G. A. Karagulyan
Let $Ω$ be any set of directions (unit vectors) on the plane. We study maximal operators defined by \md0 M_Ωf(x)=\sup_{δ>0, ω\in Ω} \frac{1}{2δ}\int_{-δ}^δ|f(x+tω)|dt. \emd for the generalized lacunary sets $Ω$ associated with an integer $μ>0$. It is proved the following sharp inequality: $$ \|M_Ωf(x)\|_2\lesssim \sqrtμ \|f\|_2. $$
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Weighted estimates for lacunary maximal functions on homogeneous groups
该研究聚焦齐次群上广义缺项极大函数的加权估计问题，推导得出两类缺项极大函数的改进加权估计，为相关调和分析问题提供了新的结果。
We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First, we obtain an $H^1$ to $L^{1,\infty}$ bound for lacunary maximal operators under a dimensional assumption on the underlying measure and an assumption on an $L^p$ regularity bound for some $p>1$. Secondly, we obtain a necessary and sufficient condition for $L^2$ boundedness of lacunary maximal operator associated to averages over convex curves in the plane. Finally we prove an $L^p$ regularity result for such averages. We formulate various open problems.
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。
Let $Ω$ be any set of directions (unit vectors) on the plane. In this paper we study maximal operator of the one dimensional maximal function computed in the directions of $Ω$ We are interested in extensions of lacunary sets of directions, to collections we call $N$--lacunary, for integers $N$. We proceed by induction. Say that $Ω$ is 1--lacunary iff $Ω$ is an ordinary lacunary set of vectors. Every $N+1$--lacunary set can be obtained from some $N$--lacunary $Ω_N$ adding some points to $Ω_N$. Between each two neighbor points $a,b\inΩ_N$ we can add a 1--lacunary sequence (finite or infinite). W…
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。
Lacunary elliptic maximal operator on the Heisenberg group
本文研究 Heisenberg 群上 lacunary 椭圆极大算子的 L^p 有界性，将估计从斜对称矩阵推广至任意矩阵，并给出有界性的充要条件。
If mu is a smooth density on a hypersurface in R^d whose curvature never vanishes to infinite order, and A is a d-by-d matrix whose eigenvalues all have absolute value greater than 1, then the maximal function given by convolving f with dilates of mu by powers of A, and taking the maximum, is bounded from a corresponding version of H^1 to weak L^1.
We obtain positive and negative results concerning lacunary discrete maximal operators defined by dilations of sufficiently nonsingular hypersurfaces arising from Diophantine equations in many variables. Our negative results show that this problem differs substantially from that of lacunary discrete maximal operators defined along a nonsingular hypersurface. Our positive results are improvements over bounds for the corresponding full maximal functions which were initially studied by Magyar. In order to obtain positive results, we use an interpolation technique of the second author to reduce pr…
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 maximal functions on homogeneous groups
本文利用 Ricci--Stein 经典论证，证明齐性群上缺项膨胀测度极大函数的 $L^p$ 有界性，推广了 Korányi 球面及水平球面平均的最新结果，并通过实例展示广泛适用性。
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…
Fourier decay and endpoint estimates for lacunary maximal operators
该研究针对与紧支测度相关的缺项极大算子，构造了具有多项式傅里叶衰减的测度，证明其关联算子不满足弱型(1,1)，否定了相关公开问题。
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。
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。
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。
We prove bounds for the truncated directional Hilbert transform in $L^p(\mathbb{R}^2)$ for any $1<p<\infty$ under a combination of a Lipschitz assumption and a lacunarity assumption. It is known that a lacunarity assumption alone is not sufficient to yield boundedness for $p=2$, and it is a major question in the field whether a Lipschitz assumption alone suffices, at least for some $p$.
Let $H_k$ be the one dimensional Hilbert transform computed in the direction $(1,2^k)$ in the plane. We show that the maximal operator $\sup_k |H_kf|$ maps $L^p$ of the plane into itself for $1<p<\infty$. The same result with the Hilbert transform replaced by the one dimensional maximal function was proved by Nagel, Stein and Wainger in 1978.
On the almost everywhere convergence of two-parameter ergodic averages along directional rectangles
研究平面矩形上的双参数遍历平均的几乎处处收敛性，证明了当矩形边斜率属于有限缺项集时平均在L^p中几乎处处收敛，并构造了非缺项方向集上平均不收敛的例子。
In the present paper, we study the geometric discrepancy with respect to families of rotated rectangles. The well-known extremal cases are the axis-parallel rectangles (logarithmic discrepancy) and rectangles rotated in all possible directions (polynomial discrepancy). We study several intermediate situations: lacunary sequences of directions, lacunary sets of finite order, and sets with small Minkowski dimension. In each of these cases, extensions of a lemma due to Davenport allow us to construct appropriate rotations of the integer lattice which yield small discrepancy.
