Cartan's lemma (potential theory) - Alchetron, the free social encyclopedia

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 ) = 1 2 π ∫ C log ⁡ | z − ζ | d μ ( ζ )

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

∑ i r i < 5 H

and

u ( z ) ≥ n 2 π log ⁡ H e

for all z outside the union of these discs.

References

Cartan's lemma (potential theory) Wikipedia

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Danny McNulty

Olivia Colman