N. Asmar, Stephen J. Montgomery-Smith
In a celebrated paper, Burkholder, Gundy, and Silverstein used Brownian motion to derive a maximal function characterization of H^p spaces for 0 < p < infinity. In this paper, we show that their method extends to higher dimensions and yields a dimension-free weak type (1,1) estimate for a conjugate function on the N-dimensional torus.
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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.
A Direct Proof of Hardy-Littlewood Maximal Inequality for Operator-valued Functions
本文针对$2<p<\infty$的情形，给出了算子值Hardy-Littlewood极大不等式的一个直接证明。
The Gaussian Conjugate Rogers-Shephard Inequality
本文推导了一个关于高斯测度的共轭罗杰斯-谢普德不等式，并扩展了相关高斯不等式。
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。
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.
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.
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…
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.
We prove a sharp integral inequality which connects the dyadic maximal operator with the Hardy operator. We also give some applications of this inequality.
Let $M$ be a von Neumann algebra with a faithful normal finite trace $t$, and $H^\infty$ be a finite, maximal, subdiagonal of $M$. Fundamental theorems on conjugate functions for weak* Dirichlet algebras are shown to be a bounded linear map from $L^p(M,t)$ into $L^p(M,t)$ for $1<p<\infty$, and to be a continuous linear map from $L^1(M,t)$ into $L^{1,\infty}(M,t)$. We also obtain that if a positive operator $a$ is such that $a\log^{+}a \in L^1(M,t)$, then its conjugate belongs to $L^1(M,t)$.
Maximal monotonicity and cyclic involutivity of multi-conjugate convex functions
本文研究多个凸函数情形下循环对合性与极大单调性的成立条件，证明在一维实数直线上该性质保持成立，而在多维空间中一般需额外正则性假设，并通过实例揭示对偶与多重共轭凸函数及不同维度空间之间的本质差异。
Noncommutative spherical maximal inequality associated with automorphisms
该研究建立冯·诺依曼代数上与自同构相关的非交换球面极大不等式，实现了Magyar-Stein-Wainger离散球面极大不等式向非交换领域的拓展。
A noncommutative maximal inequality for Fejér means on totally disconnected non-abelian groups
本文在完全不连通非交换紧群上，对共轭类不变的非交换L_p值函数建立了Fejér均值的非交换极大不等式，推广了Gát的经典结果，并显式计算了估计的界。
In a recent paper in Journal of Convex Analysis the authors studied, in non-reflexive Banach spaces, a class of maximal monotone operators, characterized by the existence of a function in Fitzpatrick's family of the operator which conjugate is above the duality product. This property was used to prove that such operators satisfies a restricted version of Brondsted-Rockafellar property. In this work we will prove that if a single Fitzpatrick function of a maximal monotone operator has a conjugate above the duality product, then all Fitzpatrick function of the operator have a conjugate above the…
We prove a local version of Fefferman-Stein inequality for the local sharp maximal function, and a local version of John-Nirenberg inequality for locally BMO functions, in the framework of locally homogeneous spaces, in the sense of Bramanti-Zhu [Manuscripta Math. 138 (2012), no. 3-4, 477-528].
A hybrid method for countable equilibrium, variational inequality and maximal monotone inclusion problems with fixed point constraints
提出一种混合投影方法，在一致光滑和一致凸的实Banach空间中，逼近可数族广义非扩张型映射的不动点集、可数多个均衡问题解集、可数多个变分不等式问题解集和可数多个极大单调算子零点集的公共元，并证明强收敛性。
Young's inequality for Banach function spaces and its application to the maximal regularity estimate
该研究给出了Young不等式成立的Banach函数空间的充要条件，并将其应用于相关Besov空间上热方程的极大正则性估计。
A noncommutative maximal inequality for ergodic averages along arithmetic sets
本文在非交换 $L_p$ 空间中，对沿算术集合 $\{k^t\}$ 的遍历平均建立了极大不等式，并给出了 $p$ 的适用范围。
In this note, we establish a novel maximal inequality of the 2D Young integral $\int_a^b\int_c^d FdG$ in terms of the $(p,q)$-bivariation norms of the section functions $x\mapsto F(x,y)$ and $y\mapsto F(x,y)$ where $G:[a,b]\times [c,d]\rightarrow \mathbb{R}$ is a controlled path satisfying finite $(p,q)$-variation conditions. The proof is reminiscent from the Young's original ideas \cite{young1} in defining two-parameter integrals in terms of $(p,q)$-finite bivariations. Our result complements the standard maximal inequality established by Towghi \cite{towghi1} in terms of joint variations. We…
In this paper, we study the behavior of the bounds of matrix-valued maximal inequality in $\mathbb{R}^n$ for large $n$. The main result of this paper is that the $L_p$-bounds ($p>1$) can be taken to be independent of $n$, which is a generalization of Stein and Strömberg's resut in the scalar-valued case. We also show that the weak type $(1,1)$ bound has similar behavior as Stein and Stömberg's.
