Boundedness of higher order Riesz transforms associated with Schrodinger type operator on generalized Morrey spaces

Abstract

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.

Boundedness of high order commutators of Marcinkiewicz integrals associated with Schrödinger operators

“…In [2], Chen and Jin showed the boundedness of µ L j and [b, µ L j ] on the Morrey spaces related to certain nonnegative potentials. In [9], we introduced the generalized Morrey space related to nonnegative potential V, which covers the general Morrey space; see [2,7,8,11].…”

On boundedness property of singular integral operators associated to a Schrödinger operator in a generalized Morrey space and applications

“…They have showed that the singular integral operators are not only bounded in weighted Lebesgue spaces but also in weighted Morrey spaces. In addition, lots of Morrey type spaces associated to a Schrödinger operator have been also studied (see [2,19,21,27,33]) to extend the wellknown Morrey spaces. In recent years the problem related to Schrödinger operator has attracted a great deal of attention of many mathematicians; see [3,4,5,6,7,9,10,22,29,35] and references therein.…”