与平均算子相关的非交换极大微分变换
Noncommutative maximal differential transforms associated to averaging operators
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中文总结 AI 辅助
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。
中文摘要 AI 辅助
本文建立了算子族$(T_N)_N$的非交换极大弱型$(1,1)$与强型$(p,p)$估计，该算子族由$$T_Nf=\sum_{k=N_1}^{N_2}ν_{k}(M_{k}-\mathsf{E}_k)f,$$定义，其中$M_k$表示二进Hardy–Littlewood平均算子，$\mathsf{E}_{k}$是关于边长为$2^{-k}$的二进立方体的条件期望，$N=(N_1,N_2)$满足$N_1<N_2$与$(ν_{k})\in\ell_{\infty}$。我们方法的主要创新点是提出了针对非光滑核的非交换Cotlar型不等式，这一结果即便在经典调和分析中也属新结论。作为应用，我们得到了与平均算子相关的非交换极大微分变换的有界性理论。
英文摘要
In this paper, we establish the noncommutative maximal weak type $(1,1)$ and strong type $(p,p)$ estimates for the family of operators $(T_N)_N$, defined by $$T_Nf=\sum_{k=N_1}^{N_2}ν_{k}(M_{k}-\mathsf{E}_k)f,$$ where $M_k$ denotes the dyadic Hardy--Littlewood average operator, $\mathsf{E}_{k}$ is the conditional expectation with respect to the dyadic cubes of side-length $2^{-k}$, $N=(N_1,N_2)$ with $N_1<N_2$ and $(ν_{k})\in\ell_{\infty}$. The main novelty of our approach is the development of a noncommutative Cotlar-type inequality for non-smooth kernels, a result that is new even in classical harmonic analysis. As an application, we obtain the boundedness theory of the noncommutative maximal differential transforms for averaging operators.
