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Harmonic measures of sides of a slit perpendicular to the domain boundary. (English) Zbl 1259.30008

Let the functions \(f(z,t)\), \(t\geq 0\), normalized near infinity by \(f(z, t)= z+ 2t/z+ O(1/z^2)\), map subdomains \(\mathbb{D}_t\) of the upper half-plane \(\mathbb{H}\) into \(\mathbb{R}\), and satisfy the chordal Löwner equation \[ \frac{df(z, t)}{dt} =\frac{2}{f(z,t) \lambda(t)}, \] where \(\lambda(t)\) is a continuous real-valued driving term.
In the case when the subdomains \(\mathbb{D}_t\) are a decreasing family consisting of \(\mathbb{H}\) minus an increasing vertical slit from the origin on \(\partial\mathbb{H}\), the authors show that the harmonic measures of the two sides of the slit are asymptotically equal. They also look at singular solutions of the chordal Löwner equation when the driving term has a higher Lipschitz order.

MSC:

30C35 General theory of conformal mappings
30C20 Conformal mappings of special domains
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References:

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