Marcinkiewicz integrals with rough kernel associated with Schrӧdinger operators and commutators on generalized vanishing local Morrey spaces

F. Gürbüz

Let L=-Δ+V(x) be a Schrӧdinger operator, where Δ is the Laplacian on Rn, while nonnegative potential V(x) belonging to the reverse Hӧlder class. In this paper, using the some conditions on φ( x.r), we dwell on the boundedness of Marcinkiewicz integrals with rough kernel associated with schrӧ}dinger operators and commutators generated by these operators and local Campanato functions both on generalized local Morrey spaces and on generalized vanishing local Morrey spaces, respectively. As an application of the above results, the boundedness of parametric Marcinkiewicz integral and its commutator both on generalized local Morrey spaces and on generalized vanishing local Morrey spaces is also obtained. 

Tbilisi Mathematical Journal, Vol. 11(3) (2018), pp. 133-156