F. Nazarov, S. Petermichl, K. A. Škreb, S. Treil
The convex body maximal operator is a natural generalisation of the Hardy Littlewood maximal operator. In the presence of a matrix weight it is not bounded.
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We prove sharp $L^1$ inequalities for the dyadic maximal function $M_Tϕ$ when $ϕ$ satisfies certain $L^1$ and $L^{\infty}$ conditions
In this paper, we show that dyadic paraproducts $π_b$ with $b$ in dyadic BMO are bounded on matrix weighted $L^p(W)$ if $W$ is a matrix $\text{A}_p$ weight.
Sharp upper bounds for integral quantities related to the Bellman function of three variables of the dyadic maximal operator
本文研究二进极大算子三积分变量 Bellman 函数相关积分量，给出其精确上界。
On a sharp upper bound related to the Bellman function of three integral variables of the dyadic maximal operator
该研究聚焦于二进极大算子的三积分变量Bellman函数相关积分量的精确上界，探究其性质，该上界已由文献[11]确定。
Let $S$ be the Lie group $\mathrm{R}^n\ltimes \mathrm{R}^+$ endowed with the left-invariant Riemannian symmetric space structure and the right Haar measure $ρ$, which is a Lie group of exponential growth. Hebisch and Steger in [Math. Z. 245(2003), 37--61] proved that any integrable function on $(S,ρ)$ admits a Calderón--Zygmund decomposition which involves a particular family of sets, called Calderón--Zygmund sets. In this paper, we first show the existence of a dyadic grid in the group $S$, which has {nice} properties similar to the classical Euclidean dyadic cubes. Using the properties of th…
We provide an alternative proof and expression of the Bellman function of the dyadic maximal operator in connection with the Dyadic Carleson Imbedding Theorem, which appears in [10]. We also evaluate the Bellman function of four variables of the dyadic maximal operator
On a constant related to the Bellman function of three integral variables of the dyadic maximal operator: Part A
本文研究二元极大算子三个积分变量Bellman函数相关常数关于第二变量的单调性。
On a constant related to the Bellman function of three integral variables of the dyadic maximal operator: Part B
本文研究二进极大算子三变量Bellman函数相关常数的行为，重点分析该常数关于其第一个变量的单调性性质。
We give a new proof of the sharp weighted $L^2$ inequality ||T||_{L^2(w)} \leq c [w]_{A_2} where $T$ is the Hilbert transform, a Riesz transform, the Beurling-Ahlfors operator or any operator that can be approximated by Haar shift operators. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner to estimate the oscillation of dyadic operators.
Using Wilson's Haar basis in $\R^n$, which is different than the usual tensor product Haar functions, we define its associated dyadic paraproduct in $\R^n$. We can then extend "trivially" Beznosova's Bellman function proof of the linear bound in $L^2(w)$ with respect to $[w]_{A_2}$ for the 1-dimensional dyadic paraproduct. Here trivial means that each piece of the argument that had a Bellman function proof has an $n$-dimensional counterpart that holds with the same Bellman function. The lemma that allows for this painless extension we call the good Bellman function Lemma. Furthermore the argum…
On the weighted weak-type constant for the dyadic square function
本文通过显式构造例子，证明了二进平方函数的加权弱型估计在A_2特征上是最优的。
We prove a sharp integral inequality which connects the dyadic maximal operator with the Hardy operator. We also give some applications of this inequality.
We prove a sharp integral inequality for the dyadic maximal operator and give as an application another proof for the computation of its Bellman function of three variables.
We prove a necessary condition that has every extremal sequence for the Bellman function of the dyadic maximal operator.This implies the weak-Lp uniqueness for such a sequence.
We obtain sharp estimates for the quasi norm of the maximal function of f when it satisfies certain conditions.
We give a characterization of the extremal sequences for the Bellman function of the dyadic maximal operator.In fact we prove that they behave approximately like eigenfunctions of this operator for a specific eigenvalue.
We provide sharp weak estimates for the distribution function of Mϕwhen on ϕwe impose L1, Lq and Lp,1 restrictions. Here M is the dyadic maximal operator associated to a tree T on a non-atomic probability measure space.
We obtain sharp estimates for the localized distribution function of Mϕ, when ϕbelongs to Lp,\inf where M is the dyadic maximal operator. We obtain these estimates given the L1 and Lq norm, q < p and certain weak Lp-conditions.
For p>1 we find the Bellman function of two variables associated with the dyadic maximal operator on Rn.Actually we do that in the more general setting of tree-like maximal operators.We provide a simple and elementary proof,different from those in [4] and [9].
