# Sharp embedding between Besov-Triebel-Sobolev spaces and modulation spaces

Yufeng Lu School of Mathematics Sciences, Peking University, Beijing 100871,
PR China<EMAIL_ADDRESS>
###### Abstract

The embedding relations between Besov-Triebel-Sobolev spaces and modulation
spaces are determined explicitly. We extend the results in [1],[2],[3] to the
most general cases. And we give the sharp embedding between Fourier $L^{p}$
spaces and modulation spaces.

##  1 Introduction

The modulation spaces $M_{p,q}^{s}$ are one of the function spaces introduced
by Feichtinger [4] in the 1980s using the short-time Fourier transform to
measure the decay and the regularity of the function in a different way from
the usual $L^{p}$ Sobolev spaces or Besov-Triebel spaces. The precise
definitions will be given in Section 2. Roughly speaking, the Besov-Triebel
spaces mostly use the dyadic decompositions of the frequency space while the
modulation spaces using the uniform decompositions of the frequency space.
Therefore, these spaces may have many different properties in some cases.
Also, they have some properties similarly. So, the relationships between the
modulation spaces and Besov-Triebel spaces are important.

Many basic properties of the modulation spaces had been studies in [5, 6, 7]
such as the Banach spaces properties and the dual spaces of modulation spaces.
And the modulation spaces also have some applications in pseudo-differential
operators in [8, 9, 10, 11, 12],[13]. Modulation spaces also have many
applications in the analysis of partial differential equations. For example,
the Schrodinger and wave semigroups which are not bouned on neither $L^{p}$ or
$B^{s}_{p,q}$ for $p\neq 2$, are bounded on $M_{p,q}^{s}$(see [14]). So, the
modulation space is a good space for the initial data of the Cauchy problem
for nonlinear disperse equations(see [15, 2, 16, 17]) .

As for embedding properties of modulation spaces, Toft in [18] discussed some
of the sufficient conditions of the embedding between modulation spaces and
Besov spaces, also, Toft in [13] discussed the embedding between the $L^{p}$
spaces and the Fourier transform of $L^{p}$ space with the modulation spaces.
Then, Okoudjou in [19] gave some sufficient conditions for some cases of
embedding between Sobolev-Besov spaces and modulation spaces by the means of
the short-time Fourier transform. Later, Sugmito and Tomita in [1] gave the
sharp embedding between modulation spaces and Besov spaces when $1\leqslant
p,q\leqslant\infty$ by means of the scaling of modulation spaces, then Wang
and Huang in [2] extended the condition to $0<p,q\leqslant\infty$ by means of
the uniform decomposition of frequency space. But they all discuss the case
when the indexes of physics space and frequency space of modulation spaces and
Besov spaces are the same. Later, Kobayashi and Sugimoto in [3] studied the
sharp embedding between $L^{p}$-Sobolev space and modulation spaces using the
method similar to [20]. As for Triebel spaces, Guo, Wu and Zhao in [21] gave
the sharp condition of the embedding $F_{p,r}\hookrightarrow M_{p,q}^{s}$ when
$0<p\leqslant 1$.

In this article, we study the sharp conditions of the embedding between Besov-
Triebel-Sobolev spaces and modulation spaces. Let us take Besov spaces for
example, in Theorem 4.1, we get the sufficient and necessary conditions of the
embedding $B_{p_{0},q_{0}}^{s}\hookrightarrow M_{p,q}$, which is the most
general case of $0<p,q,p_{0},q_{0}\leqslant\infty$, also in Theorem 4.2, we
get the sufficient and necessary conditions of the embedding of the opposite
side. In the case of $0<p,q<1$, the dual of Besov spaces and modulation spaces
are different, so we cannot just get the opposite side by the method of
duality. Instead, we use the method of the uniform decompositions of frequency
space to get the result as desired.

We explain the organization of this paper. We give some basic notations and
properties of the spaces we need in the next preliminary section. In section
3, we give some lemmas we need in the proof, especially for the case of
$0<p,q\leqslant\infty$ we discuss in section 4. The embedding between Besov
spaces and modulation spaces is considered in section 4. We mainly use the
method of uniform decomposition in [2]. Then we study the embedding between
$L^{p}$ Sobolev spaces and modulation spaces in section 5, as we can see,
there are four conditions totally. In section 6, we consider the embedding
Triebel spaces and modulation spaces which is sometimes equivalent to the
embedding of Sobolev spaces by Littlewood-Paley theory. In the last section,
we give the sharp embedding relationship between the Fourier $L^{r}$ spaces
and modulation spaces.

