BOUNDEDNESS OF VARIATION OPERATORS ASSOCIATED WITH THE HEAT SEMIGROUP GENERATED BY HIGH ORDER SCHRÖDINGER TYPE OPERATORS

doi:10.1007/s10473-020-0504-z

Acta mathematica scientia,Series B ›› 2020, Vol. 40 ›› Issue (5): 1215-1228.doi: 10.1007/s10473-020-0504-z

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BOUNDEDNESS OF VARIATION OPERATORS ASSOCIATED WITH THE HEAT SEMIGROUP GENERATED BY HIGH ORDER SCHRÖDINGER TYPE OPERATORS

Suying LIU¹, Chao ZHANG²

1. Department of Applied Mathematics, Northwest Polytechnical University, Xi'an 710072, China;
2. School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China

Received:2019-09-19 Revised:2020-01-15 Online:2020-10-25 Published:2020-11-04
Contact: Chao ZHANG E-mail:zaoyangzhangchao@163.com
Supported by:
The first author was supported by the National Natural Science Foundation of China (11701453) and Fundamental Research Funds for the Central Universities (31020180QD05). The second author was supported by the National Natural Science Foundation of China (11971431, 11401525), the Natural Science Foundation of Zhejiang Province (LY18A010006), and the first Class Discipline of Zhejiang-A (Zhejiang Gongshang University-Statistics).

Abstract

Abstract: In this article, we derive the $L^p$-boundedness of the variation operators associated with the heat semigroup which is generated by the high order Schrödinger type operator $(-\Delta)^2+V^2$ in $\mathbb R^n(n\ge 5)$ with $V$ being a nonnegative potential satisfying the reverse Hölder inequality. Furthermore, we prove the boundedness of the variation operators on associated 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 $(-\Delta)^2.$

Key words: Variation operators, high order Schrödinger type operators, heat semigroup

CLC Number:

42B35

Suying LIU, Chao ZHANG. BOUNDEDNESS OF VARIATION OPERATORS ASSOCIATED WITH THE HEAT SEMIGROUP GENERATED BY HIGH ORDER SCHRÖDINGER TYPE OPERATORS[J].Acta mathematica scientia,Series B, 2020, 40(5): 1215-1228.

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BOUNDEDNESS OF VARIATION OPERATORS ASSOCIATED WITH THE HEAT SEMIGROUP GENERATED BY HIGH ORDER SCHRÖDINGER TYPE OPERATORS

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