Jose A. Barrionuevo, Lucas Oliveira
Using variants of the TT* method we give a self-contained proof of the result of Alfonseca, Soria and Vargas on maximal operators on arbitrary directions in $\rr^2$. We also give a sharp $L^2$ estimate for a maximal function extending a Theorem of Cordoba.
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We give a straighforward proof of the two weight estimates of the generalized maximal operator under Sawyer type testing conditions. The proof relies on the Martingale Carleson Embedding Theorem.
The main aim of this paper is to investigate $\left(H_{p},L_{p}\right)$- type inequalities for the the maximal operators of Nörlund logaritmic means, for $0<p<1.$
A note on maximal operators for moment curves
本文研究d≥3维欧氏空间中与矩曲线相关的极大算子，推导其Lᵖ映射性质，并在特殊情形下得到最优估计，属于调和分析领域的理论注记。
We prove a maximal Fourier restriction theorem for the sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^{d}$ for any dimension $d\geq 3$ in a restricted range of exponents given by the Stein-Tomas theorem. The proof consists of a simple observation. When $d=3$ the range corresponds exactly to the full Stein-Tomas one, but is otherwise a proper subset when $d>3$. We also present an application regarding the Lebesgue points of functions in $\mathcal{F}(L^p)$ when $p$ is sufficiently close to 1.
Let $0<α<n$ and $M_α$ be the fractional maximal function. The nonlinear commutator of $M_α$ and a locally integrable function $b$ is given by $[b,M_α](f)=bM_α(f)-M_α(bf)$. In this paper, we mainly give some necessary and sufficient conditions for the boundedness of $[b,M_α]$ on variable Lebesgue spaces when $b$ belongs to Lipschitz or $BMO(\rn)$ spaces, by which some new characterizations for certain subclasses of Lipschitz and $BMO(\rn)$ spaces are obtained.
In this paper maximal commutators and commutators of maximal functions with functions of bounded mean oscillation are investigated. New pointwise estimates for them are proved.
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.
We prove some new $L^p$ estimates for maximal Bochner-Riesz operator in the plane.
In this paper we prove that the Hardy-Littlewood maximal operator is bounded on Morrey spaces $\mathcal{M}_{1,λ}(\rn)$, $0 \le \la < n$ for radial, decreasing functions on $\rn$
A note on the maximal operator on Banach function spaces
本文肯定地回答了 Nieraeth (2023) 提出的关于重新缩放的 Banach 函数空间上极大算子的两个猜想，并给出了判别 Banach 函数空间上的极大算子在其伴随空间上亦有界的新准则。
A note on generalized spherical maximal means
本文针对广义球面极大平均算子，给出了文献[4]中定理1.1(i)的另一种证明，即当$n\geq2$且$M^\alpha$在$L^p$上有界时，$\operatorname{Re}\alpha$需满足的下界条件。
We study the two weight quantitative estimates for the commutator of maximal functions and the maximal commutators with respect to the symbol in weighted BMO space on spaces of homogeneous type. These commutators turn out to be controlled by the sparse operators in the setting of space of homogeneous type. The lower bound of the maximal commutator is also obtained.
There are two parts for this paper. In the first part, we extend some results in a recent paper by Du, Guth, Li and Zhang to a more general class of phase functions. The main methods are Bourgain-Demeter's $l^2$ decoupling theorem and induction on scales. In the second part, we prove some positive results for the maximal extension operator for hypersurfaces with positive principal curvatures. The main methods are sharp $L^2$ estimates by Du and Zhang, and the bilinear method by Wolff and Tao.
On Maximal Functions Associated to Families of Curves in the Plane
本文提出统一框架研究平面曲线族极大函数，证明尖锐L^p算子界，并应用于受限投影和Furstenberg型集合维数问题。
The purpose of this note is to provide a summary of the recent work of the authors on two variations of the pointwise convergence problem for the solutions to the fractional Schrödinger equations; convergence along a tangential line and along a set of lines, as exhibiting some new results in each setting. For the former case, we make a simple observation on a path along a tangential curve of exponential order. We discuss counterexamples for the latter case that show some of the known smooth regularities are essentially optimal.
