Positive differentials, theta functions and $H^2$ Hardy kernels

Abstract:

Let $R$ be a planar regular region whose Schottky double $\hat {R}$ has genus $g$ and set $\hat {T}_0=\{z\in \mathbb {C}^g|\sqrt {-1} z\in \mathbb {R}^g \}$. For fixed $a\in R$ we determine the range of the function $F(e)=\theta (a-\bar {a}+e)/\theta (e) (e\in \hat {T}_0)$ where $\theta (z)$ is the Riemann theta function on $\hat {R}$. Also we introduce two weighted Hardy spaces to study the problem when the matrix $(\frac {\partial ^2\log F}{\partial z_i\partial z_j}(e))$ is positive definite. The proof relies on new theta identities using Fay’s trisecants formula.