Cartan's lemma (potential theory)
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.[1]
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:
Given H ∈ (0, 1), there exist discs of radius ri such that
and
for all z outside the union of these discs.
Notes
- B.Ya. Levin, Lectures on Entire Functions
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