Fractional Bloom boundedness and compactness of commutators

Let T be a non-degenerate Calderon-Zygmund operator and let b : R-d -> C be locally integrable. Let 1 < p <= q < infinity and let mu p epsilon A(p) and lambda(q) epsilon A(q), where A(p) denotes the usual class of Muckenhoupt weights. We show that vertical bar vertical bar[b, T]vertical bar vertical bar(L mu p -> L lambda q) similar to vertical bar vertical bar b vertical bar vertical bar(BMO nu alpha), [b, T] epsilon K(L-mu(p), L-lambda(q)) iff b epsilon VMO nu alpha, where L-mu(p) = L-p((mu)p) and alpha/d = 1/p - 1/q, the symbol K stands for the class of compact operators between the given spaces, and the fractional weighted (alpha)(BMO nu) and VMO nu alpha spaces are defined through the following fractional oscillation and Bloom weight: O-nu(alpha)(b; Q) = nu(Q)(-alpha/d)(1/nu(Q) integral(Q) |b - < b > Q|), nu = ( mu/lambda)beta, beta = (1 + alpha/d)(-1). The key novelty is dealing with the off-diagonal range p < q, whereas the case p = q was previously studied by Lacey and Li. However, another novelty in both cases is that our approach allows complex-valued functions b, while other arguments based on the median of b on a set are inherently real-valued.