Cartan's lemma (potential theory)

Not to be confused with Cartan's theorem or Cartan's lemma.

In potential theory, a branch of mathematics, Cartan's lemma, named after Henri Cartan, is a bound on the measure and complexity of the set on which a logarithmic Newtonian potential is small.

Statement of the lemma

The following statement can be found in Levin's book.

Let μ be a finite positive Borel measure on the complex plane C with μ(C) = n. Let u(z) be the logarithmic potential of μ:

$u(z)={\frac {1}{2\pi }}\int _{\mathbf {C} }\log |z-\zeta |\,d\mu (\zeta )$

Given H ∈ (0, 1), there exist discs of radii r_i such that

$\sum _{i}r_{i}<5H$

and

$u(z)\geq {\frac {n}{2\pi }}\log {\frac {H}{e}}$

for all z outside the union of these discs.