Calderon-Zygmund theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of good metrics, we introduce a new approach for general measure spaces which admit a Markov semigroup satisfying purely algebraic assumptions. We shall construct an abstract form of "Markov metric" governing the Markov process and the naturally associated BMO spaces, which interpolate with the Lp-scale and admit endpoint inequalities for Calderon-Zygmund operators. Motivated by noncommutative harmonic analysis, this approach gives the first form of Calderon-Zygmund theory for arbitrary von Neumann algebras, but is also valid in classical settings like Riemannian manifolds with nonnegative Ricci curvature or doubling/nondoubling spaces. Other less standard commutative scenarios like fractals or abstract probability spaces are also included. Among our applications in the noncommutative setting, we improve recent results for quantum Euclidean spaces and group von Neumann algebras, respectively linked to noncommutative geometry and geometric group theory.

中文翻译：

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代数卡尔德龙-齐格蒙德理论

Calderon-Zygmund 理论传统上是在满足额外规律性的度量空间上发展起来的。由于缺乏好的度量，我们为一般测度空间引入了一种新方法，该方法允许满足纯代数假设的马尔可夫半群。我们将构建一个抽象形式的“马尔可夫度量”来管理马尔可夫过程和自然关联的 BMO 空间，它用 Lp 尺度进行插值，并承认 Calderon-Zygmund 算子的端点不等式。受非交换调和分析的启发，这种方法给出了任意冯诺依曼代数的 Calderon-Zygmund 理论的第一种形式，但也适用于经典设置，如具有非负 Ricci 曲率或加倍/非加倍空间的黎曼流形。还包括其他不太标准的交换场景，如分形或抽象概率空间。在我们在非对易设置中的应用中，我们改进了量子欧几里得空间和群冯诺依曼代数的最新结果，分别与非对易几何和几何群论相关。