##  2 Perliminaries

###  2.1 Basic natation

The following notation will be used throughout this article. For
$0<p,q\leqslant\infty$, we denote

$\displaystyle\sigma(p,q)$
$\displaystyle:=d\left(0\wedge(\frac{1}{q}-\frac{1}{p})\wedge(\frac{1}{q}+\frac{1}{p}-1)\right);$
$\displaystyle\tau(p,q)$
$\displaystyle:=d\left(0\vee(\frac{1}{q}-\frac{1}{p})\vee(\frac{1}{q}+\frac{1}{p}-1)\right),$

where $a\wedge b=\min\left\\{a,b\right\\},a\vee b=\max\left\\{a,b\right\\}$.

We write $\mathscr{S}({\mathbb{R}}^{d})$ to denote the Schwartz space of all
complex-valued rapidly decreasing infinity differentiable functions on
${\mathbb{R}}^{d}$, and $\mathscr{S}^{\prime}({\mathbb{R}}^{d})$ to denote the
dual space of $\mathscr{S}({\mathbb{R}}^{d})$, all called the space of all
tempered distributions. For simplification, we omit ${\mathbb{R}}^{d}$ without
causing ambiguity. The Fourier transform is defined by
${\mathscr{F}}f(\xi)=\hat{f}(\xi)=\int_{{\mathbb{R}}^{d}}f(x)e^{-ix\xi}d\xi$,
and the inverse Fourier transform by
${\mathscr{F}^{-1}}f(x)=(2\pi)^{-d}\int_{{\mathbb{R}}^{d}}f(\xi)e^{ix\xi}d\xi$.
For $1\leqslant p<\infty$, we define the $L^{p}$ norm:

$\displaystyle\left\|f\right\|_{p}=\left(\int_{{\mathbb{R}}^{d}}\left|f(x)\right|^{p}\right)^{1/p}$

and
$\left\|f\right\|_{\infty}=ess\sup_{x\in{\mathbb{R}}_{d}}\left|f(x)\right|$.
We also define the $L^{p}$ Sobolev norm :

$\displaystyle\left\|f\right\|_{W^{s,p}}=\left\|(I-\triangle)^{s/2}f\right\|_{p}.$

Recall that the Sobolev spaces is defined by
$W^{s,p}=\left\\{f\in\mathscr{S}^{\prime}:\left\|f\right\|_{W^{s,p}}<\infty\right\\}$.
And the Fourier $L^{p}$ is defined by
${\mathscr{F}}L^{p}=\left\\{f\in\mathscr{S}^{\prime}:\left\|\hat{f}\right\|_{p}<\infty\right\\}$.

We use the notation $I\lesssim J$ if there is an independently constant C such
that $I\leqslant CJ$. Also we denote $I\approx J$ if $I\lesssim J$ and
$J\lesssim I$. For $1\leqslant p\leqslant\infty$, we denote
$1/p+1/p^{\prime}=1$, for $0<p<1$, denote $p^{\prime}=\infty$.

###  2.2 Modulation spaces

Let $0<p,q\leqslant\infty$, the short time Fourier tranform of $f$ respect to
a window function $g\in\mathscr{S}$ is defined as (see [4]):

$\displaystyle
V_{g}f(x,\xi)=\int_{{\mathbb{R}}^{d}}f(t)\overline{g(t-x)}e^{-it\xi}dt.$

And then for $1\leqslant p,q\leqslant\infty$, we denote

$\displaystyle\left\|f\right\|_{M_{p,q}^{s}}=\left\|V_{g}f(x,\xi)\left\langle\xi\right\rangle^{s}\right\|_{L_{\xi}^{q}L_{x}^{p}},$

where $\left\langle\xi\right\rangle=1+\left|\xi\right|$. Modulation space
$M_{p,q}^{s}$ are defined as the space of all tempered distribution
$f\in\mathscr{S}^{\prime}$ for which $\left\|f\right\|_{M_{p,q}^{s}}$ is
finite.

Also, we know another equivalent definition of modulation by uniform
decomposition of frequency space (see [6]).

