A CHARATERIZATION OF COMMUTATORS FOR PARABOLIC SINGULAR INTEGRALS

In this paper, the authors give a characterization of the L(P)-boundedness of the commutators for the parabolic singular integrals. More precisely, the authors prove that if b is an element of BMO(phi)(R(n), rho), then the commutator [b, T] is a bounded operator from L(P) (R(n)) to the Orlicz space L(psi)(R(n)), where the kernel function Omega has no any smoothness on the unit sphere S(n-1). Conversely, if assuming on Omega a slight smoothness on S(n-1), then the boundedness of [b,T] from L(P)(R(n)) to L(psi)(R(n)) implies that b is an element of BMO(phi)(R(n), p). The results in this paper improve essentially and extend some known conclusions.