A Class Of Singular Integral Operators On The Fock Space

Bargmann transform is a unitary transformation from L2(R)to Fock space.The chassical operators on L2(R)such as Fourier transform and Hilbert transform corre-spond to operators on Fock space under Bargmann transform.The classical Hilbert transform corresponds to the integral operator(Sφf)(z)= ∫ f(w)ezwφ(z-w)e-|w|e-|w|dA(w)on the Fock space,among them,φ(z)=∫0zeu2/2du.We will mainly study the bounded-ness of integral operators of the form(Sφf)(z)= ∫Cf(w)ezwφ(z-w)e-|w|2dA(w)on the Fock space,and give some necessary and sufficient conditions for Sφ ’s bound-edness.In particular.We give a complete characterization of the boundedness of Sφwhen we give φ(z)= eaz2+bz.