Let $\mu$ be a compactly supported borel probability measure on $\mathbb C$ then it's logaruthmic potential is,



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> $P\_{\mu}(z)= \int\_{\mathbb C} log|z-w|d\mu(w)$
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It's well known that $P\_\mu$ is subharmonic on $\mathbb C$ and harmonic on (Supp ${\mu}$)$^C$.


In general $P\_\mu$ need not be continuous on $\mathbb C$. So under what conditions (necessary and sufficient or at least sufficient) on measure $\mu$, $P\_{\mu}$ is continuous?


Thanks for any reference or help.



Measure for which it's logarithmic potential is continuous
A necessary and sufficient condition was given in 



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> M.G. Arsove, Continuous potentials and linear mass distributions, SIAM
>  Rev. 2, 1960, 177-184.
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Let $\mu\_{r}(z)$ be the total mass of the positive measure $\mu$ on the ball of radius $r$ about $z$.
The potential $P\_{\mu}$ is continuous at $z\_{0}\in\mathbb{C}$ if and only if
$$
\lim \_{r \rightarrow 0}\left\{\limsup\_{z \rightarrow z\_{0}} \int\_{0}^{r} \frac{\mu\_{t}(z)}{t} d t\right\}=0.
$$
Moreover, when $P\_{\mu}$ is continuous at $z\_{0}$, the limsup is a lim and
$$
\lim \_{z \rightarrow z\_{0}} \int\_{0}^{r} \frac{\mu\_{t}(z)}{t} d t=\int\_{0}^{r} \frac{\mu\_{t}\left(z\_{0}\right)}{t} d t.
$$
Thus, a sufficient condition for continuity at $z\_{0}$ is that there exists $\alpha>0$ such that, in a neighborhood of $z\_{0}$, one has $\mu\_{r}(z)\leq Mr^{\alpha}$, for some constant $M$. In particular, if $\mu$ has a $L^{p}$ density with $p>1$, the potential $P\_{\mu}$ is continuous.