Let $\sigma$ be a smooth cut-off function adapted to the unit cube
$[-1/2,1/2]^{d}$ and $\sigma=0$ outside the cube $[-3/4,3/4]^{d}$, we write
$\sigma_{k}=\sigma(\cdot-k)$, and assume that

$\displaystyle\sum_{k\in{\mathbb{Z}}^{d}}\sigma_{k}(\xi)\equiv 1,\
\forall\xi\in{\mathbb{R}}^{d}.$

Denote $\sigma_{k}(\xi)=\sigma(\xi-k)$, and
$\Box_{k}={\mathscr{F}^{-1}}\sigma_{k}{\mathscr{F}}$, then we have the
following equivalent norm of modulation space:

$\displaystyle\left\|f\right\|_{M_{p,q}^{s}}=\left\|\left\langle
k\right\rangle^{s}\left\|\Box_{k}f\right\|_{L_{x}^{p}}\right\|_{\ell_{k}^{q}({\mathbb{Z}}^{d})}.$

We simply write $M_{p,q}$ instead of $M_{p,q}^{0}$. One can prove the
$M_{p,q}^{s}$ norm is independent of the choice of cut-off function $\sigma$.
Also $M_{p,q}^{s}$ is a quasi Banach space and when $1\leqslant
p,q\leqslant\infty$, $M_{p,q}^{s}$ is a Banach space. When $p,q<\infty$, then
$\mathscr{S}$ is dense in $M_{p,q}^{s}$. Also, $M_{p,q}^{s}$ has some basic
properties, we list them in the following lemma(see [2, 22]).

######  Lemma 2.1.

For
$s,s_{0},s_{1},\in{\mathbb{R}},0<p,p_{0},p_{1},q,q_{0},q_{1}\leqslant\infty$,

(1)

if $s_{0}\leqslant s_{1},p_{1}\leqslant p_{0},q_{1}\leqslant q_{0}$, we have
$M_{p_{1},q_{1}}^{s_{1}}\hookrightarrow M_{p_{0},q_{0}}^{s_{0}}$;

(2)

when $p,q<\infty$, the dual space of $M_{p,q}^{s}$ is
$M_{p^{\prime},q^{\prime}}^{-s}$;

(3)

the interpolation spaces theorem is true for $M_{p,q}^{s}$, i.e. for
$0<\theta<1$ when

$s=(1-\theta)s_{0}+\theta
s_{1},\quad\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}},\quad\frac{1}{q}=\frac{1-\theta}{q_{0}}+\frac{\theta}{q_{1}},$

we have
$\left(M_{p_{0},q_{0}}^{s_{0}},M_{p_{1},q_{1}}^{s_{1}}\right)_{\theta}=M_{p,q}^{s}$;

(4)

when $q_{1}<q,s+d/q>s_{1}+d/q_{1}$, then we have $M_{p,q}^{s}\hookrightarrow
M_{p,q_{1}}^{s_{1}}$.

###  2.3 Besov-Triebel spaces

Let $0<p,q\leqslant\infty,s\in{\mathbb{R}},$ choose
$\psi:{\mathbb{R}}^{d}:\rightarrow[0,1]$ be a smooth radial bump function
adapted to the ball $B(0,2)$: $\psi(\xi)=1$ as $\left|\xi\right|\leqslant 1$
and $\psi(\xi)=0$ as $\left|\xi\right|\geqslant 2$. We denote
$\varphi(\xi)=\psi(\xi)-\psi(2\xi)$, and $\varphi_{j}(\xi)=\varphi(2^{-j}\xi)$
for $1\leqslant j,j\in{\mathbb{Z}}$, $\varphi_{0}(\xi)=1-\sum_{j\geqslant
1}\varphi_{j}(\xi)$. We say that
$\triangle_{j}={\mathscr{F}^{-1}}\varphi_{j}{\mathscr{F}}$ are the dyadic
decomposition operators. The Besov spaces $B^{s}_{p,q}$ and the Triebel spaces
$F^{s}_{p,q}$ are definied in the following way :

$\displaystyle
B^{s}_{p,q}=\left\\{f\in\mathscr{S}^{\prime}({\mathbb{R}}^{d}):\left\|f\right\|_{B^{s}_{p,q}}=\left\|2^{js}\left\|\triangle_{j}f\right\|_{L_{x}^{p}}\right\|_{\ell_{j\geqslant
0}^{q}}<\infty\right\\},$ $\displaystyle
F^{s}_{p,q}=\left\\{f\in\mathscr{S}^{\prime}({\mathbb{R}}^{d}):\left\|f\right\|_{F^{s}_{p,q}}=\left\|\left\|2^{js}\triangle_{j}f\right\|_{\ell_{j\geqslant
0}^{q}}\right\|_{L_{x}^{p}}<\infty\right\\}.$

