Cartan's lemma (potential theory)

Short description: Mathematical 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:

[math]\displaystyle{ u(z) = \frac{1}{2\pi}\int_\mathbf{C} \log|z-\zeta|\,d\mu(\zeta) }[/math]

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

[math]\displaystyle{ \sum_i r_i \lt 5H }[/math]

and

[math]\displaystyle{ u(z) \ge \frac{n}{2\pi}\log \frac{H}{e} }[/math]

for all z outside the union of these discs.