The embedding property of the scaling limit of modulation spaces

Modulation spaces Mp,qs were introduced by Feichtinger [3] in 1983. Later, considering the scaling property of the modulation spaces, Sugimoto and Wang [11] defined the scaling limit of the modulation spaces, which contains both the modulation spaces and Bényi and Oh's modulation spaces [1], and these spaces also have some applications in nonlinear Schrödinger equations. So, it is important to consider the relationship between these new spaces and some classical Banach spaces such as Lp spaces, Fourier Lp spaces and Besov-Triebel-Sobolev spaces. We study the embedding properties of the scaling limit of the modulation spaces, including the homogeneous case and nonhomogeneous case.