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Balayage of Measures with Respect to (Sub-)Harmonic Functions

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Abstract

We investigate some properties of balayage, or, sweeping (out), of measures with respect to subclasses of subharmonic functions. The following issues are considered: relationships between balayage of measures with respect to classes of harmonic or subharmonic functions and balayage of measures with respect to significantly smaller classes of specific classes of functions; integration of measures and balayage of measures; sensitivity of balayage of measures to polar sets, etc.

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REFERENCES

  1. D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer Monogr. Math. (Springer, London, 2001).

  2. T. Yu. Bayguskarov, G. R. Talipova, and B. N. Khabibullin, ‘‘Subsequences of zeros for classes of entire functions of exponential type, allocated by restrictions on their growth,’’ SPb. Math. J. 28, 127–151 (2017).

    MathSciNet  Google Scholar 

  3. J. Bliedner and W. Hansen, Potential Theory. An Analytic and Probabilistic Approach to Balayage (Springer, Berlin, 1986).

    Google Scholar 

  4. N. Bourbaki, ‘‘Integration of measures,’’ in Elements of Mathematics (Springer, Berlin, Heidelberg, 2004).

    Book  Google Scholar 

  5. M. Brelot, ‘‘Historical introduction,’’ in Potential Theory, Lectures Given at a Summer School of the Centro Internazionale Matematico Estivo, Italy, July 2–10, 1969, Ed. by M. Brelot, Vol. 49 of C.I.M.E. Summer Schools (Springer, Berlin, Heidelberg, 2010), pp. 1–22.

  6. B. J. Cole and Yh. J. Ransford, ‘‘Subharmonicity without upper semicontinuity,’’ J. Funct. Anal. 147, 420–442 (1997).

    Article  MathSciNet  Google Scholar 

  7. J. L. Doob, Classical Potential Theory and Its Probabilistic Counterpart, Vol. 262 of Grundlehren der mathematischen Wissenschaften (Springer, New York, 1984).

  8. T. W. Gamelin, Uniform Algebras and Jensen Measures (Cambridge Univ. Press, Cambridge, 1978).

    MATH  Google Scholar 

  9. S. J. Gardiner, Harmonic Approximation (Cambridge Univ. Press, Cambridge, 1995).

    Book  Google Scholar 

  10. P. M. Gauthier, ‘‘Uniform approximation,’’ in Complex Potential Theory (Kluwer Academic, Netherlands, 1994), pp. 235–271.

    Book  Google Scholar 

  11. P. M. Gautier, ‘‘Subharmonic extensions and approximations,’’ Can. Math. Bull. 37 (5), 46–53 (1994).

    Article  MathSciNet  Google Scholar 

  12. W. Hansen and I. Netuka, ‘‘Jensen measures in potential theory,’’ Potential Anal. 37, 79–90 (2011).

    Article  MathSciNet  Google Scholar 

  13. L. L. Helms, Introduction to Potential Theory (Wiley Interscience, New York, London, Sydney, Toronto, 1969).

    MATH  Google Scholar 

  14. W. K. Hayman and P. B. Kennedy, Subharmonic Functions (Academic, London etc., 1976), Vol. 1.

  15. B. N. Khabibullin, ‘‘Dual representation of superlinear functionals and its applications in function theory. II,’’ Izv. Math. 65, 1017–1039 (2001).

    Article  MathSciNet  Google Scholar 

  16. B. N. Khabibullin, ‘‘Criteria for (sub-)harmonicity and continuation of (sub-)harmonic functions,’’ Sib. Math. J. 44, 713–728 (2003).

    Article  MathSciNet  Google Scholar 

  17. B. N. Khabibullin, ‘‘Zero sequences of holomorphic functions, representation of meromorphic functions, and harmonic minorants,’’ Sb. Math. 198, 261–298 007).

  18. B. N. Khabibullin, A. P. Rozit, and E. B. Khabibullina, ‘‘Order versions of the Hahn–Banach theorem and envelopes. II. Applications to the function theory,’’ Itogi Nauki Tekh., Ser. Sovrem. Mat. Pril. Temat. Obz. 162, 93–135 (2019).

    Google Scholar 

  19. B. N. Khabibullin and A. V. Shmelyova, ‘‘Balayage of measures and subharmonic functions on a system of rays. I. Classic case,’’ SPb. Math. J. 31, 117–156 (2020).

    MathSciNet  Google Scholar 

  20. P. Koosis, The Logarithmic Integral. II (Cambridge Univ. Press, Cambridge, 1992).

    Book  Google Scholar 

  21. N. S. Landkof, Foundations of Modern Potential Theory, Vol. 180 of Grundlehren der mathematischen Wissenschaften (Springer, New York, Heidelberg, 1972).

  22. P.-A. Meyer, Probability and Potentials (Blaisdell, Waltham, MA, Toronto, London, 1966).

    MATH  Google Scholar 

  23. B. N. Khabibullin and E. B. Menshikova, ‘‘On the distribution of zero sets of holomorphic functions. II,’’ Funct. Anal. Appl. 53, 65 (2019).

    Article  MathSciNet  Google Scholar 

  24. Th. Ransford, ‘‘Potential Theory in the Complex Plane’’ (Cambridge Univ. Press, Cambridge, 1995).

    Book  Google Scholar 

  25. Th. J. Ransford, ‘‘Jensen measures,’’ in Approximation, Complex Analysis and Potential Theory (Montréal, QC, 2000) (Kluwer, Dordrecht, 2001), pp. 221–237.

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Funding

This research was supported by a grant from the Russian Science Foundation (project no. 18-11-00002).

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Authors and Affiliations

  1. Bashkir State University, 420074, Ufa, Bashkortostan, Russia

    B. N. Khabibullin

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  1. B. N. Khabibullin
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Correspondence to B. N. Khabibullin.

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(Submitted by F. G. Avkhadiev)

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Khabibullin, B.N. Balayage of Measures with Respect to (Sub-)Harmonic Functions. Lobachevskii J Math 41, 2179–2189 (2020). https://doi.org/10.1134/S1995080220110116

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  • DOI: https://doi.org/10.1134/S1995080220110116

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