Algebraic Calderón-Zygmund theory

Calderón-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 which admit Markov semigroup purely algebraic assumptions. We shall construct an abstract form ‘Markov metric’ governing process and naturally associated BMO spaces, interpolate with Lp-scale endpoint inequalities operators. Motivated by noncommutative harmonic analysis, this gives first arbitrary von Neumann algebras, but is also valid in classical settings like Riemannian manifolds nonnegative Ricci curvature or doubling/nondoubling spaces. Other less standard commutative scenarios fractals probability are included. Among our applications setting, improve recent results quantum Euclidean group respectively linked to geometry geometric theory.