ENDPOINT ESTIMATES FOR COMMUTATORS OF RIESZ TRANSFORMS ASSOCIATED WITH SCHRÖDINGER OPERATORS

Source abstract

Abstract In this paper, we discuss the H 1 L -boundedness of commutators of Riesz transforms associated with the Schrödinger operator L =−△+ V , where H 1 L ( R n ) is the Hardy space associated with L . We assume that V ( x ) is a nonzero, nonnegative potential which belongs to B q for some q > n /2. Let T 1 = V ( x )(−△+ V ) −1 , T 2 = V 1/2 (−△+ V ) −1/2 and T 3 = ∇ (−△+ V ) −1/2 . We prove that, for b ∈ BMO ( R n ) , the commutator [ b , T 3 ] is not bounded from H 1 L ( R n ) to L 1 ( R n ) as T 3 itself. As an alternative, we obtain that [ b , T i ] , ( i =1,2,3 ) are of ( H 1 L , L 1 weak ) -boundedness.