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 introduce the notion of \textit{Perron capacity} of a set of slopes $Ω\subset \mathbb{R}$. Precisely, we prove that if the Perron capacity of $Ω$ is finite then the directional maximal operator associated $M_Ω$ is not bounded on $L^p(\mathbb{R}^2)$ for any $1 < p < \infty$. This allows us to prove that the set $$Ω_{ \boldsymbol{e}} =\left\{ \frac{\cos n}{n}: n\in \mathbb{N}^* \right\}$$ is not finitely lacunary which answers a question raised by A. Stokolos.
Weak $L^2$ bound of the lacunary Carleson operator for the non-linear Fourier transform
本文证明了在位势属于$L^1$的条件下，与$SU(1,1)$值非线性傅里叶变换相关的缺项Carleson极大函数具有弱$L^2$有界性。
$L^p$ improving properties and maximal estimates for certain multilinear averaging operators
本文研究曲率超曲面上多线性 lacunary 极大平均的 $L^p$ 估计，利用 $L^1$ 改进估计与非线性 Brascamp-Lieb 不等式获得新结果，并给出双线性球面极大函数的新证明。
Singular integrals along lacunary directions in $\mathbb{R}^n$
本文针对所有维度中沿有限阶 lacunary 方向集的奇异积分，给出了尖锐的基数估计，通过二维楔形覆盖 n 维锥的新方法，推广了先前仅限二维和三维的结果。
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.
We discuss the L^p-boundedness of maximal singular integrals in the plane over a finite set V of N directions. Logarithmic bounds are established for a set V of arbitrary structure in the 2<=p<infinity range. Sharp bounds are proved for lacunary and Vargas sets of directions. The latter include the case of uniformly distributed directions and the finite truncations of the Cantor set. We make use of both classical harmonic analysis methods and product-BMO based time-frequency analysis techniques. As a further application of the latter, we derive an L^p almost orthogonality principle for Fourier…
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。
We prove a weighted inequality which controls conic Fourier multiplier operators in terms of lacunary directional maximal operators. By bounding the maximal operators, this enables us to conclude that the multiplier operators are bounded on $L^p(\mathbb{R}^3)$ with $1<p<\infty$.
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。
Maximal directional singular integrals
本文综述极大方向平均与方向奇异积分，探讨其与Kakeya–Nikodym现象、微分理论及Carleson算子的联系，以几乎正交性为统一工具，指出尖锐界的获得之处及需新思想的开放问题。
In this paper we introduce and study a bilinear spherical maximal function of product type in the spirit of bilinear Calderón-Zygmund theory. This operator is different from the bilinear spherical maximal function considered by Geba et al. We deal with lacunary and full versions of this operator, and we prove weighted estimates with respect to bilinear weights. Our approach involves sparse forms following ideas by Lacey, but we also use other techniques to handle the particular triplet $(2,2,1)$.
On the maximal directional Hilbert transform in three dimensions
本文在三维空间中建立了有限阶拉库纳方向集上极大希尔伯特变换 L^p 范数的锐利增长率，回答了 Parcet 和 Rogers 的问题，并通过模型极大算子与加权外推方法给出了首个高维锐利估计。
The Fefferman-Stein type inequalities for strong maximal operator and directional maximal operator are verified with composition of the Hardy-Littlewood maximal operator in the plane.
A sharp estimate for the Hilbert transform along finite order lacunary sets of directions
本文证明沿有限阶 lacunary 方向集的极大 Hilbert 变换的 $L^p$ 范数与方向数的对数平方根相当，并证明沿相应向量场的截断 Hilbert 变换的 $L^p$ 有界性，推广了已有结果。
We completely characterize the boundedness of planar directional maximal operators on L^p. More precisely, if Omega is a set of directions, we show that M_Omega, the maximal operator associated to line segments in the directions Omega, is unbounded on L^p, for all p < infinity, precisely when Omega admits Kakeya-type sets. In fact, we show that if Omega does not admit Kakeya sets, then Omega is a generalized lacunary set, and hence M_Omega is bounded on L^p, for p>1.
We study a two-dimensional discrete directional maximal operator along the set of the prime numbers. We show existence of a set of vectors, which are lattice points in a sufficiently large annulus, for which the $\ell^2$ norm of the associated maximal operator with supremum taken over all large scales grows with an epsilon power in the number of vectors. This paper is a follow-up to a prior work on the discrete directional maximal operator along the integers by the first and third author.
