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

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.