In this paper, we introduce and analyze in detail certain Sobolev-type spaces using harmonic analysis associated with the linear canonical transform \(\mathcal {L}^{M}f\) . We derive the Plancherel theorem for these spaces, which establishes a correspondence between the norms in these spaces and those of the transformed functions. In particular, we establish embedding theorems for these spaces, which are results that show how a Sobolev space can be "embedded" into another functional space. Additionally, we study some important properties of these spaces. These results enhance the understanding of Sobolev spaces in the context of the linear canonical transform and pave the way for new applications in harmonic analysis and partial differential equations.