Survey on bilinear spherical averages and associated maximal operators
本文综述了双线性球面平均及其相关最大算子的最新研究进展，探讨了 $L^p$ 绑定条件、开放性问题及可能的推广方向。
Almost-orthogonality principles for certain directional maximal functions
本文为线段平均和方向奇异积分相关的极大函数建立几乎正交性原理，并由此在方向集均匀分布时得到尖锐的$L^2$界。
We show that for any infinite set of unit vectors $U$ in $\ZR^2$ the maximal operator defined by $$ H_Uf(x)=\sup_{u\in U}\bigg|\pv\int_{-\infty}^\infty \frac{f(x-tu)}{t}dt\bigg|,\quad x\in \ZR^2, $$ is not bounded in $L^2(\ZR^2)$.
We introduce a notion of (finite order) lacunarity in higher dimensions for which we can bound the associated directional maximal operators in $L^p(\mathbb{R}^n)$, with $p>1$. In particular, we are able to treat the classes previously considered by Nagel--Stein--Wainger, Sjögren--Sjölin and Carbery. Closely related to this, we find a characterisation of the sets of directions which give rise to bounded maximal operators. The bounds enable Lebesgue type differentiation of integrals in $L_{\text{loc}}^p(\mathbb{R}^n)$, replacing balls by tubes which point in these directions.
We prove $L^p\to L^q$ estimates for the local maximal operator associated with dilates of the Kóranyi sphere in Heisenberg groups. These estimates are sharp up to endpoints and imply new bounds on sparse domination for the corresponding global maximal operator. We also prove sharp $L^p\to L^q$ estimates for spherical means over the Korányi sphere, which can be used to improve the sparse domination bounds for the associated lacunary maximal operator.
Bilinear spherical maximal function on the Heisenberg group
本文研究了海森堡群上的双线性球极平均函数，推导了单尺度双线性平均算子、完整双线性球极极大算子及双线性空隙极大算子的映射估计，其中完整极大算子的结果是最佳的。
We develop a notion of finite order lacunarity for direction sets in $\mathbb R^{d+1}$. Given a direction set $Ω$ that is sublacunary according to this definition, we construct random examples of Euclidean sets that contain unit line segments with directions from $Ω$ and enjoy analytical features similar to those of traditional Kakeya sets of infinitesimal Lebesgue measure. This generalizes to higher dimensions a planar result due to Bateman. Combined with earlier work of Alfonseca, Bateman, Parcet and Rogers, this notion of lacunarity and Kakeya-type sets also yields a characterization in all…
For any dimension $n \geq 2$, we consider the maximal directional Hilbert transform $\mathscr{H}_U$ on $\mathbb R^n$ associated with a direction set $U \subseteq \mathbb S^{n-1}$: \[ \mathscr{H}_Uf(x) := \frac{1}π \sup_{v \in U} \Bigl| \text{p.v.} \int f(x - tv) \, \frac{dt}{t}\Bigr|.\] The main result in this article asserts that for any exponent $p \in (1, \infty)$, there exists a positive constant $C_{p,n}$ such that for any finite direction set $U \subseteq \mathbb S^{n-1}$, \[||\mathscr{H}_U||_{p \rightarrow p} \geq C_{p,n} \sqrt{\log \#U}, \] where $\#U$ denotes the cardinality of $U$. A…
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.
