COMPACTNESS FOR COMMUTATORS OF MARCINKIEWICZ INTEGRALS IN MORREY SPACES

In this paper the authors give a characterization of the compactness for the commutator [b, mu(Omega)] in the Morrey spaces L(p,lambda)(R(n)), where mu(Omega) denotes the Marcinkiewicz integral. More precisely, the authors prove that if b is an element of VMO(R(n)), the BMO(R(n))-closure of C(c)(infinity)(R(n)), then the commutators [b, mu(Omega)] is a compact operator in the Mon-ey spaces L(p,lambda)(R(n)) for 1 < p < infinity and 0 < lambda < n. Conversely, if b is an element of BMO(R(n)) and [b, mu(Omega)] is a compact operator in L(p,lambda)(R(n)) for some p is an element of (1, infinity) and lambda is an element of (0, n), then b is an element of VMO(R(n)). In the above results, the kernel function Omega of the operator mu(Omega) is assumed to satisfy a very weak condition on S(n-1).