Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

LIOUVILLE THEOREMS FOR THE MULTIDIMENSIONAL FRACTIONAL BESSEL OPERATORS

  • Galli, Vanesa (CEMIM - Departamento de Matematica - FCEyN Universidad Nacional de Mar del Plata) ;
  • Molina, Sandra (CEMIM - Departamento de Matematica - FCEyN Universidad Nacional de Mar del Plata) ;
  • Quintero, Alejandro (CEMIM - Departamento de Matematica - FCEyN Universidad Nacional de Mar del Plata)
  • Received : 2021.10.30
  • Accepted : 2022.01.19
  • Published : 2022.10.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we establish Liouville type theorems for the fractional powers of multidimensional Bessel operators extending the results given in [6]. In order to do this, we consider the distributional point of view of fractional Bessel operators studied in [12].

Keywords

References

  1. A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10 (1960), 419-437. http://projecteuclid.org/euclid.pjm/1103038401 103038401
  2. J. J. Betancor, A. J. Castro, and P. R. Stinga, The fractional Bessel equation in Holder spaces, J. Approx. Theory 184 (2014), 55-99. https://doi.org/10.1016/j.jat.2014.05.003
  3. K. Bogdan, T. Kulczycki, and A. Nowak, Gradient estimates for harmonic and qharmonic functions of symmetric stable processes, Illinois J. Math. 46 (2002), no. 2, 541-556. http://projecteuclid.org/euclid.ijm/1258136210 https://doi.org/10.1215/ijm/1258136210
  4. W. Chen, L. D'Ambrosio, and Y. Li, Some Liouville theorems for the fractional Laplacian, Nonlinear Anal. 121 (2015), 370-381. https://doi.org/10.1016/j.na.2014.11.003
  5. J. Delsarte, Sur une extension de la formule de Taylor, Journal de Mathematiques Pures et Appliquees. Neuvieme Serie 17 (1938), 213-231.
  6. V. Galli, S. Molina, and A. Quintero, A Liouville theorem for some Bessel generalized operators, Integral Transforms Spec. Funct. 29 (2018), no. 5, 367-383. https://doi.org/10.1080/10652469.2018.1441295
  7. I. I. Hirschman, Jr., Variation diminishing Hankel transforms, J. Analyse Math. 8 (1960/61), 307-336. https://doi.org/10.1007/BF02786854
  8. Y. Y. Li, Local gradient estimates of solutions to some conformally invariant fully nonlinear equations, C. R. Math. Acad. Sci. Paris 343 (2006), no. 4, 249-252. https://doi.org/10.1016/j.crma.2006.06.008
  9. C. Martinez and M. Sanz, The theory of fractional powers of operators, North-Holland Mathematics Studies, 187, North-Holland Publishing Co., Amsterdam, 2001.
  10. C. Martinez, M. Sanz, and F. Periago, Distributional fractional powers of the Laplacean. Riesz potentials, Studia Math. 135 (1999), no. 3, 253-271.
  11. S. Molina, A generalization of the spaces Hµ, H'µ and the space of multipliers, in Proceedings of the Seventh "Dr. Antonio A. R. Monteiro" Congress of Mathematics (Spanish), 49-56, Univ. Nac. del Sur, Bahia Blanca, 2003.
  12. S. M. Molina, Distributional fractional powers of similar operators with applications to the Bessel operators, Commun. Korean Math. Soc. 33 (2018), no. 4, 1249-1269. https://doi.org/10.4134/CKMS.c170440
  13. S. Molina and S. E. Trione, n-dimensional Hankel transform and complex powers of Bessel operator, Integral Transforms Spec. Funct. 18 (2007), no. 11-12, 897-911. https://doi.org/10.1080/10652460701511244
  14. F. Oberhettinger, Tables of Bessel Transforms, Springer-Verlag, New York, 1972.
  15. H. H. Schaefer, Topological Vector Spaces, Graduate Texts in Mathematics, Vol. 3, Springer-Verlag, New York, 1971.
  16. G. N. Watson, A Treatise on the Theory of Bessel Functions, reprint of the second (1944) edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995.
  17. A. H. Zemanian, A distributional Hankel transformation, SIAM J. Appl. Math. 14 (1966), 561-576. https://doi.org/10.1137/0114049
  18. R. Zhuo, W. Chen, X. Cui, and Z. Yuan, A Liouville theorem for the fractional Laplacian, arXiv:1401.7402 (2014)