## Taiwanese Journal of Mathematics

### INTEGRAL REPRESENTATIONS AND GROWTH PROPERTIES FOR A CLASS OF SUPERFUNCTIONS IN A CONE

#### Abstract

An integral representation for a class of superfunctions, associated with the Schrödinger operator, is investigated. Meanwhile, growth properties of them are also proved outside of some exceptional sets.

#### Article information

Source
Taiwanese J. Math., Volume 15, Number 5 (2011), 2213-2233.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406431

Digital Object Identifier
doi:10.11650/twjm/1500406431

Mathematical Reviews number (MathSciNet)
MR2880401

Zentralblatt MATH identifier
1238.31002

#### Citation

Qiao, Lei; Deng, Guan-Tie. INTEGRAL REPRESENTATIONS AND GROWTH PROPERTIES FOR A CLASS OF SUPERFUNCTIONS IN A CONE. Taiwanese J. Math. 15 (2011), no. 5, 2213--2233. doi:10.11650/twjm/1500406431. https://projecteuclid.org/euclid.twjm/1500406431

#### References

• A. Ancona, First eigenvalues and comparison of Green's functions for elliptic operators on manifolds or domains, J. d'Anal. Math., 72 (1997), 45-92.
• V. S. Azarin, Generalization of a theorem of Hayman on subharmonic functions in an $m$-dimensional cone, Amer. Math. Soc. Translation, 80 (1969), 119-138.
• E. F. Beckenbach, Generalized convex functions, Bull. Amer. Math. Soc., 43 (1937), 363-371.
• L. Bragg and J. Dettman, Function theories for the Yukawa and Helmholtz equations, Rocky Mountain J. Math., 25 (1995), 887-917.
• M. Bramanti, Potential theory for stationary Schrödinger operators: a survey of results obtained with non-probabilistic methods, Le Math., 47 (1992), 25-61.
• R. Courant and D. Hilbert, Methods of mathematical physics, Vol. 1, Interscience Publishers, New York, 1953.
• M. Essén, H. L. Jackson and P. J. Rippon, On minimally thin and rarefied sets in ${\bf R}^{p},~p\geq2$, Hiroshima Math. J., 15 (1985), 393-410.
• D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer Verlag, Berlin, 1977.
• P. Hartman, Ordinary differential equations, Wiley, New York, London, Sydney, 1964.
• A. I. Kheyfits, The Riesz-Herglotz formula for generalized harmonic functions and their boundary behavior (Russian), Dokl. Akad. Nauk SSSR, 321 (1991), 263-265; translation in Soviet Math. Dokl., 44 (1992), 688-691.
• B. Ya. Levin and A. I. Kheyfits, Asymptotic behavior of subfunctions of the stationary Schrödinger operator, Preprint, http://arXiv/abs/math/021132896, 2002, p. 96.
• C. Miranda, Partial differential equations of elliptic type, Springer, Berlin, 1970.
• L. Nirenberg, Existence theorems in partial differential equations, NYU Notes, 1954.
• M. Reed and B. Simon, Methods of modern mathematical physics, Vol. 3, Acad Press, London-New York-San Francisco, 1970.
• A. Russakovskii, On asymptotic behavior of subfunctions of finite lower order of the Schrödinger operator, Sib. Math. J., 30 (1989), 160-170.
• D. Siegel and E. O. Talvila, Uniqueness for the $n$-dimensional half space Dirichlet problem, Pacific J. Math., 175 (1996), 571-587.
• B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc., 7 (1982), 447-526.
• E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ, 1970.
• G. M. Verzhbinskii and V. G. Maz'ya, Asymptotic behavior of solutions of elliptic equations of the second order close to a boundary. I, Sibirsk. Mat. J., 12 (1971), 874-899.