Trace Operators for Modulation, α-Modulation and Besov Spaces

The α-modulation spaces M p,q , introduced by Gröbner in [10] are a class of function spaces that contain Besov spaces B p,q (α = 1) and modulation spaces M s p,q (α = 0) as special cases. There are two kinds of basic coverings on Euclidean R which is very useful in the theory of function spaces and their applications, one is the uniform covering R = ⋃ k∈Zn Qk, where Qk denote the unit cube with center k; another is the dyadic covering R = ⋃ k∈N{ξ : 2 k−1 6 |ξ| < 2} ⋃ {ξ : |ξ| 6 1}. Roughly speaking, these decompositions together with the frequency-localized techniques yield the frequencyuniform decomposition operator k ∼ F χQkF and the dyadic decomposition operator ∆k ∼ F χ{ξ:|ξ|∼2k}F , respectively. The tempered distributions acted on these decomposition operators and equipped with the l(L(R)) norms, we then obtain Feichtinger’s modulation spaces and Besov spaces, respectively. During the past twenty years, the third covering was independently found by Feichtinger and Gröbner [3, 4, 10], and Päivärinta and Somersalo [12]. This covering, so called α-covering has a moderate scale which is rougher than that of the uniform covering and is thinner than that of the dyadic covering. Applying the α-covering