## Successive averages and harmonic functions.(English)Zbl 0879.31002

For every $$r>0$$ let $$m_r$$ be the normalized Lebesgue measure on the ball $$B(0, r)$$ in $$\mathbb{R}^d$$. This paper is motivated by the following question asked by G. Choquet in 1994: Let $$f$$ be a continuous real function on $$\mathbb{R}^d$$ and let $$r_1$$, $$r_2$$, $$r_3$$, $$\dots$$ be strictly positive real numbers; under what conditions on $$f$$ and the sequence $$(r_n)$$ does $$(f\ast m_{r_1}\ast m_{r_2}\ast\cdots\ast m_{r_n})$$ converge to a harmonic function?
Among other things, the authors prove the following result. If $$\sum^{\infty}_{n=1}r^2_n=\infty$$, then the following holds: (1) for every superharmonic function $$u$$ on $$\mathbb{R}^d$$ admitting a harmonic minorant the sequence $$(u\ast m_{r_1}\ast m_{r_2} \ast\cdots\ast m_{r_n})$$ decreases locally uniformly to the greatest harmonic minorant of $$u$$; (2) for every continuous $$f$$ with comapct support the sequence $$(f\ast m_{r_1}\ast m_{r_2}\ast\cdots\ast m_{r_n})$$ converges uniformly to zero; and (3) for every bounded $$f$$ such that the average $$\int fdm_r$$ tends to some real number $$A$$ as $$r\rightarrow\infty$$, the sequence $$(f\ast m_{r_1}\ast m_{r_2}\ast\cdots\ast m_{r_n})$$ converges locally uniformly to $$A$$.
The paper also contains similar results when $$m_r$$ is replaced by more general measures.
Reviewer: R.Song (Ann Arbor)

### MSC:

 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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### References:

  Bauer, H., Probability Theory (1996), Berlin: de Gruyter, Berlin · Zbl 0868.60001  Benyamini, Y.; Weit, Y., Functions satisfying the mean value property in the limit, J. Analyse Math, 52, 167-198 (1989) · Zbl 0675.31002  Bliedtner, J.; Hansen, W., Potential Theory—An Analytic and Probabilistic Approach to Balayage (1986), Berlin, Heidelberg, New York, Tokyo: Springer, Berlin, Heidelberg, New York, Tokyo · Zbl 0706.31001  Choquet, G., Successive averages, Potential Theory—ICPT 94, 478-478 (1996), Berlin: Walter de Gruyter, Berlin
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