Bellman functions and two weight inequalities for Haar multipliers

Weighted inequalities for multivariable dyadic paraproducs

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…

$t$-Haar乘子在$ L^2 $上的尖锐界

Sharp bounds for $t$-Haar multipliers on $L^2$

本文研究了$ t $-Haar乘子在$ L^2 $空间上的norm界，通过分析权重函数的特性，得出了与$ C^d_{2t} $和$ A^d_q $特征相关的尖锐界，特别指出当权重函数属于$ C^d_{2t} \cap A_{\infty}^d $时，$ w^{2t} $属于$ A_q^d $。

Bilinear Multipliers of Small Lebesgue spaces

Let $G$ be a locally compact abelian metric group with Haar measure $λ$ and $\hat{G}$ its dual with Haar measure $μ,$ and $λ( G) $ is finite. Assume that$~1<p_{i}<\infty $, $p_{i}^{\prime }=\frac{ p_{i}}{p_{i}-1}$, $( i=1,2,3) $ and $θ\geq 0$. Let $ L^{(p_{i}^{\prime },θ}( G) ,$ $( i=1,2,3) $ be small Lebesgue spaces. A bounded measurable function $m( ξ,η) $ defined on $\hat{G}\times \hat{G}$ is said to be a bilinear multiplier on $G$ of type $[ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }] _{θ}$ if the bilinear operator $B_{m}$ associated with the symbol $m$, \begin{equation} B_{m}(f,g)…

鞅上一般算子的Bellman函数方法

Bellman function method for general operators on martingales

本文将Bellman函数方法推广到鞅上一般算子（不限于鞅变换），构造单一Bellman型函数编码Gundy外推定理中几乎所有算子的$L^p$-有界性，并以Haar变换和Walsh系统的Rubio de Francia不等式相关算子为例加以说明。

双参数 Haar 系统 Hardy 空间上的乘子

Multipliers on bi-parameter Haar system Hardy spaces

本文研究双参数 Haar 系统 Hardy 空间上的 Haar 乘子，证明任意有界 Haar 乘子都可使 Capon 投影的线性组合近似 1-投影分解通过该乘子，并刻画了 Capon 投影无界时的分解性质。

Rajchman测度的$H^1(\t^d)$乘子范数、Zafran型现象及Walsh-Stieltjes级数

The $H^1(\T^d)$-multiplier norm of Rajchman measures, Zafran-type phenomena, and Walsh--Stieltjes series

该论文研究复Rajchman测度作为傅里叶乘子在$H^1(\t^d)$零均值部分的卷积，得到乘子范数下界，揭示了Zafran型现象，还关联到Walsh-Stieltjes系数的相关结果。