In this work we extend the theory of the classical Hardy space $H^1$ to the rational Dunkl setting. Specifically, let $Δ$ be the Dunkl Laplacian on a Euclidean space $\mathbb{R}^N$. On the half-space $\mathbb{R}_+\times\mathbb{R}^N$, we consider systems of conjugate $(\partial_t^2+Δ_{\mathbf{x}})$-harmonic functions satisfying an appropriate uniform $L^1$ condition. We prove that the boundary values of such harmonic functions, which constitute the real Hardy space $H^1$, can be characterized in several different ways, namely by means of atoms, Riesz transforms, maximal functions or Littlewood-…
In this paper we study sharp pointwise inequalities for maximal operators. In particular, we strengthen DeVore's inequality for the moduli of smoothness and a logarithmic variant of Bennett--DeVore--Sharpley's inequality for rearrangements. As a consequence, we improve the classical Stein--Zygmund embedding deriving $\dot{B}^{d/p}_\infty L_{p,\infty}(\mathbb{R}^d) \hookrightarrow \text{BMO}(\mathbb{R}^d)$ for $1 < p < \infty$. Moreover, these results are also applied to establish new Fefferman--Stein inequalities, Calderón--Scott type inequalities, and extrapolation estimates. Our approach is …
In this paper we generalize the Hájek-Rényi-Chow maximal inequality for submartingales to $L^p$ type Riesz spaces with conditional expectation operators. As applications we obtain a submartingale convergence theorem and a strong law of large numbers in Riesz spaces. Along the way we develop a Riesz space variant of the Clarkson's inequality for $1\le p\le 2$.
The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximal monotone operators provided that Rockafellar's constraint qualification holds. In this note, we provide a new maximal monotonicity result for the sum of a maximal monotone relation and the subdifferential operator of a proper, lower semicontinuous, sublinear function. The proof relies on Rockafellar's formula for the Fenchel conjugate of the sum as well as some results on the Fitzpatrick function.
Sharp two-weight inequality for fractional maximal operators
本文利用抽象 Bellman 函数方法，研究树状概率空间上分数次极大算子在 Sawyer 测试条件下的精确双权范数界，并揭示其与精确 Sobolev 嵌入定理的联系。
Let (e^{tA})_{t \geq 0} be a C_0-contraction semigroup on a 2-smooth Banach space E, let (W_t)_{t \geq 0} be a cylindrical Brownian motion in a Hilbert space H, and let (g_t)_{t \geq 0} be a progressively measurable process with values in the space γ(H,E) of all γ-radonifying operators from H to E. We prove that for all 0<p<\infty there exists a constant C, depending only on p and E, such that for all T \geq 0 we have \E \sup_{0\le t\le T} || \int_0^t e^{(t-s)A} g_s dW_s \ ||^p \leq C \mathbb{E} (\int_0^T || g_t ||_{γ(H,E)}^2 dt)^\frac{p}{2}. For p \geq 2 the proof is based on the observation …
This paper is devoted to the study of various maximal ergodic theorems in noncommutative $L_p$-spaces. In particular, we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive contractions on $L_p$ and the analogue of Stein's maximal inequality for symmetric positive contractions. We also obtain the corresponding individual ergodic theorems. We apply these results to a family of natural examples which frequently appear in theory of von Neumann algebras and in quantum probability.
Noncommutative maximal inequalities for polynomial ergodic averages
本文证明了沿多项式序列的非交换极大遍历不等式，扩展了p的范围，通过Bourgain主弧策略及非交换Stein外插、组合方法和多线性Doob极大不等式等新工具实现。
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