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Riesz Transforms Associated with Higher-Order Schrödinger Type Operators

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Abstract

Let L = L 0 + V be a Schrödinger type operator, where L 0 is a higher order elliptic operator with bounded complex coefficients in divergence form and V is a signed measurable function. Under the strongly subcritical assumption on V, we study the L q boundedness of Riesz transform ∇m L −1/2 for q ≤ 2 based on the off-diagonal estimates of semigroup e −tL. Furthermore, the authors impose extra regularity assumptions on V to obtain the L q boundedness of Riesz transform ∇m L −1/2 for some q > 2. In particular, these results are applied to the more interesting Schrödinger operators L = P(D) + V, where P(D) is any homogeneous positive elliptic operator with constant coefficients.

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Acknowledgements

The authors would like to express their sincere gratitude to the reviewing referee for his/her many constructive comments which help us to improve the previous version. The first author is supported by NSFC (No. 11661061 and No. 11671163). The second author is supported by NSFC (No. 11371057, No. 11471033 and No. 11571160). The third author is supported by NSFC (No. 11371158 and No. 11771165) and the program for Changjiang Scholars and Innovative Research Team in University (No. IRT13066).

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Authors and Affiliations

  1. Department of Mathematics, Hubei Key Laboratory of Mathematical Science, Central China Normal University, Wuhan, 430079, China

    Qingquan Deng & Xiaohua Yao

  2. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems (BNU), Ministry of Education Beijing Normal University, Beijing, 100875, China

    Yong Ding

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  1. Qingquan Deng
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  2. Yong Ding
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  3. Xiaohua Yao
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Correspondence to Qingquan Deng.

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Deng, Q., Ding, Y. & Yao, X. Riesz Transforms Associated with Higher-Order Schrödinger Type Operators. Potential Anal 49, 381–410 (2018). https://doi.org/10.1007/s11118-017-9661-7

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