Cartan's lemma (potential theory)

Template:Short description Template:Distinguish 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:

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

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

$\sum _i{r}_i<5H$

and

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

for all z outside the union of these discs.

Notes

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