Three-dimensional Fractional Fourier Transform In L~1 Space And Its Applications

FRFT is a generalization of the traditional Fourier transform,especially chirp signals have good time-frequency domain characteristics.In this paper,we will study the L1(R3)theory of the 3-d fractional Fourier transform(the 3-d FRFT).In view of the special structure of the FRFT,we will study properties of L1(R)through introducing a suitable chirp operator.For 1≤q<p’<∞,q=1/(1+λ)and 1/p+1/p’=1,we show that the dual of central block space hq,p’ is the central Money space Bp,λ.As applications,we characterize the central BMO space CMOp’.We show that the commutator[Hβ,b]generated by the fractional Hardy operator Hβ and the locally integrable functionb is a bounded linear operator from the central Morrey space Bp,λ1 to the central Morrey space Bq,λ2 where-n<β<n,1<p,q<∞,-1/p≤λ1<0 and-1/q<λ2=λ1+β/n<0.The corresponding result for the adjoint operator of[Hβ,b]is also obtained.