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A Comparison Theorem for Subharmonic Functions

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Abstract

In this article, we prove an extension of the mean value theorem and a comparison theorem for subharmonic functions. These theorems are used to answer the question whether we can conclude that two subharmonic functions which agree almost everywhere on a surface with respect to the surface measure must coincide everywhere on that surface. We prove that this question has a positive answer in the case of hypersurfaces, and we also provide a counterexample in the case of surfaces of higher co-dimension. We also apply these results to Ahlfors–David sets and we prove other versions of the main results in terms of measure densities.

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Acknowledgements

This work forms part of the author’s doctoral dissertation under the supervision of Professor Dinh Tien Cuong and Professor Pham Hoang Hiep. The author is grateful to receive valuable comments and strong support from his advisors and Dr. Do Hoang Son. The author is also immensely grateful to the referees for helpful comments and suggestions. The author would like to thank IMU and TWAS for supporting his PhD studies through the IMU Breakout Graduate Fellowship.

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Authors and Affiliations

  1. Institute of Mathematics, Vietnam Academy of Science and Technology, 18, Hoang Quoc Viet, Hanoi, Vietnam

    Thai Duong Do

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  1. Thai Duong Do
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Correspondence to Thai Duong Do.

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Do, T.D. A Comparison Theorem for Subharmonic Functions. Results Math 74, 176 (2019). https://doi.org/10.1007/s00025-019-1098-4

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  • DOI: https://doi.org/10.1007/s00025-019-1098-4

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