Sparse domination implies vector-valued sparse domination

Emiel Lorist, Zoe Nieraeth*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

3 Citations (Scopus)
29 Downloads (Pure)


We prove that scalar-valued sparse domination of a multilinear operator implies vector-valued sparse domination for tuples of quasi-Banach function spaces, for which we introduce a multilinear analogue of the UMD condition. This condition is characterized by the boundedness of the multisublinear Hardy-Littlewood maximal operator and goes beyond examples in which a UMD condition is assumed on each individual space and includes e.g. iterated Lebesgue, Lorentz, and Orlicz spaces. Our method allows us to obtain sharp vector-valued weighted bounds directly from scalar-valued sparse domination, without the use of a Rubio de Francia type extrapolation result. We apply our result to obtain new vector-valued bounds for multilinear Calderón-Zygmund operators as well as recover the old ones with a new sharp weighted bound. Moreover, in the Banach function space setting we improve upon recent vector-valued bounds for the bilinear Hilbert transform.

Original languageEnglish
Pages (from-to)1107–1141
Number of pages35
JournalMathematische Zeitschrift
Issue number1
Publication statusPublished - 2022


  • Banach function space
  • Bilinear Hilbert transform
  • Muckenhoupt weights
  • Multilinear
  • Sparse domination
  • UMD


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