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 $(-\Delta)^2.$

中文翻译：

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与高阶薛定谔型算子产生的热半群相关的变分算子的有界性

在本文中，我们推导了与由高阶 Schr\"odinger 类型算子 $(-\Delta)^2+V^2$ 生成的热半群相关的变分算子的 $L^p$-有界性。此外，我们证明了Morrey空间上变分算子的有界性，在证明主要结果时，我们总是利用与双调和算子$(-\Delta)^2产生的热半群相关的变分不等式。 $