Stable STFT phase retrieval and Poincar\'e inequalities

In recent work [P. Grohs and M. Rathmair. Stable Gabor Phase Retrieval and Spectral Clustering. Communications on Pure and Applied Mathematics (2018)] and [P. Grohs and M. Rathmair. Stable Gabor phase retrieval for multivariate functions. Journal of the European Mathematical Society (2021)] the instabilities of Gabor phase retrieval problem, i.e. reconstructing $f\in L^2(\mathbb{R})$ from its spectrogram $|\mathcal{V}_g f|$ where $$\mathcal{V}_g f(x,\xi) = \int_{\mathbb{R}} f(t)\overline{g(t-x)}e^{-2\pi i \xi t}\,\mbox{d}t,$$ have been classified in terms of the connectivity of the measurements. These findings were however crucially restricted to the case where the window $g(t)=e^{-\pi t^2}$ is Gaussian. In this work we establish a corresponding result for a number of other window functions including the one-sided exponential $g(t)=e^{-t}\mathbb{1}_{[0,\infty)}(t)$ and $g(t)=\exp(t-e^t)$. As a by-product we establish a modified version of Poincar\'e's inequality which can be applied to non-differentiable functions and may be of independent interest.