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Growth estimates for modified Green potentials in the upper-half space. (English) Zbl 1220.31008
The authors construct a modified Green potential in the upper-half space of the n-dimensional Euclidean space. The motivation is the modification of the Poisson kernel by subtracting some special harmonic polynomials which allowed Finkelstein and Scheinberg to solve the Dirichlet problem for the upper half-space with arbitrary continuous data. In the present article, the authors modify the Green function of order α for the upper half-space by subtracting some combinations of suitable ultraspherical (Gegenbauer) polynomials from the kernel. The behavior at infinity of this modified Green function is also given. The finding is that outside a Borel set with controlled capacity, the Green function grows slower than an appropriate power of the absolute value of the argument times an appropriate power of the last (positive) coordinate of the argument. A geometric approach using certain covering with balls of the aforementioned Borel set is also provided.
MSC:
31B10Integral representations of harmonic functions (higher-dimensional)
31C05Generalizations of harmonic (subharmonic, superharmonic) functions
References:
[1]Aikawa, H.: On the behavior at infinity of non-negative superharmonic functions in a half space, Hiroshima math. J. 11, 425-441 (1981) · Zbl 0468.31002
[2]Deng, G. T.: Integral representations of harmonic functions in half spaces, Bull. sci. Math. 131, 53-59 (2007) · Zbl 1111.31002 · doi:10.1016/j.bulsci.2006.03.008
[3]Essén, M.; Jackson, H. L.: On the covering properties of certain exceptional sets in a half space, Hiroshima math. J. 10, 233-262 (1980) · Zbl 0447.31003
[4]Finkelstein, M.; Scheinberg, S.: Kernels for solving problems of Dirichlet type in a half-plane, Adv. math. 18, No. 1, 108-113 (1975) · Zbl 0309.31001 · doi:10.1016/0001-8708(75)90004-3
[5]Fuglede, B.: Le théorème du minimax et la théorie fine du potential, Ann. inst. Fourier (Grenoble) 15, 65-88 (1965) · Zbl 0128.33103 · doi:10.5802/aif.196 · doi:numdam:AIF_1965__15_1_65_0
[6]Kheyfits, A.: Representation of the analytic functions of infinite order in a half-plane, Izv. akad. Nauk armjan. SSR, ser. Mat. 6, No. 6, 472-476 (1971)
[7]Levin, B. Ya.: Lectures on entire functions, Translations of mathematical monographs (1996) · Zbl 0856.30001
[8]Mizuta, Y.: On the behavior at infinity of Green potentials in a half space, Hiroshima math. J. 10, 607-613 (1980) · Zbl 0451.31010
[9]Mizuta, Y.: Boundary limits of Green potentials of general order, Proc. amer. Math. soc. 101, No. 1, 131-135 (1987) · Zbl 0659.31007 · doi:10.2307/2046563
[10]Mizuta, Y.: On the boundary limits of Green potentials of functions, J. math. Soc. Japan 40, No. 4, 583-594 (1988) · Zbl 0695.31006 · doi:10.2969/jmsj/04040583
[11]Mizuta, Y.; Shimomura, T.: Growth properties for modified Poisson integrals in a half space, Pacific J. Math. 212, No. 2, 333-346 (2003) · Zbl 1056.31004 · doi:10.2140/pjm.2003.212.333
[12]Siegel, D.; Talvila, E.: Sharp growth estimates for modified Poisson integrals in a half space, Potential anal. 15, 333-360 (2001) · Zbl 0987.31003 · doi:10.1023/A:1011817130061
[13]Stein, E. M.: Singular integrals and differentiability properties of functions, (1979)
[14]Szegö, G.: Orthogonal polynomials, American mathematical society colloquium publications 23 (1975)