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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2605.08562 |
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| _version_ | 1866913115219689472 |
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| author | He, Xiang Li Qianjun Fu, Zunwei |
| author_facet | He, Xiang Li Qianjun Fu, Zunwei |
| contents | Littlewood--Paley theory is a fundamental tool for frequency localization, square-function control, and multiplier analysis, yet a systematic counterpart in the fractional Fourier transform (FrFT) setting has remained incomplete. We develop a unified FrFT Littlewood--Paley framework based on the observation that, for a fixed $α\notinπ\mathbb Z$, a broad class of FrFT-side operators are exact chirp conjugates of their classical Fourier counterparts through $$ M_αf(x)=e^{iπ|x|^2\cotα}f(x). $$
Within this unified framework we present: the FrFT multiplier identity; Littlewood--Paley square-function estimates and the converse theorem; sharp dyadic interval decompositions; Marcinkiewicz and Mihlin--H"ormander multiplier results; maximal, rough square-function, and almost-orthogonality estimates; twisted dyadic martingale geometry; inhomogeneous Sobolev, Besov, and Triebel--Lizorkin descriptions; Calderón reproducing formulae; pullback spaces and FrFT Riesz--Bessel operators; BMO, Carleson, sharp-maximal, and Hardy-space; twisted product estimates, multilinear bounds, and a Kato--Ponce theorem; fractional order-shifting in Lipschitz spaces; and the classical limit and singular boundary laws for the fractional parameter. The recurring theme is that a large class of FrFT operators are exact chirp conjugates of their classical counterparts, so most estimates are inherited with the same constants after one time identification of the rescaled symbols. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_08562 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Recent progress of Littlewood-paley Theory with chirp function He, Xiang Li Qianjun Fu, Zunwei Functional Analysis Classical Analysis and ODEs Littlewood--Paley theory is a fundamental tool for frequency localization, square-function control, and multiplier analysis, yet a systematic counterpart in the fractional Fourier transform (FrFT) setting has remained incomplete. We develop a unified FrFT Littlewood--Paley framework based on the observation that, for a fixed $α\notinπ\mathbb Z$, a broad class of FrFT-side operators are exact chirp conjugates of their classical Fourier counterparts through $$ M_αf(x)=e^{iπ|x|^2\cotα}f(x). $$ Within this unified framework we present: the FrFT multiplier identity; Littlewood--Paley square-function estimates and the converse theorem; sharp dyadic interval decompositions; Marcinkiewicz and Mihlin--H"ormander multiplier results; maximal, rough square-function, and almost-orthogonality estimates; twisted dyadic martingale geometry; inhomogeneous Sobolev, Besov, and Triebel--Lizorkin descriptions; Calderón reproducing formulae; pullback spaces and FrFT Riesz--Bessel operators; BMO, Carleson, sharp-maximal, and Hardy-space; twisted product estimates, multilinear bounds, and a Kato--Ponce theorem; fractional order-shifting in Lipschitz spaces; and the classical limit and singular boundary laws for the fractional parameter. The recurring theme is that a large class of FrFT operators are exact chirp conjugates of their classical counterparts, so most estimates are inherited with the same constants after one time identification of the rescaled symbols. |
| title | Recent progress of Littlewood-paley Theory with chirp function |
| topic | Functional Analysis Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2605.08562 |