# On boundedness property of singular integral operators associated to a Schr\"odinger operator in a generalized Morrey space and applications

## TL;DR

This paper establishes the boundedness of Riesz transforms linked to Schr"odinger operators in a new generalized Morrey space, leading to improved regularity results for Schr"odinger equations.

## Contribution

It introduces a new weighted Morrey space framework and proves the boundedness of associated singular integral operators, extending previous results to more general potentials.

## Key findings

- Boundedness of Riesz transforms in the new Morrey space.
- Regularity results for Schr"odinger equation solutions.
- Applicability to potentials satisfying reverse H"older's inequality.

## Abstract

In this paper, we provide the boundedness property of the Riesz transforms associated to the Schr\"odinger operator $\mathcal{L}=-\Delta + \mathbf{V}$ in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential $\V$ considered in this paper is a non-negative function satisfying the suitable reverse H\"older's inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schr\"odinger equations in the new Morrey space.