One can prove that the Besov-Triebel norms defined by different dyadic
decompositions are all equivalent(see [23]), so without loss of generality, we
can assume that when $1\leqslant j$, $\varphi_{j}(\xi)=1$ on
$D_{j}:=\left\\{\xi\in{\mathbb{R}}^{d}:\frac{3}{4}2^{j}\leqslant\left|\xi\right|\leqslant\frac{5}{4}2^{j}\right\\}$
for convenience. Also, Besov-Triebel spaces have some basic properties known
already(see [23, 7, 22]).

######  Lemma 2.2.

Let
$s,s_{1},s_{2}\in{\mathbb{R}},0<p,p_{1},p_{2},q,q_{1},q_{2}\leqslant\infty$,

(1)

if $q_{1}\leqslant q_{2}$, we have $B_{p,q_{1}}^{s}\hookrightarrow
B_{p,q_{2}}^{s}$, $F_{p,q_{1}}^{s}\hookrightarrow F_{p,q_{2}}^{s}$;

(2)

$\forall\varepsilon>0$, we have $B_{p,q_{1}}^{s+\varepsilon}\hookrightarrow
B_{p,q_{2}}^{s}$, $F_{p,q_{1}}^{s+\varepsilon}\hookrightarrow
F_{p,q_{2}}^{s}$;

(3)

$B_{p,p\wedge q}^{s}\hookrightarrow F^{s}_{p,q}\hookrightarrow B_{p,p\vee
q}^{s}$;

(4)

if $p_{1}\leqslant p_{2},s_{1}-d/p_{1}=s_{2}-d/p_{2}$, we have
$B_{p_{1},q}^{s_{1}}\hookrightarrow B_{p_{2},q}^{s_{2}}$;

(5)

if $p_{1}<p_{2},s_{1}-d/p_{1}=s_{2}-d/p_{2}$, we have
$F_{p_{1},q_{1}}^{s_{1}}\hookrightarrow F_{p_{2},q_{2}}^{s_{2}}$;

(6)

when $1\leqslant p,q<\infty$, the dual space of $B^{s}_{p,q}$ is
$B_{p^{\prime},q^{\prime}}^{-s}$, the dual space of $F^{s}_{p,q}$ is
$F_{p^{\prime},q^{\prime}}^{-s}$;

(7)

the interpolation spaces theorem is true for $B^{s}_{p,q}$ and $F^{s}_{p,q}$,
i.e. for $0<\theta<1$ when

$s=(1-\theta)s_{0}+\theta
s_{1},\quad\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}},\quad\frac{1}{q}=\frac{1-\theta}{q_{0}}+\frac{\theta}{q_{1}},$

we have
$\left(B_{p_{0},q_{0}}^{s_{0}},B_{p_{1},q_{1}}^{s_{1}}\right)_{\theta}=B_{p,q}^{s}$,
$\left(F_{p_{0},q_{0}}^{s_{0}},F_{p_{1},q_{1}}^{s_{1}}\right)_{\theta}=F_{p,q}^{s}$
;

(8)

when $1<p<\infty$, $F_{p,2}^{s}=W^{s,p}$, $F_{1,2}^{s}\hookrightarrow
W^{s,1},W^{s,\infty}\hookrightarrow F_{\infty,2}^{s}$.

##  3 Some lemmas

In this section, we give some useful lemmas and propositions proved already,
which will be used in our proof.

The following Bernstein’s inequality is very useful in time-frequency analysis
(see [22]) :

######  Lemma 3.1.

[Bernstein’s inequality] Let $0<p\leqslant
q\leqslant\infty,b>0,\xi_{0}\in{\mathbb{R}}^{d}$. Denote
$L_{B(\xi,b)}^{p}=\left\\{f\in L^{p}:\mbox{ supp }\hat{f}\subseteq
B(\xi,R)\right\\}$. Then there exists $C(d,p,q)>0$, such that

$\displaystyle\left\|f\right\|_{q}\leqslant
C(d,p,q)R^{d(1/p-1/q)}\left\|f\right\|_{p}$

holds for all $f\in L_{B(\xi,b)}^{p}$ and $C(d,p,q)$ is independent of $b>0$
and $\xi_{0}\in{\mathbb{R}}^{d}$.

