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The Generalised Laplace Operator and the Topological Characteristic of Removable \( {\overline{S}} \) - Singular Sets of Subharmonic Functions

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Abstract

The class of harmonic and subharmonic functions have been studied by many authors, defined in different ways, by the Laplace differential operator, averaging, generalised Laplace operators, etc. The well-known theorem of Blaschke-Privalov gives an excellent criterion for subharmonicity in terms of the generalised Laplace operators: an upper semi-continuous in the domain \(D \subset \mathbb {R}^{n}\) function u(x), \(u(x)\not \equiv -\infty ,\) is subharmonic if and only if \( {\overline{\bigtriangleup }} u(x)\ge 0 \quad \forall x^0\in D{\setminus } u_{-\infty }.\) One of the notable results is Privalov’s theorem, where he got more deeper result with an exceptional set E: if the function u(x), \(u(x)\not \equiv -\infty \), is upper semi-continuous in the domain \(D \subset {\mathbb {R}}^{n}\) and the following two conditions hold:

  1. (1)

    \( {\overline{\bigtriangleup }}_B u(x^0)\ge 0 \quad \forall x^0\in D {\setminus } [E \cup u_{-\infty }]\), where \(E \subset D\) is a closed in D set, \(mes E =0\);

  2. (2)

    \({\overline{\bigtriangleup }}_B u(x^0) > - \infty \quad \forall x^0\in E{\setminus } P, \) where \(P \subset E \) is some polar set.

Then the function u(x) is subharmonic in D. The purpose of this paper is to characterise completely this type of exceptional sets. For this, we introduce the so-called \( {\underline{S}} \) and \( {\overline{S}} \) singular-sets, which are directly related to the exceptional set of I. Privalov. We prove: \(E \in {\underline{S}}\) if and only if \(mes E =0;\) \(E \in {\overline{S}}\) if and only if \(E^\circ =\emptyset .\)

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Data Availibility

Some of the data we use can be found in http://www.mathnet.ruplatform. This data is free online. For example all of the works of I.I.Privalov can be found in http://www.mathnet.ru/eng/person35683, other data also can be found in the same way. The work of Sh.Shopulatov can be found in https://uzmj.mathinst.uz/archive--2018-...-.html. The authors did not have any special access privileges that others would not have. If some data can not be found and if needed, anyone could request the authors. The authors have the copy of all published works and can be sent if requested.

References

  1. Blaschke, W.: Ein Mittelwertsatz und eine kennzeichnende Eigenschaft des logarithmischen Potentials. Ber. Verh. S\(\ddot{a}\)chs. Akad. Wiss., 68, 3–7 (1916)

  2. Brelot, M.: Elements de la Theorie Classique Du Potentiel, 4e edn, pp. 24–28. Centre du documentation universitaire, Paris (1969)

    Google Scholar 

  3. Pokrovski, A.V.: Conditions for subharmonicity and subharmonic extensions of functions. Sb. Math. 208(8), 1225–1245 (2017)

    Article  MathSciNet  Google Scholar 

  4. Privalov, I.I.: Sur les fonctions harmoniques. Rec. Math. XXXII(3), 464–469 (1925)

    MATH  Google Scholar 

  5. Privalov, I.I.: On a theorem of S. Saks. Rec. Math. [Mat. Sb.] 9(2), 457–460 (1941)

    MathSciNet  Google Scholar 

  6. Privalov, I.I.: To the definition of a subharmonic function. News USSR Acad. Sci. 5(4–5), 281–284 (1941)

    MathSciNet  MATH  Google Scholar 

  7. Privalov, I.I.: Quelques applications de 1 operateur generalise de Laplace. Rec. Math. [Mat. Sb.] 11(3), 149–154 (1942)

    Google Scholar 

  8. Sadullaev, A.: Pluripotential Theory and its Applications, pp. 42–52. Palmarium Academic Publishing, Germany (2012)

    Google Scholar 

  9. Sadullaev, A., Madrakhimov, R.M.: Smoothness of subharmonic functions. Math. USSR-Sb. 69(1), 179–195 (1991)

    Article  MathSciNet  Google Scholar 

  10. Saks, S.: On the operators of Blaschke and Privaloff for subharmonic functions. Rec. Math. Mat. Sb. 9(2), 451–456 (1941)

    MathSciNet  MATH  Google Scholar 

  11. Shopulatov, Sh.Sh: On a weak criterion for the subharmonicity of functions in \({\mathbb{R}}^{n}\). Uzb. Math. J. 2, 136–141 (2018)

  12. Shopulatov, Sh.Sh.: On the properties of singular removable sets of subharmonic functions. Rep. AS RUz 2, 3–6 (2019)

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Acknowledgements

We would like to express our deep gratitude to Karim Rakhimov for his assistance in the work.

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Correspondence to Shomurod Shopulatov.

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Communicated by Dan Volok.

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Sadullaev, A., Shopulatov, S. The Generalised Laplace Operator and the Topological Characteristic of Removable \( {\overline{S}} \) - Singular Sets of Subharmonic Functions. Complex Anal. Oper. Theory 15, 50 (2021). https://doi.org/10.1007/s11785-021-01102-w

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  • DOI: https://doi.org/10.1007/s11785-021-01102-w

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