How to find riesz measure of a function?

Question

How to find the Riesz measure for some given subharmonic function? For example, let $u(z)=u(Re(z))$ be a continuous piecewise linear function, what is it's riesz measure? My teacher says that it has form $\mu(z)=\sum_{n=1}^\infty\delta(x-x_n)dy$, where $x_n$ are endpoints, but I can't figure out why is it so.

Answer 1

This can't be true. There are constants missing in the sum. First consider a (continuous) piecewise linear function on $\mathbb{R}$ and compute the second order distributional derivative. For example, take $f(x) = |x|$. Then $f'(x) = 2\theta(x)-1$ and $f''(x) = 2\delta(x)$ (in the distributional sense). Here $\theta$ is the Heaviside function.

Similarly, if $f$ is continuous and piecewise linear, then $f'' = \sum c_n\delta(x-x_n)$ for suitable contants $c_n$. (More precisely, $c_n = f'_+(x_n)-f'_-(x_n)$ is the change of the derivatives as we cross a "corner" in the graph of $f$.)

Hence, in two dimensions, if $u(x,y) = f(x)$, where $f$ is piecewise linear, then $$ \mu_u = \Delta u = \sum c_n\delta(x-x_n)\,dy. $$