The author considers the analogue of the Gabor wave-front set in the setting of anisotropic Gelfand-Shilov ultradistributions of Beurling type \(\Sigma^{s\,'}_t(\mathbb{R}^d)\), \(s+t>1\). It is defined as follows. Let \(u\in\Sigma^{s\,'}_t(\mathbb{R}^d)\) and let \(\psi\) be a test function in \(\Sigma^s_t(\mathbb{R}^d)\). Then \((x_0,\xi_0)\in\mathbb{R}^d\times (\mathbb{R}^d\backslash\{0\})\) does not belong to the anisotropic \(t,s\)-Gelfand-Shilov wave front set \(WF^{t,s}(u)\) of \(u\) if there is an open set \(U\subseteq \mathbb{R}^d\times (\mathbb{R}^d\backslash\{0\})\) containing \((x_0,\xi_0)\) such that \[ \sup_{\lambda>0,\, (x,\xi)\in U} e^{r\lambda}|V_{\psi}u(\lambda^tx,\lambda^s\xi)|<\infty\text{ for all }r>0; \] here, \(V_{\psi}u\) stands for the short-time Fourier transform of \(u\) with window \(\psi\). The main result of the article describes the propagation of singularities with respect to this wave front set of an operator \(\mathcal{K}:\Sigma^{s\,'}_t(\mathbb{R}^d)\rightarrow \Sigma^{s\,'}_t(\mathbb{R}^d)\); i.e., it shows that for \(u\in\Sigma^{s\,'}_t(\mathbb{R}^d)\), \(WF^{t,s}(\mathcal{K}u)\) is included in a set built, in a precise manner, out of \(WF^{t,s}(K)\) and \(WF^{t,s}(u)\) where \(K\) is the kernel of \(\mathcal{K}\).
In the last section, the author applies this result to a class of evolution equations.