Sharp estimates of uniform harmonic majorants in the plane (Q580537)

S-Y. A. Chang and D. Marshall have proved that an analytic function f in the unit disk with Dirichlet integral \(\leq \pi\) satisfies \[ \int^{\pi}_{-\pi}\exp (| f(e^{i\theta})-f(0)|^ 2)d\theta \leq C<\infty. \] This is the critical endpoint improvement of a theorem from Beurling's thesis. The present author considerably generalizes the Chang-Marshall theorem by proving that if \(\Omega\) is a plane domain containing the origin with area \(\leq \pi\) then \(\exp | w|^ 2\) has a harmonic majorant in \(\Omega\). In fact, this result is just a special case of his general theorem, which states that certain functions \(\Phi\) (\(| w|)\) have harmonic majorants in \(\Omega\) when \(\Omega\) is not too large. For example, if \(0\in \Omega\) and \[ \int^{1}_{0}r\theta (r)dr+\lambda^ 2\int^{\infty}_{1}t^{2\lambda -1}\theta (t)dt=\pi, \] where \(\lambda >0\) and \(\theta\) (r) is the angular measure of \(\Omega \cap (| w| =r)\), then \(\exp (| w|^{2\lambda})\) has a harmonic majorant in \(\Omega\). The proof is technically quite interesting. It involves Moser's integrability theorem, symmetrization, Ahlfors' distortion theorem, and the reduction of harmonic measure estimates in the possibly multiply connected \(\Omega\) to some in simply connected domains.