# Mean values and associated measures of $\delta $-subharmonic functions

Mathematica Bohemica (2002)

- Volume: 127, Issue: 1, page 83-102
- ISSN: 0862-7959

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topWatson, Neil A.. "Mean values and associated measures of $\delta $-subharmonic functions." Mathematica Bohemica 127.1 (2002): 83-102. <http://eudml.org/doc/249014>.

@article{Watson2002,

abstract = {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.},

author = {Watson, Neil A.},

journal = {Mathematica Bohemica},

keywords = {superharmonic; $\delta $-subharmonic; Riesz measure; spherical mean values; superharmonic function; -subharmonic function; Riesz measure; spherical mean values},

language = {eng},

number = {1},

pages = {83-102},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {Mean values and associated measures of $\delta $-subharmonic functions},

url = {http://eudml.org/doc/249014},

volume = {127},

year = {2002},

}

TY - JOUR

AU - Watson, Neil A.

TI - Mean values and associated measures of $\delta $-subharmonic functions

JO - Mathematica Bohemica

PY - 2002

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 127

IS - 1

SP - 83

EP - 102

AB - 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.

LA - eng

KW - superharmonic; $\delta $-subharmonic; Riesz measure; spherical mean values; superharmonic function; -subharmonic function; Riesz measure; spherical mean values

UR - http://eudml.org/doc/249014

ER -

## References

top- Domination, uniqueness and representation theorems for harmonic functions in half-spaces, Ann. Acad. Sci. Fenn. Ser. A.I. Math. 6 (1981), 161–172. (1981) Zbl0441.31003MR0639973
- Mean values and associated measures of superharmonic functions, Hiroshima Math. J. 13 (1983), 53–63. (1983) Zbl0512.31009MR0693550
- A general form of the covering principle and relative differentiation of additive functions, Proc. Cambridge Phil. Soc. 41 (1945), 103–110. (1945) Zbl0063.00352MR0012325
- A general form of the covering principle and relative differentiation of additive functions II, Proc. Cambridge Phil. Soc. 42 (1946), 1–10. (1946) Zbl0063.00353MR0014414
- Some non-negativity theorems for harmonic functions, Ann. Acad. Sci. Fenn. Ser. A.I. 452 (1969), 1–8. (1969) MR0265620
- Potentiels sur un ensemble de capacité nulle. Suites de potentiels, C. R. Acad. Sci. Paris 244 (1957), 1707–1710. (1957) Zbl0086.30601MR0087757
- Classical Potential Theory and its Probabilistic Counterpart, Springer, New York, 1984. (1984) Zbl0549.31001MR0731258
- The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1985. (1985) Zbl0587.28004MR0867284
- Geometric Measure Theory, Springer, Berlin, 1969. (1969) Zbl0176.00801MR0257325
- Some properties of the Riesz charge associated with a $\delta $-subharmonic function, Potential Anal. 1 (1992), 355–371. (1992) Zbl0766.31010MR1245891
- Sets of regular increase of entire functions, Teor. Funkts., Funkts. Anal. Prilozh. 40 (1983), 36–47. (Russian) (1983) Zbl0601.30036MR0738442
- Hahn decomposition for the Riesz charge of $\delta $-subharmonic functions, Math. Scand. 83 (1998), 277–282. (1998) Zbl1023.31005MR1673934
- Two definitions of fractional dimension, Math. Proc. Cambridge Phil. Soc. 91 (1982), 57–74. (1982) Zbl0483.28010MR0633256
- Superharmonic extensions, mean values and Riesz measures, Potential Anal. 2 (1993), 269–294. (1993) Zbl0785.31002MR1245245
- 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. (1994) Zbl0793.31001MR1246891
- Domination and representation theorems for harmonic functions and temperatures, Bull. London Math. Soc. 27 (1995), 467–472. (1995) Zbl0841.31007MR1338690

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