Successive averages and harmonic functions (Q1361062)

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.