We compute the Bellman function of three integral variables associated to the dyadic maximal operator on a subset of its domain. Additionally, we provide an upper bound for the whole domain of its definition.
We provide some new estimates for Bellman type functions for the dyadic maximal opeator on $R^n$ and of maximal operators on martingales related to weighted spaces. Using a type of symmetrization principle, introduced for the dyadic maximal operator in earlier works of the authors we introduce certain conditions on the weight that imply estimate for the maximal operator on the corresponding weighted space. Also using a well known estimate for the maximal operator by a double maximal operators on different m easures related to the weight we give new estimates for the above Bellman type function…
We precisely evaluate Bellman type functions for the dyadic maximal opeator on $R^n$ and of maximal operators on martingales related to local Lorentz type estimates. Using a type of symmetrization principle, introduced for the dyadic maximal operator in earlier works of the authors we precisely evaluate the supremum of the Lorentz quasinorm of the maximal operator on a function $ϕ$ when the integral of $ϕ$ is fixed and also the same Lorentz quasinorm of $ϕ$ is fixed. Also we find the corresponding supremum when the integral of $ϕ$ is fixed and several weak type conditions are given.
The dyadic paraproduct is bounded in weighted Lebesgue spaces $L_p(w)$ if and only if the weight $w$ belongs to the Muckenhoupt class $A_p^d$. However, the sharp bounds on the norm of the dyadic paraproduct are not known even in the simplest $L_2(w)$ case. In this paper we prove the linear bound on the norm of the dyadic paraproduct in the weighted Lebesgue space $L_2(w)$ using Bellman function techniques and extrapolate this result to the $L_p(w)$ case.
We prove that the extremal sequences for the Bellman function of the dyadic maximal operator behave approximately as eigenfunctions of this operator for a specific eigenvalue. We use this result to prove the analogous one with respect to the Hardy operator.
We precisely compute the Bellman function of two variables of the dyadic maximal operator in relation to Kolmogorov inequality. In this way we give an alternative proof of the results in [5].Additionally, we characterize the sequences of functions that are extremal for this Bellman function.The proof of this is based on that is given in this paper for the Bellman function we are interested in.
Properties of a maximal function for vector-valued martingales were studied by the author in an earlier paper. Restricting here to the dyadic setting, we prove the equivalence between (weighted) L^p inequalities and weak type estimates, and discuss an extension to the case of locally finite Borel measures on R^n. In addition, to compensate for the lack of an L^\infty inequality, we derive a suitable BMO estimate. Different dyadic systems in different dimensions are also considered.
We give a characterization of the extremal sequences for the Bellman function of three variables of the dyadic maximal operator in relation to Kolmogorov's inequality. In fact we prove that they behave approximately like eigenfunctions of this operator for a specific eigenvalue. For this approach we use the one introduced in [4], where the respective Bellman function has been precisely evaluated.
We provide a description for the Bellman function related to the Carleson Imbedding theorem, first mentioned in [4], with the use of the Hardy operator.
A multiparameter integral inequality for the dyadic maximal operator and applications
本文证明了二进极大算子的尖锐多参数积分不等式，改进单参数结果，并用于确定三变量Bellman函数的定义域及下界。
Towards direct $L^2$-bounds for maximal partial sums of Walsh--Fourier series: The case of dyadic partial sums
本文提出了一种直接获得二进制部分和沃尔什-傅里叶级数最大部分和 $L^2$ 界的方法，并扩展到一般线性化部分和算子的界证明。
We introduce atoms for dyadic atomic $H^1$ for which the equivalence between atomic and maximal function defnitions is dimension independent. We give the sharp, up to $\log(d)$ factor, estimates for the $H^1 \to L^1$ norm estimates for the special maximal function.
Uncentered Fractional Maximal functions and mean oscillation spaces associated with dyadic Hausdorff content
本文研究了非中心分数极大函数在关联于二进Hausdorff内容的平均振荡空间上的作用，改进了其在BMO和VMO函数上的现有结果，并证明了多维非中心极大函数在相应空间上的有界性。
We prove a sharp integral inequality for the dyadic maximal operator due to which the evaluation of the Bellman function of this operator with respect to two variables is possible, as can be seen in [3]. Our inequality of interest is proved in this article by a simpler and more immediate way. We also study a stability result in connection with this inequality, that is we provide a necessary and sufficient condition, for a sequence of functions,under which we obtain equality in the limit. The proof of this result is based on the proof of the related inequality which we present in this article.
