Sentence examples for modulation spaces are from inspiring English sources

Notice that the dilation properties of modulation spaces are employed in the step (ii).

Our results show that modulation spaces are good substitutions for Lebesgue spaces.

Modulation spaces are Banach spaces whose definitions are independent of the choice of the window g (see [2, 3]).

As an application, mapping properties of unimodular Fourier multiplier ei|D|α between Lp-Sobolev spaces and modulation spaces are discussed.

In the latter case modulation spaces are Banach spaces, whereas they are merely quasi-Banach spaces if a Lebesgue parameter is smaller than one.

In this paper, we are mainly concerned about the boundedness of the hypersingular integral operators along curves on weighted modulation spaces M s p, q ( R n ) for 0 < p < 1, 0 < q ≤ ∞ and s ∈ R. From our results, we will see that modulation spaces are good substitutions for Lebesgue spaces.

The inclusion relations between the Lp-Sobolev spaces and the modulation spaces is determined explicitly.

By [36, Proposition 2.8] it suffices to prove the result in the Weyl case (A=1/2), and then (3.4) in terms of symplectic modulation spaces is begin{aligned} Vert a #bVert _{mathcal {M}^{p_0,q_0}_{(omega _0)}} lesssim Vert aVert _{mathcal {M}^{p_1,q_1}_{(omega _1)}}Vert bVert _{mathcal {M}^{p_2,q_2}_{(omega _2)}}, quad a, b in mathscr {S} ({mathbf {R}^{2d}}).

It is worth to point out that the modulation space is a better substitution to study the strongly singular integrals because there is no restriction on the index p. Here we will consider the strongly singular integrals along homogeneous curves (T_{n, beta, gamma}) on the α-modulation spaces.

The Fourier multipliers for the modulation spaces have been developed in many papers [14, 17, 24 26] where the so-called unimodular Fourier multipliers were studied and applied into PDEs.

Recently Hardy type modulation spaces have been proposed in [13].