Also, by using the Bernstein’s inequality, we can get the following Young type
inequality for $0<p<1$:

######  Lemma 3.2 ([7]).

Let $0<p<1,R_{1},R_{2}>0,\xi_{1},\xi_{2}\in{\mathbb{R}}^{d}$, then there
exists $C(d,p)>0$, such that

$\displaystyle\left\|\left|f\right|*\left|g\right|\right\|_{p}\leqslant
C(d,p)(R_{1}+R_{2})^{d(1/p-1)}\left\|f\right\|\left\|g\right\|$

holds for all $f\in L_{B(\xi_{1},R_{1})}^{p},g\in L_{B(\xi_{2},R_{2})}^{p}$.

By the definition of $\Box_{k}$ and $\triangle_{j}$ operators, we know that
they are all convolution type operators, so by the Lemma above and the usual
Young’s inequality, we have:

######  Corollary 3.3.

For $0<p\leqslant\infty,1\leqslant j,\ell,k\in{\mathbb{Z}}^{d}$, there exists
$C>0$ independent of $k,j$, such that

$\displaystyle\left\|\triangle_{j}\Box_{k}f\right\|$ $\displaystyle\leqslant
C\left\|\Box_{k}f\right\|_{p},$
$\displaystyle\left\|\triangle_{j}\triangle_{\ell}f\right\|_{p}$
$\displaystyle\leqslant C\left\|\triangle_{j}f\right\|_{p}.$

Finally we recall a result known already, which is the special case of Theorem
4.1 and Theorem 4.2.

######  Proposition 3.4 ([2]).

For $0<p,q\leqslant\infty,s\in{\mathbb{R}}$, we have

(1)

$B^{s}_{p,q}\hookrightarrow M_{p,q}$ if and only if $s\geqslant\tau(p,q)$;

(2)

$M_{p,q}\hookrightarrow B^{s}_{p,q}$ if and only if $s\leqslant\sigma(p,q)$.

######  Proposition 3.5.

([3]) Let $1\leqslant p,q\leqslant\infty,s\in{\mathbb{R}}$, then
$W^{s,p}\hookrightarrow M_{p,q}$ if and only if one of the following cases is
satisfied:

(1)

$1<p\leqslant q,s\geqslant\tau(p,q)$;

(2)

$p>q,s>\tau(p,q)$;

(3)

$p=1,q=\infty,s\geqslant\tau(1,\infty)$;

(4)

$p=1,q\neq\infty,s>\tau(1,q)$.

##  4 Embedding between Besov spaces and modulation spaces

In this section, we consider the embedding with the form
$B^{s}_{p_{0},q_{0}}\hookrightarrow M_{p,q}$. If we got the sufficient and
necessary condition of this embedding, then the embedding
$B_{p_{1},q_{1}}^{s_{1}}\hookrightarrow M_{p,q}^{s}$ can be done in the same
way.

Our main result is:

######  Theorem 4.1.

For $0<p,q,p_{0},q_{0}\leqslant\infty,s\in{\mathbb{R}}$, the embedding
$B^{s}_{p_{0},q_{0}}\hookrightarrow M_{p,q}$ is true if and only if one of the
following two conditions is satisfied:

(1)

$p_{0}\leqslant p,q_{0}\leqslant q,s\geqslant\tau(p_{0},q)$;

(2)

$p_{0}\leqslant p,q_{0}>q,s>\tau(p_{0},q)$.

As for the other side of the embedding between Besov spaces and modulation
spaces, we also have:

######  Theorem 4.2.

For $0<p,q,p_{1},q_{1}\leqslant\infty,s\in{\mathbb{R}}$, the embedding
$M_{p,q}\hookrightarrow B^{s}_{p_{1},q_{1}}$ is true if and only if one of the
following two conditions is satisfied:

(1)

$p_{1}\geqslant p,q_{1}\geqslant q,s\leqslant\sigma(p_{1},q)$;

(2)

$p_{1}\geqslant p,q_{1}<q,s<\sigma(p_{1},q)$.

######  Remark 4.3.

If we regard the Besov space as a special $\alpha$-modulation space when
$\alpha=1$ as stated in [24], the result above is coincident with the Theorem
1.2 in [24], where the authors got the full characterization of embedding
between $\alpha$-modulation spaces.

