# Article

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Keywords:
superharmonic; $\delta$-subharmonic; Riesz measure; spherical mean values
Summary:
Let $u$ be a $\delta$-subharmonic function with associated measure $\mu$, and let $v$ be a superharmonic function with associated measure $\nu$, on an open set $E$. For any closed ball $B(x,r)$, of centre $x$ and radius $r$, contained in $E$, let ${\mathcal M}(u,x,r)$ denote the mean value of $u$ over the surface of the ball. We prove that the upper and lower limits as $s,t\rightarrow 0$ with $0<s<t$ of the quotient $({\mathcal M}(u,x,s)-{\mathcal M}(u,x,t))/({\mathcal M}(v,x,s)-{\mathcal M}(v,x,t))$, lie between the upper and lower limits as $r\rightarrow 0+$ of the quotient $\mu (B(x,r))/\nu (B(x,r))$. This enables us to use some well-known measure-theoretic results to prove new variants and generalizations of several theorems about $\delta$-subharmonic functions.
References:
[1] D. H. Armitage: Domination, uniqueness and representation theorems for harmonic functions in half-spaces. Ann. Acad. Sci. Fenn. Ser. A.I. Math. 6 (1981), 161–172. DOI 10.5186/aasfm.1981.0602 | MR 0639973 | Zbl 0441.31003
[2] D. H. Armitage: Mean values and associated measures of superharmonic functions. Hiroshima Math. J. 13 (1983), 53–63. DOI 10.32917/hmj/1206133537 | MR 0693550 | Zbl 0512.31009
[3] A. S. Besicovitch: A general form of the covering principle and relative differentiation of additive functions. Proc. Cambridge Phil. Soc. 41 (1945), 103–110. MR 0012325 | Zbl 0063.00352
[4] A. S. Besicovitch: A general form of the covering principle and relative differentiation of additive functions II. Proc. Cambridge Phil. Soc. 42 (1946), 1–10. MR 0014414 | Zbl 0063.00353
[5] A. M. Bruckner, A. J. Lohwater, F. Ryan: Some non-negativity theorems for harmonic functions. Ann. Acad. Sci. Fenn. Ser. A.I. 452 (1969), 1–8. MR 0265620
[6] G. Choquet: Potentiels sur un ensemble de capacité nulle. Suites de potentiels. C. R. Acad. Sci. Paris 244 (1957), 1707–1710. MR 0087757 | Zbl 0086.30601
[7] J. L. Doob: Classical Potential Theory and its Probabilistic Counterpart. Springer, New York, 1984. MR 0731258 | Zbl 0549.31001
[8] K. J. Falconer: The Geometry of Fractal Sets. Cambridge University Press, Cambridge, 1985. MR 0867284 | Zbl 0587.28004
[9] H. Federer: Geometric Measure Theory. Springer, Berlin, 1969. MR 0257325 | Zbl 0176.00801
[10] B. Fuglede: Some properties of the Riesz charge associated with a $\delta$-subharmonic function. Potential Anal. 1 (1992), 355–371. DOI 10.1007/BF00301788 | MR 1245891 | Zbl 0766.31010
[11] A. F. Grishin: Sets of regular increase of entire functions. Teor. Funkts., Funkts. Anal. Prilozh. 40 (1983), 36–47. (Russian) MR 0738442 | Zbl 0601.30036
[12] M. Sodin: Hahn decomposition for the Riesz charge of $\delta$-subharmonic functions. Math. Scand. 83 (1998), 277–282. DOI 10.7146/math.scand.a-13856 | MR 1673934 | Zbl 1023.31005
[13] C. Tricot: Two definitions of fractional dimension. Math. Proc. Cambridge Phil. Soc. 91 (1982), 57–74. DOI 10.1017/S0305004100059119 | MR 0633256 | Zbl 0483.28010
[14] N. A. Watson: Superharmonic extensions, mean values and Riesz measures. Potential Anal. 2 (1993), 269–294. DOI 10.1007/BF01048511 | MR 1245245 | Zbl 0785.31002
[15] N. A. Watson: Applications of geometric measure theory to the study of Gauss-Weierstrass and Poisson integrals. Ann. Acad. Sci. Fenn. Ser. A.I. Math. 19 (1994), 115–132. MR 1246891 | Zbl 0793.31001
[16] N. A. Watson: Domination and representation theorems for harmonic functions and temperatures. Bull. London Math. Soc. 27 (1995), 467–472. DOI 10.1112/blms/27.5.467 | MR 1338690 | Zbl 0841.31007

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