We prove the following theorem: Let MSC: 42B25; 42B35

Let L 2 = (−∆) 2 + V 2 be a Schrödinger type operator, where V = 0 is a non-negative potential and belongs to the reverse Hölder class RH q for q n/2, n 5. The higher Riesz transform associated with L 2 is denoted by R = ∇ 2 L 2 ∇ 2 . In this paper, we investigate the boundedness of higher Riesz transforms and their commutators on the generalized Morrey spaces related to some non-negative potential.

Suppose L = −∆ + V is a Schrödinger operator on R n , where n 3 and the nonnegative potential V belongs to reverse Hölder class RH n . Let b belong to a new Campanato space Λ θ β (ρ), and let µ L j be the Marcinkiewicz integrals associated with L. In this paper, we establish the boundedness of the m-order, where 1/q = 1/p − mβ/n and 1 < p < n/(mβ). As an application, we obtain the boundedness of [b m , µ L j ] on the generalized Morrey spaces related to certain nonnegative potentials.

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By a counterexample, we prove that the results obtained in [1] are incorrect and there exist some theoretical mistakes in fractional complex transform.