Abstract
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 language | English |
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Pages (from-to) | 1107–1141 |
Number of pages | 35 |
Journal | Mathematische Zeitschrift |
Volume | 301 |
Issue number | 1 |
DOIs | |
Publication status | Published - 2022 |
Keywords
- Banach function space
- Bilinear Hilbert transform
- Muckenhoupt weights
- Multilinear
- Sparse domination
- UMD