Positive differentials, theta functions and 𝐻² Hardy kernels

Abstract

Let R R be a planar regular region whose Schottky double R ^ \hat {R} has genus g g and set T ^ 0 = { z ∈ C g | − 1 z ∈ R g } \hat {T}_0=\{z\in \mathbb {C}^g|\sqrt {-1}\,z\in \mathbb {R}^g \} . For fixed a ∈ R a\in R we determine the range of the function F ( e ) = θ ( a − a ¯ + e ) / θ ( e ) ( e ∈ T ^ 0 ) F(e)=\theta (a-\bar {a}+e)/\theta (e) (e\in \hat {T}_0) where θ ( z ) \theta (z) is the Riemann theta function on R ^ \hat {R} . Also we introduce two weighted Hardy spaces to study the problem when the matrix ( ∂ 2 log ⁡ F ∂ z i ∂ z j ( e ) ) (\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.