Logarithmic potentials (Q5947051)

In 1944, \textit{M. Brelot} has formulated the following result: a subharmonic function \(u\) in \(\mathbb{R}^2\) is a logarithmic potential up to an additive harmonic function iff \(\lim_{r\to\infty} [M(r,u)- \mu(B_0^r)\log r]\) exists and is finite [Ann. Sci. Éc. Norm. Supér. (3) 61, 303-332 (1944; Zbl 0061.22801)]. In the above limit \(M(r,u)\) is the mean value of \(u(x)\) on \(|x|=r\), and \(\mu(B_0^r)= \mu(x: |x|< r)\); \(\mu(x)\) is a measure associated to \(u(x)\), defined by \(d\mu(x)= (1/2\pi) \Delta u dx\). The necessary and sufficient condition for \(u\) to be a logarithmic potential contains both \(u\) and \(\mu\). In the present paper the author shows that it is possible to express \(u\) as a logarithmic potential, up to an additive harmonic function, iff \(\lim_{r\to\infty} [M(r^2,u)- 2M(r,u)]\) exists and is finite. This condition contains only \(u\).