Remarks on the Unimodular Fourier Multipliers onα-Modulation Spaces

Abstract

We study the boundedness properties of the Fourier multiplier operatoreiμ(D)onα-modulation spacesMp,qs,α (0≤α<1)and Besov spacesBp,qs(Mp,qs,1). We improve the conditions for the boundedness of Fourier multipliers with compact supports and for the boundedness ofeiμ(D)onMp,qs,α. Ifμis a radial functionϕ(|ξ|)andϕsatisfies some size condition, we obtain the sufficient and necessary conditions for the boundedness ofeiϕ(|D|)betweenMp1,q1s1,αandMp2,q2s2,α.

Unimodular multipliers on $$\alpha$$-modulation spaces: a revisit with new method

“…Among numerous references on -modulation spaces, one can see [21,27] for elementary properties of -modulation spaces, see [20] for the full characterization of embedding between -modulation spaces, see [30,33,35] for the research of boundedness of fractional integrals and see [4][5][6][7] for the study of pseudodifferential operators and nonlinear appoximation. We also point out that the boundedness of unimodular multipliers on -modulation spaces has been studied in [31,32], in which if we take = 0 , the result is in accordance to that in [22]. Denote by [t] the integer part of t ∈ ℝ .…”

“…More precisely, we establish the boundedness on -modulation spaces of e i (D) by a new method derived from [24]. In contrast to the previous results as in [31,32], our results are valid under a weaker condition, see Remark 1.1 for more details. Since the -covering for ∈ (0, 1] is not uniformly bounded as the case of modulation space ( = 0 ), we refine the technique in [24], making it more efficient and adaptable to our situation.…”

“…To compare with the results of this paper, Theorem A is stated by an equivalent form of the corresponding boundedness results in [31,32], where the potential loss has been proved to be sharp.…”

Unimodular multipliers on $α$-modulation spaces: A revisit with new method under weaker conditions

“…Among numerous references on α-modulation spaces, one can see [21,27] for elementary properties of α-modulation spaces, see [20] for the full characterization of embedding between α-modulation spaces, see [30,33,35] for the research of boundedness of fractional integrals and see [4,5,6,7] for the study of pseudodifferential operators and nonlinear appoximation. We also point out that the boundedness of unimodular multipliers on α-modulation spaces has been studied in [32,31], in which if we take α = 0, the result is accordance to that in [22]. Denote by [t] the integer part of t ∈ R. The main boundedness result on α-modulation space of unimodular multipliers can be stated as follows.…”

“…Denote by [t] the integer part of t ∈ R. The main boundedness result on α-modulation space of unimodular multipliers can be stated as follows. Theorem A ( [32,31]…”