Self-improving boundedness of the maximal operator on quasi-Banach lattices over spaces of homogeneous type
本文证明了齐型空间上具有Fatou性质的拟Banach格上Hardy--Littlewood极大算子的自改进性质，推广了2010年欧氏空间上的有界性准则，并将其应用于变指标Lebesgue空间。
Given a general dyadic grid ${\mathscr{D}}$ and a sparse family of cubes ${\mathcal S}=\{Q_j^k\}\in {\mathscr{D}}$, define a dyadic positive operator ${\mathcal A}_{\mathscr{D},{\mathcal S}}$ by $${\mathcal A}_{\mathscr{D},{\mathcal S}}f(x)=\sum_{j,k}f_{Q_j^k}χ_{Q_j^k}(x).$$ Given a Banach function space $X({\mathbb R}^n)$ and the maximal Calderón-Zygmund operator $T_{\natural}$, we show that $$\|T_{\natural}f\|_X\le c(n,T)\sup_{\mathscr{D},{\mathcal S}}\|{\mathcal A}_{\mathscr{D},{\mathcal S}}f\|_{X}.$$ This result is applied to weighted inequalities. In particular, it implies: (i) the "two-w…
We discuss the dyadic John-Nirenberg space that is a generalization of functions of bounded mean oscillation. A John-Nirenberg inequality, which gives a weak type estimate for the oscillation of a function, is discussed in the setting of medians instead of integral averages. We show that the dyadic maximal operator is bounded on the dyadic John-Nirenberg space and provide a method to construct nontrivial functions in the dyadic John-Nirenberg space. Moreover, we prove that the John-Nirenberg space is complete. Several open problems are also discussed.
We give a new proof of the sharp one weight $L^p$ inequality for any operator $T$ that can be approximated by Haar shift operators such as the Hilbert transform, any Riesz transform, the Beurling-Ahlfors operator. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner to estimate the oscillation of dyadic operators. Our method is flexible enough to prove the corresponding sharp one-weight norm inequalities for some operators of harmonic analysis: the maximal singular integrals associated to $T$, Dyadic square functions …
Precompactness in matrix weighted Bourgain-Morrey spaces
本文研究了矩阵加权Bourgain-Morrey空间中预紧集的条件，证明了二进制平均算子的有界性，并揭示了预紧集的必要性。
We show that the discrete Hardy-Littlewood maximal functions associated with the Euclidean balls in $\mathbb Z^d$ with dyadic radii have bounds independent of the dimension on $\ell^p(\mathbb Z^d)$ for $p\in[2, \infty]$.
Boundedness of the dyadic maximal function on graded Lie groups
本文将欧氏空间二进极大函数的有界性结果推广到任意分级李群，利用群 Fourier 变换关于 Rockland 算子的阶给出 $L^p$ 有界性的判别准则。
We provide a quantitative two weight estimate for the dyadic paraproduct $π_b$ under certain conditions on a pair of weights $(u;v)$ and $b$ in $Carl_{u,v}$, a new class of functions that we show coincides with BMO when $u = v \in A^d_2$. We obtain quantitative two weight estimates for the dyadic square function and the martingale transforms under the assumption that the maximal function is bounded from $L_2(u)$ into $L_2(v)$ and $v \in RH^d_1$. Finally we obtain a quantitative two weight estimate from $L_2(u)$ into $L_2(v)$ for the dyadic square function under the assumption that the pair $(u…
The aim of this paper is to get the product Lp-estimates, weighted estimates and two-weighted estimates for rough multilinear fractional integral operators and rough multi-sublinear fractional maximal operators, respectively. The author also studies two-weighted weak type estimate on product Lp (Rn) for rough multi-sublinear fractional maximal operators. In fact, this article is the rough kernel versions of [4, 5]' s results.
