Cylindrical Carleman's formula of subharmonic functions and its application
- School of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450046, China.
Our aim in this paper is to prove the cylindrical Carleman's formula for subharmonic functions in a truncated cylinder. As an
application, we prove that if the positive part of a harmonic function in a cylinder satisfies a slowly growing condition, then
its negative part can also be dominated by a similar slowly growing condition, which
improves some classical results about harmonic functions in a cylinder.
- Cylindrical Carleman's formula
- subharmonic function
D. H. Armitage, A Nevanlinna theorem for superharmonic functions in half-spaces, with applications, J. London Math. Soc., 23 (1981), 137–157.
D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin-New York (1977)
Ü. Kuran, Harmonic majorizations in half-balls and half-spaces, Proc. London Math. Soc., 21 (1970), 614–636.
B. Y. Levin , Entire and subharmonic functions, Adv. Soviet Math., Amer. Math. Soc., Providence, RI (1992)
I. Miyamoto, Harmonic functions in a cylinder which vanish on the boundary, Japan. J. Math., 22 (1996), 241–255.
I. Miyamoto, H. Yoshida , Harmonic functions in a cylinder with normal derivatives vanishing on the boundary, Ann. Polon. Math., 74 (2000), 229–235.
I. Miyamoto, H. Yoshida, On a covering property of minimally thin sets at infinity in a cylinder, Math. Montisnigri, 20/21 (2007/08), 35–54.
L. Qiao, Asymptotic behavior of Poisson integrals in a cylinder and its application to the representation of harmonic functions, Bull. Sci. Math., 144 (2018), 39–54.
A. Y. Rashkovskiı, L. I. Ronkin, Subharmonic functions of finite order in a cone. III., Functions of completely regular growth, J. Math. Sci., 77 (1995), 2929–2940.
L. I. Ronkin, Functions of completely regular growth , Kluwer Academic Publishers Group , Dordrecht (1992)
G. V. Rozenblyum, M. Z. Solomyak, M. A. Shubin, Spectral theory of differential operators, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow (1989)
H. Yoshida, Harmonic majorization of a subharmonic function on a cone or on a cylinder , Pacific J. Math., 148 (1991), 369–395.