximal Operators on Hyperbolic Spaces
本文证明在n维实双曲空间上，缺项球面极大算子对所有n≥2和1<p≤∞在L^p上有界，并揭示双曲空间无穷远几何导致缺项集比欧氏情形更大。
We prove that for $d\ge 2$, the asymptotic order of the usual Nikolskii inequality on $\mathbb{S}^d$ (also known as the reverse Hölder's inequality) can be significantly improved in many cases, for lacunary spherical polynomials of the form $f=\sum_{j=0}^m f_{n_j}$ with $f_{n_j}$ being a spherical harmonic of degree $n_j$ and $n_{j+1}-n_j\ge 3$. As is well known, for $d=1$, the Nikolskii inequality for trigonometric polynomials on the unit circle does not have such a phenomenon.
We prove new $\ell ^{p} (\mathbb Z ^{d})$ bounds for discrete spherical averages in dimensions $ d \geq 5$. We focus on the case of lacunary radii, first for general lacunary radii, and then for certain kinds of highly composite choices of radii. In particular, if $ A _{λ} f $ is the spherical average of $ f$ over the discrete sphere of radius $ λ$, we have \begin{equation*} \bigl\lVert \sup _{k} \lvert A _{λ_k} f \rvert \bigr\rVert _{\ell ^{p} (\mathbb Z ^{d})} \lesssim \lVert f\rVert _{\ell ^{p} (\mathbb Z ^{d})}, \qquad \tfrac{d-2} {d-3} < p \leq \tfrac{d} {d-2},\ d\geq 5, \end{equation*} f…
In this paper we investigate the $L^p$ boundedness of the lacunary maximal function $ M_{\Ha}^{lac} $ associated to the spherical means $ A_r f$ taken over Koranyi spheres on the Heisenberg group. Closely following an approach used by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions leading to new unweighted and weighted estimates. The key ingredients in the proof are the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-τ_y A_rf$, where $τ_yf(x)=f(xy^{-1})$ is the right translation operator.
Lacunary $δ$-Discretised Spherical Maximal Operators
本文研究lacunary $\delta$-离散球面极大算子的$L^p$有界性及端点弱型估计，常数一致，极限恢复经典结果。
Lebesgue bounds for multilinear spherical and lacunary maximal averages
本文研究了多线性球面和空隙最大平均值的勒贝格界，通过证明球面平均算子的映射性质，建立了相应的L^p空间上的有界性结果。
Bounds for lacunary bilinear spherical and triangle maximal functions
本文研究缺项双线性球面与三角形极大函数的 $L^p\times L^q\rightarrow L^r$ 有界性，在球面情形对 $d\geq2$ 得到尖锐结果并推广至退化曲面，三角形情形在 $d\geq7$ 取得部分进展。
$L^{p}-$estimates for uncentered spherical averages and lacunary maximal functions
本文引入非中心球面平均等算子的双线性类似物，获得了$d\geq2$维度下的$L^p$估计，并在线性情形下证明了单尺度平均算子的$L^p$改进性质与缺项极大函数的$L^p$估计。
In this work we obtain the pointwise almost everywhere convergence for two families of multilinear operators: (a) truncated homogeneous singular integral operators associated with $L^q$ functions on the sphere and (b) lacunary multiplier operators of limited decay. The a.e. convergence is deduced from the $L^2\times\cdots\times L^2\to L^{2/m}$ boundedness of the associated maximal multilinear operators.
We show that the discrete lacunary spherical maximal function is bounded on $l^p(\mathbb{Z}^d)$ for all $p >\frac{d+1}{d-1}$. Our range is new in dimension 4, where it appears that little was previously known for general lacunary radii. Our technique follows that of Kesler-Lacey-Mena, using the Kloosterman refinement to improve the estimates in several places, which leads to an overall improvement in dimension 4.
In this paper we deal with lacunary and full versions of the spherical maximal function on the Heisenberg group $\mathbb{H}^n$, for $n\ge 2$. By suitable adaptation of an approach developed by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions, which lead to new unweighted and weighted estimates. In particular, we deduce the $L^p$ boundedness, for $1<p<\infty$, of the lacunary maximal function associated to the spherical means on the Heisenberg group. In order to prove the sparse bounds, we establish $ L^p-L^q $ estimates for local (single scale) variants of th…
We provide a near-complete classification of the Lorentz spaces $Λ_φ$ for which the sequence $\{S_{n}\}_{n\in \mathbb{N}}$ of partial Fourier sums is almost everywhere convergent along lacunary subsequences. Moreover, under mild assumptions on the fundamental function $φ$, we identify $Λ_φ:= L\log\log L\log\log\log\log L$ as the \emph{largest} Lorentz space on which the lacunary Carleson operator is bounded as a map to $L^{1,\infty}$. In particular, we disprove a conjecture stated by Konyagin in his 2006 ICM address. Our proof relies on a newly introduced concept of a "Cantor Multi-tower Embed…
Boundary estimates for the fractional spherical maximal function
本文研究了分数球极最大函数及其空隙版本的$ L^p-L^q $有界性条件，证明了在最大$ L^p-L^q $有界区域边界处的限制弱类型估计。
On a variant of bilinear spherical maximal function
该研究针对双线性球面平均对应的全极大函数与间隙极大函数，在任意维度下获得 $L^p$ 估计，其中高维估计依赖切片法，一维间隙极大函数的界需用到三线性平滑不等式。
Counterexamples for lacunary dilates via dyadic spike blocks
本文通过双进制尖峰块构造了Erdős关于圆周上稀疏延拓点wise行为的两个反例，展示了在c=1/2端点处Matsuyama正定理无法延伸，并回答了Erdős问题#996的否定。
Polynomial Ergodic Averages Along Short Intervals
该研究证明了短区间上次数$d\geq2$、起始时间双间隙的多项式轨道在$1<p<\infty$且$c>(d-1)/d$时的$L^p$变分估计及几乎处处收敛，还指出端点$L^1$不成立，且密集起始时间序列具强扫出性质。
One sided orthogonal polynomials and a pointwise convergence result for $SU(2)$-valued nonlinear Fourier series
本文研究$SU(2)$值非线性傅里叶级数与单位圆上正交多项式的关系，证明再生核的普适性收敛结果，并建立多项式乘积逐点收敛与零点及局部参数的联系，最后在特定条件下证明沿缺项序列的几乎处处收敛。
We extend the main result of Harrow, Kolla, and Schulman -- the existence of dimension-free $L^2$-bounds for the spherical maximal function in the hypercube -- to all $L^p, p > 1$. Our approach is motivated by the spectral technique developed by Stein and Nevo and Stein in the context of pointwise ergodic theorems on general groups. We provide an example which demonstrates that no dimension-free weak-type (1-1) bound exists at the endpoint.
Pointwise Convergence of Ergodic Averages Along Integer Cantor Sets
研究沿整数康托集的遍历平均的逐点收敛，通过证明相关遍历平均在特定条件下几乎处处收敛，解决了康托测度在自相似尺度下的稀疏微分问题。
On the endpoint estimate for discrete spherical average over sparse sequences
本文针对高度复合 regime 下的 lacunary 离散球面极大算子，建立了适配序列的 Orlicz 弱型端点估计，还给出了特定序列下的 $\ell(\log\ell)^2\to\ell^{1,\infty}$ 估计，解决了相关端点行为问题。
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