# Boundedness of variation operators associated with the heat semigroup generated by high order Schr\"odinger type operators

## TL;DR

This paper establishes the boundedness of variation operators linked to the heat semigroup generated by high order Schrödinger operators, extending results to Morrey spaces and utilizing inequalities from biharmonic heat semigroups.

## Contribution

It proves the $L^p$ and Morrey space boundedness of variation operators for high order Schrödinger-type operators, a novel extension in harmonic analysis.

## Key findings

- Boundedness of variation operators on $L^p$ spaces.
- Boundedness of variation operators on Morrey spaces.
- Use of biharmonic heat semigroup inequalities in proofs.

## Abstract

In this paper, we derive the $L^p$-boundedness of the variation operators associated with the heat semigroup which is generated by the high order Schr\"odinger type operator $(-\Delta)^2+V^2$. Further more, we prove the boundedness of the variation operators on Morrey spaces. In the proof of the main results, we always make use of the variation inequalities associated with the heat semigroup generated by the biharmonic operator $(-\Delta)^2.$