Given any finite direction set $Ω$ of cardinality $N$ in Euclidean space, we consider the maximal directional Hilbert transform $H_Ω$ associated to this direction set. Our main result provides an essentially sharp uniform bound, depending only on $N$, for the $L^2$ operator norm of $H_Ω$ in dimensions 3 and higher. The main ingredients of the proof consist of polynomial partitioning tools from incidence geometry and an almost-orthogonality principle for $H_Ω$. The latter principle can also be used to analyze special direction sets $Ω$, and derive sharp $L^2$ estimates for the corresponding ope…
Lebesgue space estimates for spherical maximal functions on Heisenberg groups
本研究证明海森堡群上余维2球面伸缩对应的局部极大算子的$L^p\ o L^q$估计（除两端点外最优），可改进全局极大算子稀疏支配界，还探讨了缺项变体及向Métivier群的推广。
Bilinear spherical maximal function with fractal dilations
本文研究了与一般集合E⊂R_+相关的双线性球极最大函数的L^p有界性，通过上Minkowski维度量化了有界范围，解决了分形情况下双线性球极最大函数在p_1=1或p_2=1时的有界性问题。
Given a family G of rectangles, to which one associates a tree [G], one defines a natural number $λ$ [G] called its analytic split and satisfying, for all 1 < p < $\infty$ log($λ$ [G]) p MG p p where MG is the Hardy-Littlewood type maximal operator associated to the family G. As an application, we completely characterize the boundeness of planar rarefied directional maximal operators on L p for 1 < p < $\infty$. Precisely, if $Ω$ is an arbitrary set of angles in [0, $π$ 4), we prove that any rarefied basis B of the directional basis R $Ω$ yields an operator MB that has the same L p-behavior th…
For a Schwartz function $f$ on the plane and a non-zero $v\in\ZR^2$ define the Hilbert transform of $f$ in the direction $v$ to be $$ H_vf(x)=\text{p.v.}\int_\ZR f(x-vy) \frac{dy}y $$ Let $ζ$ be a Schwartz function with frequency support in the annulus $1\le| ξ|\le2$. We prove that the maximal operator $$ \sup_{\abs v=1}\abs{H_vζ* f} $$ maps $L^2$ into weak $L^2$, and $L^p$ into $L^p$ for $p>2$. The $L^2$ estimate is sharp. The method of proof is based upon techniques related to the pointwise convergence of Fourier series, especially the recent proof given by Lacey and Thiele.
We prove L2 x L2 to weak L1 estimates for some novel bilinear maximal operators of Kakeya and lacunary type thus extending to this setting, the works of Cordoba and of Nagel, Stein and Wainger.
Sparse bounds for maximal triangle and bilinear spherical averaging operators
该研究通过调整Roncal等人方法，证明了双线性算子在特定$ L^p $指数范围内的稀疏界。
Maximal directional operators along algebraic varieties
本文建立了沿任意维数多项式簇上有限方向集的极大方向平均算子的 $L^2$ 范数的尖锐增长阶，推广了 Córdoba、Katz 和 Demeter 的工作，并改进了球面情形的已知界，同时给出 Kakeya 型极大函数的新估计。
In the first part of this paper we establish, in terms of so called k-tangential sets, a kind of optimal estimate for the size and structure of the set of non-differentiability of Lipshitz functions with one-sided directional derivatives. These results can be applied to many important special functions in analysis, like distance functions or different maximal functions. In the second part, having the results from the first part in our use, we focus more carefully on the differentiability properties of the classical Hardy-Littlewood maximal function. For example, we will show that if f is conti…
By a classical result of Weyl, for any increasing sequence $(n_k)_{k \geq 1}$ of integers the sequence of fractional parts $(\{n_k x\})_{k \geq 1}$ is uniformly distributed modulo 1 for almost all $x \in [0,1]$. Except for a few special cases, e.g. when $n_k=k, k \geq 1$, the exceptional set cannot be described explicitly. The exact asymptotic order of the discrepancy of $(\{n_k x\})_{k \geq 1}$ is only known in a few special cases, for example when $(n_k)_{k \geq 1}$ is a (Hadamard) lacunary sequence, that is when $n_{k+1}/n_k \geq q > 1, k \geq 1$. In this case of quickly increasing $(n_k)_{…
Sharp power-weight estimates for the fractional spherical maximal operator
针对分数阶球极大算子，本文刻画了使对应幂权估计成立的β范围，在p=q的对角情形下证实了Lee的猜想，并对缺项分数阶球极大算子证明了类似结果。
Directional maximal operators in the plane
本研究针对平面方向极大算子$D_{\Omega}$的Lebesgue有界性问题，修正了Bateman 2009年相关证明的漏洞，引入可容许有限阶缺隙性概念，建立了有限阶缺隙性、无Kakeya型集合与算子有界性的等价刻画。
We prove (essentially) sharp $L^2$ estimates for a restricted maximal operator associated to a planar vector field that depends only on the horizontal variable. The proof combines an understanding of such vector fields from earlier work of the author with a result of Nets Katz on directional maximal operators.
In $R^d$, define a maximal function in the directions $v\in \directions\subset\{x \mid \abs x=1\}$ by $$ M^\directions f(x)=\sup_{v\in\directions} \sup_{\zve} \int_{-\ze}^\ze \abs{f(x-vy)} dy. $$ For a function $f$ on $\ZR^d$, let $S_\zw f$ denote the Fourier restriction of $f$ to a region $\zw$. We are especially interested taking \zw to be a sector of $R^d$ with base points at the origin. A sector is a product of the interval $(0,\infty)$ with respect to a choice of (non orthogonal) basis. What is most important is that the basis is a subset of $\directions$. Consider a collection $\zW$ of p…
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