Weighted Solution of the Dirac Beltrami Equation with Coefficient in VMO

We study the generalized Beltrami equation D f = mu(x)(D) over bar f + h, where D is the left Dirac operator in Rn+1 acting on functions in Rn+1 and with values in the complex Clifford algebra Cl-n, (D) over bar is its conjugate, and mu is a Cl-n-valued function with compact support, with vanishing mean oscillation, satisfying parallel to mu parallel to (1,infinity) = Sigma parallel to mu(alpha)parallel to(infinity) < 1, where (mu(alpha)) are the coordinates of mu in Cl-n. Let. be a weight function in Rn+1. We prove that if omega(1/p) belongs to theMuckenhoupt class A(p,q) with 1/q = 1/p - 1/(n + 1), 1 < p < n + 1 that makes continuous the Riesz potential I-1 : L-p(omega) -> L-q(omega(q/p)), then for every h is an element of L-Cln(p) (omega), there exists a solution of the equation above satisfying partial derivative(i) f is an element of L-Cln(p) (omega) for every distributional partial derivative. If omega = 1, we prove the same result for any 1 < p < infinity.