Algebraic Calderón-Zygmund theory

Marius Junge, Tao Mei, Javier Parcet, Runlian Xia

Research output: Contribution to journalArticlepeer-review


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 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 Calderón-Zygmund operators. Motivated by noncommutative harmonic analysis, this approach gives the first form of Calderón-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.

Original languageEnglish (US)
Article number107443
JournalAdvances in Mathematics
StatePublished - Jan 6 2021


  • Calderón-Zygmund theory
  • Markov metric
  • von Neumann algebras

ASJC Scopus subject areas

  • General Mathematics


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