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Growth of harmonic functions on biregular trees

Year 2022, Volume: 9 Issue: 2, 1 - 8, 13.05.2022
https://doi.org/10.13069/jacodesmath.1056555

Abstract

On a biregular tree of degrees $q+1$ and $r+1$, we study the growth of two classes of harmonic functions. First, we prove that if $f$ is a bounded harmonic function on the tree and $x$, $y$ are two adjacent vertices, then $|f(x)-f(y)|\leq 2 (qr-1)\|f\|_\infty/((q+1)(r+1))$, thus generalizing a result of Cohen and Colonna for regular trees. Next, we prove that if $f$ is a positive harmonic function on the tree and $x$, $y$ are two vertices with $d(x,y)=2$, then $f(x)/(qr)\leq f(y)\leq qr\cdot f(x)$.

References

  • [1] V. Anandam, Harmonic functions and potentials on finite and infinite networks, Springer, Heidelberg, Bologna (2011).
  • [2] S. Axler, P. Bourdon, W. Ramey, Harmonic function theory, Springer-Verlag, New York (2001).
  • [3] N. L. Biggs, Discrete mathematics, Clarendon Press, Oxford University Press, New York (1985).
  • [4] P. Cartier, Fonctions harmoniques sur un arbre, Sympos. Math. 9 (1972) 203–270.
  • [5] J. M. Cohen, F. Colonna, The Bloch space of a homogeneous tree, Bol. Soc. Mat. Mex. 37 (1992) 63–82.
  • [6] E. Nelson, A proof of Liouville’s theorem, Proc. Amer. Math. Soc. 12(6) (1961) 995.
  • [7] W. Woess, Random walks on infinite graphs and groups, Cambridge University Press (2000).
Year 2022, Volume: 9 Issue: 2, 1 - 8, 13.05.2022
https://doi.org/10.13069/jacodesmath.1056555

Abstract

References

  • [1] V. Anandam, Harmonic functions and potentials on finite and infinite networks, Springer, Heidelberg, Bologna (2011).
  • [2] S. Axler, P. Bourdon, W. Ramey, Harmonic function theory, Springer-Verlag, New York (2001).
  • [3] N. L. Biggs, Discrete mathematics, Clarendon Press, Oxford University Press, New York (1985).
  • [4] P. Cartier, Fonctions harmoniques sur un arbre, Sympos. Math. 9 (1972) 203–270.
  • [5] J. M. Cohen, F. Colonna, The Bloch space of a homogeneous tree, Bol. Soc. Mat. Mex. 37 (1992) 63–82.
  • [6] E. Nelson, A proof of Liouville’s theorem, Proc. Amer. Math. Soc. 12(6) (1961) 995.
  • [7] W. Woess, Random walks on infinite graphs and groups, Cambridge University Press (2000).
There are 7 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Francisco Javier Gonzalez Vieli This is me

Publication Date May 13, 2022
Published in Issue Year 2022 Volume: 9 Issue: 2

Cite

APA Gonzalez Vieli, F. J. (2022). Growth of harmonic functions on biregular trees. Journal of Algebra Combinatorics Discrete Structures and Applications, 9(2), 1-8. https://doi.org/10.13069/jacodesmath.1056555
AMA Gonzalez Vieli FJ. Growth of harmonic functions on biregular trees. Journal of Algebra Combinatorics Discrete Structures and Applications. May 2022;9(2):1-8. doi:10.13069/jacodesmath.1056555
Chicago Gonzalez Vieli, Francisco Javier. “Growth of Harmonic Functions on Biregular Trees”. Journal of Algebra Combinatorics Discrete Structures and Applications 9, no. 2 (May 2022): 1-8. https://doi.org/10.13069/jacodesmath.1056555.
EndNote Gonzalez Vieli FJ (May 1, 2022) Growth of harmonic functions on biregular trees. Journal of Algebra Combinatorics Discrete Structures and Applications 9 2 1–8.
IEEE F. J. Gonzalez Vieli, “Growth of harmonic functions on biregular trees”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 9, no. 2, pp. 1–8, 2022, doi: 10.13069/jacodesmath.1056555.
ISNAD Gonzalez Vieli, Francisco Javier. “Growth of Harmonic Functions on Biregular Trees”. Journal of Algebra Combinatorics Discrete Structures and Applications 9/2 (May 2022), 1-8. https://doi.org/10.13069/jacodesmath.1056555.
JAMA Gonzalez Vieli FJ. Growth of harmonic functions on biregular trees. Journal of Algebra Combinatorics Discrete Structures and Applications. 2022;9:1–8.
MLA Gonzalez Vieli, Francisco Javier. “Growth of Harmonic Functions on Biregular Trees”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 9, no. 2, 2022, pp. 1-8, doi:10.13069/jacodesmath.1056555.
Vancouver Gonzalez Vieli FJ. Growth of harmonic functions on biregular trees. Journal of Algebra Combinatorics Discrete Structures and Applications. 2022;9(2):1-8.