For a special case when $p_{0}=q_{0}=2$, which means that
$B^{s}_{p_{0},q_{0}}=H^{s}$, the classical Sobolev spaces, then we have

######  Corollary 4.4.

Let $0<p,q\leqslant\infty,s\in{\mathbb{R}}$, we have

(1)

the embedding $H^{s}\hookrightarrow M_{p,q}$ is true if and only if one of the
following two cases is true:

  1. 1.

$2\leqslant p,2\leqslant q,s\geqslant 0$;

  2. 2.

$2\leqslant p,2>q,s>d(1/q-1/2)$.

(2)

The embedding $M_{p,q}\hookrightarrow H^{s}$ is true if and only if the
following two cases is true:

  1. 1.

$p\leqslant 2,q\leqslant 2,s\leqslant 0$;

  2. 2.

$p\leqslant 2,q>2,s<d(1/q-1/2)$.

###  4.1 Proof of Theorem 4.1

In this subsection, we prove the Theorem 4.1. For sufficiency, by using the
lemmas above, we can easily prove it. As for necessity, it may be much more
difficult. Here, we mostly use the way in [[2]] to prove it.

####  4.1.1 Sufficiency

Use Lemma 2.2, we know that for any case in 4.1, we have
$B^{s}_{p_{0},q_{0}}\hookrightarrow B_{p_{0},q}^{\tau(p_{0},q)}$; then by
Proposition 3.4, we know that $B_{p_{0},q}^{\tau(p_{0},q)}\hookrightarrow
M_{p_{0},q}$; also by Lemma 2.1, we have $M_{p_{0},q}\hookrightarrow M_{p,q}$.

####  4.1.2 Necessity

If we have $B^{s}_{p_{0},q_{0}}\hookrightarrow M_{p,q}$, then we have

$\displaystyle\left\|f\right\|_{M_{p,q}}\lesssim\left\|f\right\|_{B^{s}_{p_{0},q_{0}}},\
\forall f\in B^{s}_{p_{0},q_{0}}.$ (4.1)

Then we can take some different kinds of $f$ into (4.1) to get some
restrictions of indexes.

(1)

Choose $\eta\in\mathscr{S}$, such that supp $\eta\subset[-1/8,1/8]^{d}$,
denote $f={\mathscr{F}^{-1}}\eta$, for $\forall\lambda\leqslant 1$, take
$f_{\lambda}(x)=f(\lambda x)$ into (4.1). By the choose of $\eta$, we know
that supp ${\mathscr{F}}f_{\lambda}\subset\lambda[-1/8,1/8]^{d}$, which means
that

$\displaystyle\Box_{k}f_{\lambda}=\begin{cases*}f_{\lambda},&$k=0$;\\\
0,&$k\neq
0$;\end{cases*}\triangle_{j}f_{\lambda}=\begin{cases*}f_{\lambda},&$k=0$;\\\
0,&$k\neq 0$.\end{cases*}$

So, take $f_{\lambda}$ into (4.1), we have
$\left\|f_{\lambda}\right\|_{p}\lesssim\left\|f_{\lambda}\right\|_{p_{0}}$,
which means that $\lambda^{-d/p}\lesssim\lambda^{-d/p_{0}}$. Take
$\lambda\rightarrow 0$, we have $p_{0}\leqslant p$.

(2)

Denote $A_{j}:=\left\\{k\in{\mathbb{Z}}^{d}:k+[-3/4,3/4]^{d}\subseteq
D_{j}\right\\}$, for $\ell\geqslant 10$, choose $k_{\ell}\in A_{\ell}$, take
$f_{\ell}={\mathscr{F}^{-1}}\eta(\cdot-k_{\ell})$, $\eta$ is just the function
in (1). Then we know that supp ${\mathscr{F}}f_{\ell}\subseteq
k_{\ell}+[-1/8,1/8]^{d}\subseteq D_{\ell}$, so we have

$\displaystyle\Box_{k}f_{\ell}=\begin{cases*}f_{\ell},&$k=k_{\ell}$;\\\
0,&$k\neq
k_{\ell};$\end{cases*}\triangle_{j}f_{\ell}=\begin{cases*}f_{\ell},&$j=\ell$;\\\
\triangle_{j}f_{\ell},&$0<\left\|j-\ell\right\|_{\infty}\leqslant 3$;\\\
0,&$\left\|j-\ell\right\|_{\infty}>3.$\end{cases*}$

