鲁比奥·德·弗朗西亚外推定理与稀疏界

中文摘要 AI 辅助

稀疏界已被证明是简化理解调和分析中许多重要算子的有力工具。算子存在稀疏界意味着该算子在Muckenhoupt类权重下是加权有界的。如同Calderón-Zygmund算子的例子，这一蕴含关系可证明加权有界性对Muckenhoupt权重的特征常数具有最优依赖。本文针对次可加算子证明了一个部分逆命题：以权重特征常数为任意先验控制的算子范数的加权有界性，蕴含稀疏形式界的存在。该结果表明，算子范数对权重特征常数的近最优依赖源于加权有界性本身，而非依赖次可加算子的特定结构。此外，它使Rubio de Francia外推定理可扩展至次线性算子的弱型端点，并导出算子解析族的稀疏形式界的插值结果。作为应用，我们得到了粗糙奇异积分算子的新稀疏形式界。

英文摘要

Sparse bounds have proved to be a powerful tool which simplify the understanding of many important operators in harmonic analysis. The existence of a sparse bound for an operator implies the weighted boundedness of the operator for weights within the Muckenhoupt classes. As in the example of Calderón-Zygmund operators, this implication can prove weighted boundedness with the best possible dependency on the characteristic constant of the Muckenhoupt weight. In this paper, we prove a partial converse for sub-additive operators by showing that weighted boundedness, with an arbitrary a priori control of the operator norm in terms of the characteristic constants of the weight, implies the existence of sparse form bounds. This result implies that a near-optimal dependency of the operator norm on characteristic constants of the weights follows from the weighted boundedness itself, rather than relying on the particular structure of the sub-additive operator. Moreover, it allows Rubio de Francia's extrapolation theorem to be extended to the weak-type end-point for sub-linear operators and leads to an interpolation result for sparse form bounds for analytic families of operators. As an application, we obtain new sparse form bounds for rough singular integral operators.

补充信息

备注

Theorem 1.6 is not true, as the example of the iterated maximal function, which is not weak-type $(1,1)$, shows. The error occurs in (26) and (31), as we cannot bound the $A_s$ characteristic constant of the weight by $Ω$, as is required by (9). The author thanks Emiel Lorist for kindly pointing this out and for sharing a weaker but correct version of Theorem 1.6