Operator-valued Fourier multipliers of bounded s-variation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918277946540032 |
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| author | Deng, Chenxi Lorist, Emiel Veraar, Mark |
| author_facet | Deng, Chenxi Lorist, Emiel Veraar, Mark |
| contents | In this paper, we establish an operator-valued Fourier multiplier theorem in weighted Lebesgue spaces, Besov and Triebel--Lizorkin spaces, assuming the multiplier has $\mathcal{R}$-bounded range and satisfies an $\ell^r$-summability condition on its bounded $s$-variation seminorms over dyadic intervals. The exponents $r$ and $s$ reflect the relationship between the geometric properties of the underlying Banach spaces (type and cotype) and the boundedness of Fourier multiplier operators. As our main tool we prove a weighted vector-valued variational Carleson inequality and deduce an estimate of Littlewood--Paley--Rubio de Francia type. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_04803 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Operator-valued Fourier multipliers of bounded s-variation Deng, Chenxi Lorist, Emiel Veraar, Mark Functional Analysis Primary: 42A45, Secondary: 42B25, 42B35, 46B09, 46B20, 46E35, 46E40 In this paper, we establish an operator-valued Fourier multiplier theorem in weighted Lebesgue spaces, Besov and Triebel--Lizorkin spaces, assuming the multiplier has $\mathcal{R}$-bounded range and satisfies an $\ell^r$-summability condition on its bounded $s$-variation seminorms over dyadic intervals. The exponents $r$ and $s$ reflect the relationship between the geometric properties of the underlying Banach spaces (type and cotype) and the boundedness of Fourier multiplier operators. As our main tool we prove a weighted vector-valued variational Carleson inequality and deduce an estimate of Littlewood--Paley--Rubio de Francia type. |
| title | Operator-valued Fourier multipliers of bounded s-variation |
| topic | Functional Analysis Primary: 42A45, Secondary: 42B25, 42B35, 46B09, 46B20, 46E35, 46E40 |
| url | https://arxiv.org/abs/2601.04803 |