Riesz measure of a subharmonic function

Builds on Subharmonic function Laplacian Radon measure Open set Harmonic function

Definition

Let $n \ge 2$ , let $\Omega \subseteq \R^n$ be an open set, and let $u$ be subharmonic on $\Omega$ and not identically $-\infty$ on any connected component of $\Omega$ . Then $u$ is locally integrable on $\Omega$ . Moreover, there is a unique positive Radon measure $\mu_u$ on $\Omega$ such that $\int_\Omega u \, \Delta\psi \, d\lambda = \int_\Omega \psi \, d\mu_u \qquad \text{for all } \psi \in C_c^\infty(\Omega),$ where $\lambda$ is Lebesgue measure and $\Delta$ is the Laplacian. The measure $\mu_u$ is the Riesz measure of $u$ . In other words, $\mu_u = \Delta u$ in the sense of distributions.

Remarks

Some authors normalise differently, for example by $\frac{1}{2\pi}\Delta u$ in the plane, so that $\log\abs{z}$ has Riesz measure $\delta_0$ . If $u$ is of class $C^2$ , then $\mu_u = (\Delta u)\, \lambda$ . The function $u$ is harmonic on an open subset $V$ of $\Omega$ if and only if $\mu_u(V) = 0$ .

Example: let $u(x) = \abs{x}^2$ on $\R^n$ . Integration by parts gives $\int \abs{x}^2 \Delta\psi \, d\lambda = \int \Delta(\abs{x}^2)\, \psi \, d\lambda = 2n \int \psi \, d\lambda$ , so $\mu_u = 2n\,\lambda$ . Example: for $u(z) = \log\abs{z}$ on $\C = \R^2$ , Green's formula gives $\mu_u = 2\pi\,\delta_0$ .

Non-example: for $v(x) = -\abs{x}^2$ , the distribution $\Delta v = -2n\,\lambda$ is a negative measure. This matches the fact that $v$ is not subharmonic, so the hypothesis of the definition fails and $v$ has no Riesz measure.

Tag	`FRKAPMVZ7M`
URI	`m2://FRKAPMVZ7M`
Kind	definition
Writing	W0 stub
MSC	31B05, 31A05
Defines	Riesz measure of a subharmonic function
Uses	Subharmonic function, Laplacian, Radon measure, Open set, Harmonic function

Riesz measure of a subharmonic function

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Remarks

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