Recent progress of Littlewood-paley Theory with chirp function

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 $\alpha\notin\pi\mathbb Z$, a broad class of FrFT-side operators are exact chirp conjugates of their classical Fourier counterparts through $$ M_{\alpha}f(x)=e^{i\pi |x|^2\cot\alpha}f(x). $$ 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; Calder\'on 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.