ESTIMATES FOR THE HIGHER ORDER RIESZ TRANSFORMS RELATED TO SCHRODINGER TYPE OPERATORS

We consider the Schrodinger type operator L-k = (-Delta)(k) +V-k on R-n(n >= 2k + 1), where k = 1, 2 and the nonnegative potential V belongs to the reverse Holder class RHs with n/2 < s < n. In this paper, we establish the (L-p, L-q)-boundedness of the higher order Riesz transform T-alpha,T-beta = V-2 alpha del(2) L-2(-beta)(0 <= alpha <= 1/2 < beta <= 1,beta -alpha >= 1/2) and its adjoint operator T-alpha,T-beta* ; respectively We show that T-alpha,T-beta is bounded from Hardy type space H-L2(1) (R-n) into L-p2 (R-n) and T-alpha,T-beta* is bounded from L-p1 (R-n) into BMO type space BMOL1 (R-n) when beta - alpha > 1/2, where p(1) = n/4(beta - alpha)-2, p(2) = n/n - 4(beta - alpha) +2. Moreover, we prove that T-alpha,T-beta is bounded from BMOL1 (R-n) to itself when beta - alpha = 1/2.