# Measure for which it's logarithmic potential is continuous

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.

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.

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