1.1.2 Scope, attribution and conventions61B2D7362CC0

Classical potential theory is developed through spherical means, harmonic measures, Green functions, capacities, sweeping, and Martin boundary representations.

We work in real dimension $d\ge 2$, with ordinary Laplacian $\Delta $, unnormalized sphere area $s_{d-1}$, and unit ball volume $v_ d=s_{d-1}/d$. Put $\Phi (x)=|x|^{2-d}$ for $d\ge 3$ and $\Phi (x)=\log (1/|x|)$ for $d=2$. Then $-\Delta \Phi =c_ d\delta _0$, where $c_2=2\pi $ and $c_ d=(d-2)s_{d-1}$ for $d\ge 3$. Unless explicitly normalized otherwise, Green functions use this convention. Thus $G_{\rm Helms}=c_ dG_{\rm Foundations}$ and $C_{\rm Helms}=C_{\rm Foundations}/c_ d$. Superharmonic functions are lower semicontinuous, super-mean-valued and nontrivial on each component; positive means nonnegative unless strict positivity is stated. A potential is a nonnegative superharmonic function whose greatest harmonic minorant is zero. Measures are positive Radon measures unless signedness is specified. On noncompact spaces weak measure convergence in the source means vague convergence against $C_ c$.

The prerequisites are multivariable calculus, Lebesgue integration, measure decomposition, and compact-metric measure representation. The harmonic and subharmonic results used below are established in the preceding Pome sections.