Then by Corollary 3.3 ,we have

$\displaystyle\left\|f_{\ell}\right\|_{M_{p,q}}$
$\displaystyle=\left\|f_{\ell}\right\|_{p}\approx 1$
$\displaystyle\left\|f_{\ell}\right\|_{B^{s}_{p_{0},q_{0}}}$
$\displaystyle=\left\|2^{js}\left\|\triangle_{j}f_{\ell}\right\|_{p_{0}}\right\|_{\ell_{\left\|j-\ell\right\|_{\infty}\leqslant
3}^{q_{0}}}$ $\displaystyle\lesssim 2^{\ell
s}\left\|f_{\ell}\right\|_{p_{0}}\approx 2^{\ell s}.$

Then take the estimate above into (4.1), we have $1\lesssim 2^{\ell
s},\forall\ell\geqslant 10$, so we get $s\geqslant 0$.

(3)

Take $f_{\ell}={\mathscr{F}^{-1}}\varphi_{\ell}$, then by Corollary 3.3, we
have

$\displaystyle\left\|f_{\ell}\right\|_{M_{p,q}}$
$\displaystyle=\left\|\left\|\Box_{k}f_{\ell}\right\|_{p}\right\|_{\ell_{k}^{q}}\geqslant\left\|\left\|\Box_{k}f_{\ell}\right\|_{p}\right\|_{\ell_{k\in
A_{\ell}}^{q}}=\left\|\left\|{\mathscr{F}^{-1}}\sigma_{k}\right\|_{p}\right\|_{\ell_{k\in
A_{\ell}}^{q}}\approx 2^{\ell d/q};$
$\displaystyle\left\|f_{\ell}\right\|_{B^{s}_{p_{0},q_{0}}}$
$\displaystyle=\left\|2^{js}\left\|\triangle_{j}f_{\ell}\right\|_{p_{0}}\right\|_{\ell_{j}^{q_{0}}}=\left\|2^{js}\left\|\triangle_{j}f_{\ell}\right\|_{p_{0}}\right\|_{\ell_{\left\|j-\ell\right\|_{\infty}\leqslant
3}^{q_{0}}}\leqslant 2^{\ell s}\left\|f_{\ell}\right\|_{p_{0}}\approx
2^{\ell(s+d(1-1/p_{0}))}.$

Take the estimates above into (4.1), let $\ell\rightarrow\infty$, we have
$s\geqslant d(1/p_{0}+1/q-1)$.

(4)

When $p_{0}\geqslant 2$, choose $\eta\in\mathscr{S}$, such that supp
$\hat{\eta}\subseteq[-1/8,1/8]^{d}$, take $f_{\ell}(x)=\sum_{k\in
A_{\ell}}e^{ikx}\eta(x-k)$, then $\hat{f}_{\ell}(\xi)=\sum_{k\in
A_{\ell}}e^{-ik(\xi-k)}\hat{\eta}(\xi-k)$, so we know that supp
$\hat{f}_{\ell}\subseteq\bigcup_{k\in A_{\ell}}k+[-1/8,1/8]^{d}\subseteq
D_{\ell}$, so by the orthogonality of $\triangle_{j},\Box_{k}$, we have:

$\displaystyle\Box_{k}f_{\ell}=\begin{cases*}e^{ikx}\eta(x-k),&$k\in
A_{\ell}$;\\\ 0,&else;\end{cases*}\ \
\triangle_{j}f_{\ell}=\begin{cases*}f_{\ell},&$j=\ell$;\\\
\triangle_{j}f_{\ell},&$0<\left\|j-\ell\right\|_{\infty}\leqslant 3$;\\\
0,&$\left\|j-\ell\right\|_{\infty}>3.$\end{cases*}$

Then we have

$\displaystyle\left\|f_{\ell}\right\|_{M_{p,q}}\geqslant\left\|\left\|\Box_{k}f_{\ell}\right\|_{p}\right\|_{\ell_{k\in
A_{\ell}}^{q}}\approx 2^{\ell d/q};$
$\displaystyle\left\|f_{\ell}\right\|_{B^{s}_{p_{0},q_{0}}}=\left\|2^{js}\left\|\triangle_{j}f_{\ell}\right\|_{p_{0}}\right\|_{\ell_{j}^{q_{0}}}=\left\|2^{js}\left\|\triangle_{j}f_{\ell}\right\|_{p_{0}}\right\|_{\ell_{\left\|j-\ell\right\|_{\infty}\leqslant
3}^{q_{0}}}\leqslant 2^{\ell s}\left\|f_{\ell}\right\|_{p_{0}}\leqslant
2^{\ell
s}\left\|f_{\ell}\right\|_{2}^{2/p_{0}}\left\|f\right\|_{\infty}^{1-2/p_{0}}.$

