Some compact and non-compact embedding theorems for the function spaces defined by fractional Fourier transform

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The fractional Fourier transform is a generalization of the classical Fourier transform through an angular parameter $\alpha $. This transform uses in quantum optics and quantum wave field reconstruction, also its application provides solving some differrential equations which arise in quantum mechanics. The aim of this work is to discuss compact and non-compact embeddings between the spaces $A_{\alpha ,p}^{w,\omega }\left(\mathbb{R}^{d}\right) $ which are the set of functions in ${L_{w}^{1}\left(\mathbb{R}^{d}\right) }$ whose fractional Fourier transform are in ${L_{\omega}^{p}\left(\mathbb{R}^{d}\right) }$. Moreover, some relevant counterexamples are indicated.