Sharp estimates of unimodular multipliers on frequency decomposition spaces

Full characterization of the embedding relations between α-modulation spaces

“…In Feichtinger-Narimani [8], the authors study the Fourier multiplier between M p1,q1 and M p2,q2 , where 1 ≤ p i , q i ≤ ∞ for i = 1, 2, one can also see Feichtinger-Gröbner [4] for a general result in the frame of Banach space (with same decomposition). Recently, in order to study the behavior of unimodular multiplier on α-modulation spaces, in [24], we establish a corresponding result between M s1,α p1,q1 and M s2,α p2,q2 , where 1 ≤ p i .q i ≤ ∞, s i ∈ R. In this section, we give a full characterization of Fourier multipliers between any two α-modulation spaces, which extends all the previous results. Especially, our theorem covers the case that α 1 = α 2 and s 1 = s 2 , which allows the different decompositions and different potentials.…”

Inclusion relations between modulation and Triebel-Lizorkin spaces

“…Modulation spaces have a close relationship with the topics of time-frequency analysis (see [6]), and that they has been regarded as a appropriate spaces for the study of PDE (see [16]). As function spaces associated with the uniform decomposition, modulation spaces have many beautiful properties, for instance, the properties of product and convolution on modulation spaces (see [4,9,10]), the boundedness of unimodular multipliers on modulation spaces (see [1,14,28]). One can also see [8,24,26,27] for the study of nonlinear evolution equations on modulation spaces.…”

Fractional Integral Operators on -Modulation Spaces in the Full Range

“…We present some properties of α-modulation spaces in Section 2. See also [4,7,29] for the study of certain operators on α-modulation spaces and [5,6,[13][14][15]23] for the structure of α-modulation spaces. Let R n be the Euclidean space of dimension n. The Riesz potential operator of order β is defined as…”