A monotonicity estimate of the biharmonic Green function
Let \(\mathbb{D}\) be the unit disc in the complex plane and \(\mathbb{T} = \partial \mathbb{D}\) the unit circle. The biharmonic Green function is the function \[ \Gamma (z,\zeta) = | z - \zeta| ^2 \log\left| \frac{z - \zeta}{1 - \overline{\zeta}z}\right| ^2 + (1 - | z| ^2)(1 - | \zeta| ^2), \quad (z,\zeta) \in \mathbb{D} \times \mathbb{T}. \] The related function \(H\) defined by \[ H(z,e^{i\theta}) = \frac{(1 - | z| ^2)^2}{| e^{i\theta} - z| ^2}, \quad (z, e^{i\theta}) \in \mathbb D \times \mathbb T, \] is known as the harmonic compensator. Recently, these two functions have attracted special attention because of their relevance in Bergman space theory [cf. \textit{H. Hedenmalm, B. Korenblum} and \textit{K. Zhu}, Theory of Bergman spaces. (Springer Verlag, New York) (2000; Zbl 0955.32003)]. The main result of the paper states that the harmonic compensator \(H\) admits the estimate \[ \left( \frac{r}{r + | z| }\right)^2 H(z/r, e^{i\theta}) \leq \frac{1}{(1 + | z| )^2}H(z, e^{i\theta}),\quad | z| < r < 1, \;e^{i\theta} \in \mathbb T. \] In particular, \[ H(z(r,e^{i\theta}) = - \frac{z}{r^2} \leq \frac{4}{(1 + | z| )^2}H(z, e^{i\theta}), \quad | z| < r < 1, \;e^{i\theta} \in \mathbb T, \] that is, a sharp form of the monotonicity property for the harmonic compensator. This estimates for \(H\) leads to a corresponding estimate for \(\Gamma\). Estimates of this type yield norm estimates for dilation operators arising in approximation problems.
scientific article; zbMATH DE number 2119261