Boundary conditions for a finite Dirichlet integral in the upper half plane
Let k be an absolutely continuous function such that for some positive constant \(k_ 0\), \(k(t)\geq k_ 0\) for all real t. Suppose that \(k'\in L^ p(-\infty,\infty)\) for some \(p>1\). Suppose also that \(\lim_{t\to \infty}k(t)=\lim_{t\to -\infty}k(t)\) and that for some \(q>1\), \(k'(t)=O(| t|^{-q})\), \(| t| \to \infty\). It is shown there exists a function f, analytic in the upper half plane H with finite Dirichlet integral \[ \int_{H}\int | f'(z)|^ 2 dx dy, \] such that \(| f| =k\) almost everywhere on the real line. Examples are given which show that all the hypotheses are necessary. Also, differences between this result and what is known in the more familiar case of the unit disc are pointed out.
scientific article; zbMATH DE number 4154837