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

$P_{\mu}(z)= \int_{\mathbb C} log|z-w|d\mu(w)$

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.

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A necessary and sufficient condition was given in

M.G. Arsove, Continuous potentials and linear mass distributions, SIAM Rev. 2, 1960, 177-184.

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.

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  • $\begingroup$ Thank you so much! $\endgroup$ – Mayuresh L May 29 '19 at 21:27

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