At the beginning of Eichler and Zagier’s book on Jacobi Forms, there is the following diagram, summarizing part of the special role played by Jacobi forms of index 1.

One of the things confusing me is that by “Jacobi form” I believe they mean holomorphic Jacobi form, which are those forms with Fourier coefficients $ c(n,r)=0$ unless $ \Delta := 4n-r^{2} \geq 0$ , where I have used that the index is 1. If $ \varphi_{k, 1}( \tau , z)$ is any Jacobi form (holomorphic, weak, or otherwise) of index 1, the Fourier coefficients depend only on $ \Delta$ . So in the case where $ \varphi_{k, 1}( \tau , z)$ is holomorphic, the second correspondence above is given by

$ $ \varphi_{k, 1}( \tau , z) \longrightarrow \sum_{\substack{\Delta = 0,3 \,mod\,4\\Delta \geq 0}} c(\Delta) q^{\Delta}$ $

Quite simply, I’m wondering if this sequences of correspondences, or a similar sequence of correspondences, continue to hold in the case where $ \varphi_{k, 1}( \tau , z)$ is only a weak Jacobi form. On the one hand I feel like it should because in index 1, all that would change is that $ \Delta$ may take finitely many negative values. So would you produce a weakly holomorphic modular form in $ M_{k-\frac{1}{2}}\big(\Gamma_{0}(4)\big)$ instead?

Assuming this works in some sense, maybe some one could point out sources where this is applied to the weak Jacobi form of weight -2 and index 1,

$ $ \Theta^{2} = \frac{\vartheta^{2}(\tau, z)}{\eta^{6}(\tau)}.$ $ In this case, $ \Delta = 0,3 \, \text{mod} \,4$ and $ \Delta \geq -1$ , so I’m wondering if this gives rise to a weakly holomorphic modular form of weight -3/2 associated to some congruence subgroup.