Successive averages and harmonic functions.

*(English)*Zbl 0879.31002For 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.

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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\textit{W. Hansen} and \textit{I. Netuka}, J. Anal. Math. 71, 159--171 (1997; Zbl 0879.31002)

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##### References:

[1] | H. Bauer,Probability Theory, de Gruyter, Berlin, 1996. |

[2] | Y. Benyamini and Y. Weit,Functions satisfying the mean value property in the limit, J. Analyse Math.52 (1989), 167–198. · Zbl 0675.31002 |

[3] | J. Bliedtner and W. Hansen,Potential Theory–An Analytic and Probabilistic Approach to Balayage, Springer, Berlin, Heidelberg, New York, Tokyo, 1986. · Zbl 0706.31001 |

[4] | G. Choquet,Successive averages, inPotential Theory–ICPT 94, Proceedings of the International Conference on Potential Theory held in Kouty, Czech Republic, August 13–20, 1994, Walter de Gruyter, Berlin, 1996, p. 478. |

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