关键词: 变分算子, Schrodinger型算子, 交换子, Herz型空间, 有界性

Abstract: Let L=(-Δ)²+V² be a high order Schrodinger type operator in Rⁿ (n≥5), where a nonnegative potential V belongs to the reverse Holder class RH_q(q>n/2). Let V_ρ(e^-tL) be the variation operator associated with the high order Schrodinger type operator L. Based on the theory of atomic decompesitions of Herz type Hardy spaces, by using the properties of the Schrodinger type operators, the boundedness of the commutators composed by such variation operators and Lipschitz fuctions on L^q is proved. Furthermore, it is proved that the commutators of this kind of variation operators are bounded from Herz type Hardy spaces 
to the Herz spaces and are bounded on Morrey-Herz spaces.

Key words: variation operator, Schrodinger type operator, commutator, Herz type space, boundedness