Boas-type theorems for the free metaplectic transform

In this study, we focus on the free metaplectic transform and its implications on the properties of functions. The free metaplectic transform is a generalization of the Fourier transform that allows us to analyze the behavior of functions in the metaplectic domain. F. Moricz previously investigated the properties of functions \(f\in L^1({\mathbb {R}})\) whose Fourier transforms \(\widehat{f}\) belong to \(L^1({\mathbb {R}})\). He established certain sufficient conditions based on \(\widehat{f}\) to determine whether f belongs to the Lipschitz classes \({\text {Lip}}(\gamma )\) and \({\text {lip}}(\gamma )\), where \(0 < \gamma \le 1\), or the Zygmund classes \({\text {Zyg}}(\gamma )\) and \({\text {zyg}}(\gamma )\), where \(0 < \gamma \le 2\). In this study, our aim is to extend these findings and explore the properties of functions in relation to the free metaplectic transform.