Sparse domination implies vector-valued sparse domination
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866914815898812416 |
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| author | Lorist, Emiel Nieraeth, Zoe |
| author_facet | Lorist, Emiel Nieraeth, Zoe |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2003_02233 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Sparse domination implies vector-valued sparse domination Lorist, Emiel Nieraeth, Zoe Classical Analysis and ODEs Functional Analysis 42B25 (Primary) 46E30 (Secondary) 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. |
| title | Sparse domination implies vector-valued sparse domination |
| topic | Classical Analysis and ODEs Functional Analysis 42B25 (Primary) 46E30 (Secondary) |
| url | https://arxiv.org/abs/2003.02233 |