Subharmonic distributions on the plane

Question

A subharmonic (Schwartz) distribution on $\mathbf R^n$ is a distribution $u$ satisfying $\Delta u\ge0$. This implies $\Delta u$ is a positive Radon measure $\mu$, thus for any ball $B$ the convolution of $\chi_B\mu$ with the fundamental solution of the Laplacian $E_n$ gives a locally integrable function $U(x)=\int_B E_n(x-y)d\mu(y)$. The difference $u-U$ is harmonic on $B$, and in this way we obtain the Riesz representation $u=h+U$. In dimension $n\ge3$ the fundamental solution $E_n$ is negative, so that $U$ is upper semicontinuous by a standard application of Fatou.

My question is about the regularity of $U$ in dimension $n=2$. We have $E_2=\frac{1}{2\pi}\log\lvert x\rvert$ which changes sign. The previous argument shows that $U$ can be written as the difference of two upper semicontinuous functions. Is it possible to do better than this? I am quite sure that this is very well known stuff but it is somehow difficult to find good references about this.

(My question is spurred by the standard definition of a subharmonic function, which is assumed to be upper semicontinuous regardless of the dimension. If $n\ge3$ there is perfect conincidence with subharmonic distributions, but on the plane the two definitions seem to be different. But I am probably missing something)

Answer 1

Some good references are: Landkof, Foundations of modern potential theory. Ransford, Potential theory in the complex plane, and Hayman, Kennedy, Subharmonic functions.

The fact which is true in any dimension is the following: Let $u$ be a distribution with $\Delta u\geq 0$, (so $\Delta u$ is a non-negative measure). Then $u$ is represented by a locally integrable function, and moreover this locally integrable function is equal to a unique upper semi-continuous function almost everywhere.

One way to see this is to follow your argument, but instead of the fundamental solution $\log|x-y|$, to restrict your function to a disk and to use the Green kernel $-G(x,y)$, where $G$ is the Green function of the disk.

Alternatively you can simply use $\log|x-y|-C$ with a large positive constant, so that this is negative in your disk.