Harmonic functions and mass cancellation
Author:
J. R. Baxter
Journal:
Trans. Amer. Math. Soc. 245 (1978), 375384
MSC:
Primary 60J05; Secondary 31B05, 60J65
MathSciNet review:
511416
Fulltext PDF Free Access
Abstract 
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Abstract: If a function on an open set in has the mean value property for one ball at each point of the domain, the function will be said to possess the restricted mean value property. (The ordinary or unrestricted mean value property requires that the mean value property hold for every ball in the domain.) We specify the single ball at each point x by its radius , a function of x. Under appropriate conditions on and the function, the restricted mean value property implies that the function is harmonic, giving a converse to the mean value theorem (see references). In the present paper a converse to the mean value theorem is proved, in which the function is well behaved, but the function is only required to be nonnegative. A converse theorem for more general means than averages over balls is also obtained. These results extend theorems of D. Heath, W. Veech, and the author (see references). Some connections are also pointed out between converse mean value theorems and mass cancellation.
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 J. R. Baxter and R. V. Chacon, Potentials of stopped distributions, Illinois J. Math. 18 (1974), 649656. MR 0358960 (50:11417)
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 , Stopping times for recurrent Markov processes, Illinois J. Math. 20 (1976), 467475. MR 0420860 (54:8872)
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 W. A. Veech, A zeroone law for a class of random walks and a converse to Gauss' mean value theorem, Ann. of Math. (2) 97 (1973), 189216. MR 0310269 (46:9370)
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 , A converse to the mean value theorem for harmonic functions, Amer. J. Math. 97 (1975), 10071027. MR 0393521 (52:14330)
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 , The core of a measurable set and a problem in potential theory (preprint).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719780511416X
PII:
S 00029947(1978)0511416X
Keywords:
Restricted mean value,
invariant function,
Brownian motion
Article copyright:
© Copyright 1978
American Mathematical Society
