The Lp-to-Lq compactness of commutators with p > q

Let 1 < q < p < infinity, 1/r := 1/q - 1/p, and T be a non-degenerate Calderon-Zygmund operator. We show that the commutator [b, T] is compact from Lp(Rn) to Lq(Rn) if and only if b = a + c with a is an element of Lr(Rn) and c a constant. Since neither the corresponding Hardy-Littlewood maximal operator nor the corresponding Calderon- Zygmund maximal operator is bounded from Lp(Rn) to Lq(Rn), we take the full advantage of the compact support of the approximation element in Cc infinity(Rn), which seems to be redundant for many corresponding estimates when p q. We also extend the results to the multilinear case.Peer reviewe