Characterizations for multilinear fractional maximal and integral operators and their commutators on generalized weighted Morrey spaces and applications
本文研究了多线性分数次极大与积分算子及其交换子在广义加权Morrey空间上的有界性，证明了两种有界性刻画，并应用于Besov-Morrey空间有界性、嵌入定理及次拉普拉斯算子先验估计。
Let $({\mathcal X},ρ,μ)$ be a space of homogeneous type in the sense of Coifman and Weiss, and $Y({\mathcal X})$ a ball quasi-Banach function space on ${\mathcal X}$, which supports a Fefferman--Stein vector-valued maximal inequality, and the boundedness of the powered Hardy--Littlewood maximal operator on its associate space. The authors first introduce the Hardy space $H_{Y}^*({\mathcal X})$, associated with $Y({\mathcal X})$, via the grand maximal function, and then establish its various real-variable characterizations, respectively, in terms of radial or non-tangential maximal functions, a…
Noncommutative maximal differential transforms associated to averaging operators
研究与平均算子相关的非交换极大微分变换，建立相关算子族的非交换极大弱型与强型估计，创新提出非光滑核的非交换Cotlar型不等式，得到该类变换的有界性理论。
Dimension-free estimates for semi-commutative discrete Hardy-Littlewood maximal operators
本文在半交换$L_{p}$空间上对离散二进Hardy-Littlewood极大算子建立了无维数估计，证明了半径充分大时对所有$1<p<\infty$均成立，并导出了极大遍历不等式与双边几乎一致收敛。
Commutators of the maximal and sharp functions with weighted Lipschitz functions
本文研究了Hardy-Littlewood极大函数与加权Lipschitz函数形成的极大交换子和非线性交换子在加权Lebesgue空间上的有界性，并给出了加权Lipschitz空间的新刻画，同时将类似结果推广到锐函数的非线性交换子。
The aim of the present paper is to obtain a Sjölin-type maximal estimate for pseudo-differential operators with homogeneous symbols. The crux of the proof is to obtain a phase decomposition formula which does not involve the time traslation. The proof is somehow parallel to the paper by Pramanik and Terwilleger (P. Malabika and E. Terwilleger, A weak $L^2$ estimate for a maximal dyadic sum operator on $R^n$, Illinois J. Math, {\bf 47} (2003), no. 3, 775--813). In the present paper, we mainly concentrate on our new phase decomposition formula and the results in the Cotlar type estimate, which a…
Commutators of Maximal Operators on Weighted Morrey Spaces
本文在齐型空间中的加权Morrey空间上，利用Hardy-Littlewood极大函数、sharp极大函数和分数极大函数的交换子刻画了BMO空间。
The Hardy-Littlewood Maximal Operator on Discrete Weighted Morrey Spaces
本文引入离散加权Morrey空间，借助加权$l^1$-序列的离散Calderón-Zygmund分解，证明了离散Hardy-Littlewood极大算子在该空间及离散加权Lebesgue空间上的有界性。
In this paper, we establish BMO estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively.
In this paper we investigate the boundedness of classical operators, namely the Hardy-Littlewood maximal operator, fractional integral operators, and Calderon-Zygmund operators, on generalized weighted Morrey spaces and generalized weighted weak Morrey spaces. We prove that the operators are bounded on these spaces, under certain assumptions.
Two-weight criteria of various type for the Hardy-Littlewood maximal operator and singular integrals in variable exponent Lebesgue spaces defined on the real line are established.
We establish the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type. In addition, with the aid of interpolation theory, we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space. Finally, a concrete example shows the local version of commutators also has an independent interest.
In this paper, we first give some new characterizations of Muckenhoupt type weights through establishing the boundedness of maximal operators on the weighted Lorentz and Morrey spaces. Secondly, we establish the boundedness of sublinear operators including many interesting in harmonic analysis and its commutators on the weighted Morrey spaces. Finally, as an application, the boundedness of strongly singular integral operators and commutators with symbols in BMO space are also given.
Weighted composition--differentiation operators on mixed-norm, derivative Hardy and F(p,q,s) spaces
该研究探讨混合范数、导数Hardy空间与F(p,q,s)空间上的加权复合微分算子，通过二进准则得到其有界性等性质，还获乘子、Schatten准则及精确奇异值等结果。
In this paper, we study the maximal ergodic operator on $L^p_w(X, \mathcal{B}, μ)$ spaces, $1 \leq p < \infty$, where $(X, \mathcal{B}, μ)$ is a probability space equipped with an invertible measure preserving transformation $U$ and $w$ is an ergodic $A_p$ weight using transference method.
Two-weight norm estimates for the double Hardy transforms and strong fractional maximal functions are established in variable exponent Lebesgue spaces. Derived conditions are simultaneously necessary and sufficient in the case when the exponent of the right-hand side space is constant.
In this note besides two abstract versions of the Vitali Covering Lemma an abstract Hardy-Littlewood Maximal Inequality, generalizing weak type (1,1) maximal function inequality, associated to any outer measure and a family of subsets on a set is introduced. The inequality is (effectively) satisfied if and only if a special numerical constant called Hardy-Littelwood maximal constant is finite. Two general sufficient conditions for the finiteness of this constant are given and upper bounds for this constant associated to the family of (centered) balls in homogeneous spaces, family of dyadic cub…
For the Hardy-Littlewood maximal and Calderón-Zygmund operators, the weighted boundedness on the Lebesgue spaces are well known. We extend these to the Orlicz-Morrey spaces. Moreover, we prove the weighted boundedness on the weak Orlicz-Morrey spaces. To do this we show the weak-weak modular inequality. The Orlicz-Morrey space and its weak version contain weighted Orlicz, Morrey and Lebesgue spaces and their weak versions as special cases. Then we also get the boundedness for these function spaces as corollaries.
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