无穷维稀疏Lerner算子
Sparse Lerner operators in infinite dimensions
摘要：
我们使用几乎正交的原则，给出了一个新的和简单的证明，稀疏Lerner操作是有界的矩阵或运营商加权空间$L_W^{2}(μ)$，其中$μ$是$\R^d$上的加倍措施当且仅当重量$W$满足Muckenhoupt $A_2(μ)$ -条件，限制到稀疏收集的问题。我们的方法扩展到无限维的设置，从而允许应用程序的多参数设置。对于$A_2$ -权类，我们得到了混合$A_{2}(μ)$ - $A_{\infty}(μ)$ -条件下的上界，该上界与维数无关，并且与有限维向量环境下的最佳上界一致.作为应用，我们证明了极大Bergman投影的一个矩阵加权界，并利用Békollé-Bonami特征得到了一个新的更精确的界.此外，我们还考虑了算子值加权$L^{2}$ -空间上稀疏Lerner算子的扩张子及其对多参数的应用
英文摘要：
We use the principle of almost orthogonality to give a new and simple proof that a sparse Lerner operator is bounded on a matrix- or operator-weighted space $L_W^{2}(μ)$, where $μ$ is a doubling measure on $\R^d$ if and only if the weight $W$ satisfies the Muckenhoupt $A_2(μ)$-condition, restricted to the sparse collection in question. Our method extends to the infinite-dimensional setting, thus allowing for applications to the multi-parameter setting. For the class of Muckenhoupt $A_2$-weights, we obtain bounds in terms of mixed $A_{2}(μ)$-$A_{\infty}(μ)$-conditions, which is independent of dimension and agrees with the best known bound in the finite-dimensional vectorial setting. As an application, we prove a matrix-weighted bound for the maximal Bergman projection, where we obtain a new sharper bound in terms of the Békollé-Bonami characteristic. Furthermore, we consider commutators of sparse Lerner operators on operator-valued weighted $L^{2}$-spaces and some applications to multi-parameters
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