COMMUTATORS OF RIESZ TRANSFORMS WITH LIPSCHITZ FUNCTIONS RELATED TO MAGNETIC SCHRODINGER OPERATORS

Let A : = -(del - i (a) over right arrow) . (del - i (a) over right arrow) + V be a magnetic Schrodinger operator on L-2(R-n), n >= 2, where (a) over right arrow : = (a(1), ..., a(n)) is an element of L-loc(2) (R-n, R-n) and 0 <= V is an element of L-loc(1) (R-n). In this paper, the author shows that the commutators of the Riesz transforms L(k)A(-1/2), k is an element of {1, ..., n}, with functions in Lipschitz space Lip(alpha) (R-n) for alpha is an element of (0,1), are bounded from L-p(R-n) to L-q(R-n), where 1/p-1/q = alpha/n and L-k is the closure of partial derivative/partial derivative x(k) - ia(k) in L-2(R-n). Let rho be an admissible function modeled on the known auxiliary function determined by the Schrodinger operator -Delta+V. The author also characterizes a localized Lipschitz space Lip(alpha,rho) (R-n) in terms of the localized Riesz transforms {(R) over tilde (j)}(j=1)(n) and their adjoint operators.