Multilinear operator-valued Calderón-Zygmund theory

Francesco Di Plinio, Kangwei Li, Henri Martikainen, Emil Vuorinen

Research output: Contribution to journalArticlepeer-review

6 Citations (SciVal)

Abstract

We develop a general theory of multilinear singular integrals with operator-valued kernels, acting on tuples of UMD Banach spaces. This, in particular, involves investigating multilinear variants of the R-boundedness condition naturally arising in operator-valued theory. We proceed by establishing a suitable representation of multilinear, operator-valued singular integrals in terms of operator-valued dyadic shifts and paraproducts, and studying the boundedness of these model operators via dyadic-probabilistic Banach space-valued analysis. In the bilinear case, we obtain a T(1)-type theorem without any additional assumptions on the Banach spaces other than the necessary UMD. Higher degrees of multilinearity are tackled via a new formulation of the Rademacher maximal function (RMF) condition. In addition to the natural UMD lattice cases, our RMF condition covers suitable tuples of non-commutative Lp-spaces. We employ our operator-valued theory to obtain new multilinear, multi-parameter, operator-valued theorems in the natural setting of UMD spaces with property α.

Original languageEnglish
Article number108666
JournalJournal of Functional Analysis
Volume279
Issue number8
DOIs
Publication statusPublished - 2020 Nov 1

Subject classification (UKÄ)

  • Mathematical Analysis

Keywords

  • Calderón–Zygmund operators
  • Multilinear analysis
  • Operator-valued analysis
  • UMD spaces

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