GRADIENT ESTIMATES FOR THE COMMUTATOR WITH FRACTIONAL DIFFERENTIATION FOR SECOND ORDER ELLIPTIC PERATORS

Let L = -div(A del) be a second order divergence form elliptic operator, where A is an accretive, n x n matrix with bounded measurable complex coefficients on R-n. Let L-alpha/2 (0 < alpha < 1) denotes the fractional differential operator associated with L and (-Delta)(alpha/2) b is an element of L-n/alpha(R-n). In this article, we prove that the commutator [b, L-alpha/2] is bounded from the homogenous Sobolev space (L) over dot(alpha)(2)(R-n) to L-2(R-n).