Multi-parameter estimates via operator-valued shifts

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Abstract

We prove the mixed-norm LpLq-boundedness of a general class of singular integral operators having a multi-parameter singularity and acting on vector-valued (UMD Banach lattice-valued) functions. Moreover, families of such operators with uniform assumptions are shown to be not only uniformly bounded but R-bounded, a genuinely stronger property that is often needed in applications. Previous results of this nature only dealt with convolution-type or slightly more general paraproduct-free singular integrals. In contrast, our analysis specifically targets the array of different partial paraproducts that arise in the multi-parameter setting by interpreting them as paraproduct-valued one-parameter operators. This new point-of-view provides a conceptual simplification over the existing representation results for multi-parameter operators, which is a key to the proof of the boundedness of these operators.

Original languageEnglish
JournalProceedings of the London Mathematical Society
Volume119
Issue number6
Pages (from-to)1560-1597
Number of pages38
ISSN0024-6115
DOIs
Publication statusPublished - Dec 2019
MoE publication typeA1 Journal article-refereed

Fields of Science

  • 111 Mathematics
  • 42B20 (primary)
  • SINGULAR-INTEGRALS
  • DYADIC SHIFTS
  • CALDERON
  • REPRESENTATION
  • INEQUALITIES

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