Boundedness of bi-parameter pseudo-differential operators on bi-parameter α-modulation spaces

Though the one-parameter α-modulation spaces have been extensively studied since it was first introduced by Gröbner, not much is known about the bi-parameter α-modulation spaces. This paper is to introduce the bi-parameter α-modulation spaces and establish the boundedness of bi-parameter pseudo-differential operators on the bi-parameter α-modulation spaces. By the special relationship between α-modulation spaces and modulation spaces, we also obtain the boundedness of bi-parameter pseudo-differential operators on the bi-parameter modulation spaces.