## Tbilisi Mathematical Journal

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

Ferit Gürbüz

#### Abstract

Let $L=-\Delta+V\left( x\right)$ be a Schrödinger operator, where $\Delta$ is the Laplacian on ${\mathbb{R}^{n}}$, while nonnegative potential $V\left(x\right)$ belonging to the reverse Hölder class. In this paper, using the some conditions on $\varphi\left(x.r\right)$, we dwell on the boundedness of Marcinkiewicz integrals with rough kernel associated with schrödinger 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.

#### Article information

Source
Tbilisi Math. J., Volume 11, Issue 3 (2018), 133-156.

Dates
Accepted: 15 May 2018
First available in Project Euclid: 3 October 2018

https://projecteuclid.org/euclid.tbilisi/1538532032

Digital Object Identifier
doi:10.32513/tbilisi/1538532032

Mathematical Reviews number (MathSciNet)
MR3954200

#### Citation

Gürbüz, Ferit. Marcinkiewicz integrals with rough kernel associated with Schrödinger operators and commutators on generalized vanishing local Morrey spaces. Tbilisi Math. J. 11 (2018), no. 3, 133--156. doi:10.32513/tbilisi/1538532032. https://projecteuclid.org/euclid.tbilisi/1538532032

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