In \cite{Lee:2006:schrod-converg}, when the spatial variable $x$ is localized, Lee observed that the Schrödinger maximal operator $e^{itΔ}f(x)$ enjoys certain localization property in $t$ for frequency localized functions. In this note, we give an alternative proof of this observation by using the method of stationary phase, and then include two applications: the first is on is on the equivalence of the local and the global Schrödinger maximal inequalities; secondly the local Schrödinger maximal inequality holds for $f\in H^{3/8+}$, which implies that $e^{itΔ}f$ converges to $f$ almost everywh…
On $L^{p}$-improving bounds for maximal operators associated with curves of finite type in the plane
本文研究平面中有限型曲线极大算子的 $L^{p}$-改进性质，揭示其改进区域由曲线有限型阶数与伸缩集分形维数共同决定，并进一步分析非各向同性伸缩的影响。
A (d,k) set is a subset of R^d containing a translate of every k-dimensional plane. Bourgain showed that for k \geq k_{cr}(d), where k_{cr}(d) solves 2^{k_{cr}-1}+k_{cr} = d, every (d,k) set has positive Lebesgue measure. We give a short proof of this result which allows for an improved L^p estimate of the corresponding maximal operator, and which demonstrates that a lower value of k_{cr} could be obtained if improved mixed-norm estimates for the x-ray transform were known.
In this note, we provide various two-weight norm estimates of the multi-linear fractional maximal function and weighted maximal function between different Orlicz spaces. More precisely, we obtain Sawyer-type characterizations and norm estimates for these operators.
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…
In this note we describe some recent advances in the area of maximal function inequalities. We also study the behaviour of the centered Hardy-Littlewood maximal operator associated to certain families of doubling, radial decreasing measures, and acting on radial functions. In fact, we precisely determine when the weak type $(1,1)$ bounds are uniform in the dimension.
We investigate the relation between Carleson sequence and balayage, and use this to give an easy proof of the equivalence of the L1-norms of the maximal function and the square function in non-honogeneous martingale settings.
We obtain the off-diagonal Muckenhoupt-Wheeden conjecture for Calderón-Zygmund operators. Namely, given $1<p<q<\infty$ and a pair of weights $(u,v)$, if the Hardy-Littlewood maximal function satisfies the following two weight inequalities: $$ M : L^p(v) \rightarrow L^q(u) \quad \text{and} \quad M: L^{q'}(u^{1-q'}) \rightarrow L^{p'}(v^{1-p'}), $$ then any Calderón-Zygmund operator $T$ and its associated truncated maximal operator $T_\star$ are bounded from $L^p(v)$ to $L^q(u)$. Additionally, assuming only the second estimate for $M$ then $T$ and $T_\star$ map continuously $L^p(v)$ into $L^{q,\…
In this paper, for general plane curves $γ$ satisfying some suitable smoothness and curvature conditions, we obtain the single annulus $L^p(\mathbb{R}^2)$-boundedness of the Hilbert transforms $H^\infty_{U,γ}$ along the variable plane curves $(t,U(x_1, x_2)γ(t))$ and the $L^p(\mathbb{R}^2)$-boundedness of the corresponding maximal functions $M^\infty_{U,γ}$, where $p>2$ and $U$ is a measurable function. The range on $p$ is sharp. Furthermore, for $1<p\leq 2$, under the additional conditions that $U$ is Lipschitz and making a $\varepsilon_0$-truncation with $γ(2 \varepsilon_0)\leq 1/4\|U\|_{\te…
We study weighted boundedness of Hardy-Littlewood-type maximal function involving Orlicz functions. We also obtain some sufficient conditions for the weighted boundedness of the Hardy-Littlewood maximal function of the upper-half plane.
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.
A note on directional Lipschitz continuity in the Euclidean plane
该研究针对2017年J. M. Ash与S. Catoiu论文中关于欧几里得平面Lipschitz性质与可微性关系的猜想，证明了其更强形式，并改进了原论文的主要结果。
The Strong Matrix Weighted Maximal Operator
本文证明了强Christ-Goldberg极大函数的有界性，并给出了带双参数矩阵权重的Rubio de Francia外推定理。
The purpose of this note is to find the least weak type $(1,1)$ bound for the almost uncentered maximal operator on radial decreasing functions.
In this paper we consider weighted Morrey spaces ${\mathcal M}_{λ, {\mathcal F}}^p(w)$ adapted to a family of cubes ${\mathcal F}$, with norm $$\|f\|_{{\mathcal M}_{λ, {\mathcal F}}^p(w)}:=\sup_{Q\in {\mathcal F}}\left(\frac{1}{|Q|^λ}\int_Q|f|^pw\right)^{1/p},$$ and the question we deal with is whether a Muckenhoupt-type condition characterizes the boundedness of the Hardy--Littlewood maximal operator on ${\mathcal M}_{λ, {\mathcal F}}^p(w)$. In the case of the global Morrey spaces (when ${\mathcal F}$ is the family of all cubes in ${\mathbb R}^n$) this question is still open. In the case of t…
We discuss a planar variant of the Kakeya maximal function in the setting of a vector space over a finite field. Using methods from incidence combinatorics, we demonstrate that the operator is bounded from $L^p$ to $L^q$ when $1 \leq p \leq \frac{kn+k+1}{k(k+1)}$ and $1 \leq q \leq (n-k)p'$.
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.
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