By the orthogonality of $L^{2}$ space, we have

$\displaystyle\left\|f_{\ell}\right\|_{2}=\left(\sum_{k\in
A_{\ell}}\left\|\eta(x-k)\right\|_{2}^{2}\right)^{1/2}\approx 2^{\ell d/2}.$

Also by the fast decay of $\eta$, we have

$\displaystyle\left|f_{\ell}(x)\right|\leqslant\sum_{k\in
A_{\ell}}\left|\eta(x-k)\right|\leqslant\sum_{k\in{\mathbb{Z}}^{d}}(1+\left|x-k\right|)^{-N}\lesssim
1.$

Combine the estimates above, we have
$\left\|f\right\|_{B^{s}_{p_{0},q_{0}}}\lesssim 2^{\ell(s+d/p_{0})}$. Then
take it into (4.1), and put $\ell\rightarrow\infty$, we have $s\geqslant
d(1/q-1/p_{0})$.

(5)

Combine case(2)-(4), we have $s\geqslant\tau(p_{0},q)$. In contrast with the
necessary condition we need in Theorem 4.1, we only need to prove that if
$B_{p_{0},q_{0}}^{\tau(p_{0},q)}\hookrightarrow M_{p,q}$, then we have
$q_{0}\leqslant q$.

(5.1)

When $\tau(p_{0},q)=0$, take $f=\sum_{\ell}a_{\ell}f_{\ell}$, where $f_{\ell}$
in case (2), then by the same calculation, we have

$\displaystyle\left\|f\right\|_{M_{p,q}}\approx\left\|a_{\ell}\right\|_{\ell^{q}};\
\
\left\|f\right\|_{B_{p_{0},q_{0}}^{0}}\lesssim\left\|a_{\ell}\right\|_{\ell^{q_{0}}}.$

Take the estimates into (4.1), we have $q_{0}\leqslant q$.

(5.2)

When $\tau(p_{0},q)=d(1/p_{0}+1/q-1)$, take $f=\sum_{\ell}a_{\ell}f_{\ell}$,
where $f_{\ell}$ in case (3), then by the same calculation, we have

$\displaystyle\left\|f\right\|_{M_{p,q}}$
$\displaystyle=\left\|\left\|\sum_{\ell}a_{\ell}\Box_{k}f_{\ell}\right\|_{p}\right\|_{\ell_{k}^{q}}\geqslant\left\|\left\|\left\|\sum_{\ell}a_{\ell}\Box_{k}f_{\ell}\right\|_{p}\right\|_{\ell_{k\in
A_{j}}^{q}}\right\|_{\ell_{j}^{q}}=\left\|a_{j}2^{jd/q}\right\|_{\ell_{j}^{q}};$
$\displaystyle\left\|f\right\|_{B_{p_{0},q_{0}}^{\tau(p_{0},q)}}$
$\displaystyle=\left\|2^{j\tau}\left\|\sum_{\ell}a_{\ell}\triangle_{j}f_{\ell}\right\|_{p_{0}}\right\|_{\ell_{j}^{q_{0}}}\leqslant\sum_{\left\|\ell\right\|_{\infty}\leqslant
3}\left\|a_{j+\ell}2^{j\tau}\left\|\triangle_{j}f_{\ell+j}\right\|_{p_{0}}\right\|_{\ell_{j}^{q_{0}}}$
$\displaystyle\leqslant\sum_{\left\|\ell\right\|_{\infty}\leqslant
3}\left\|a_{\ell+j}2^{jd(1-1/p_{0})}2^{j\tau}\right\|_{\ell_{j}^{q_{0}}}\lesssim\left\|a_{j}2^{jd/q}\right\|_{\ell_{j}^{q_{0}}}.$

Take the estimates into (4.1), we have $q_{0}\leqslant q$.

(5.3)

When $\tau(p_{0},q)=d(1/q-1/p_{0}),$ take $f=\sum_{\ell}a_{\ell}f_{\ell}$,
where $f_{\ell}$ in case (4), then by the same calculation, we have

$\displaystyle\left\|f\right\|_{M_{p,q}}$
$\displaystyle=\left\|\left\|\sum_{\ell}a_{\ell}\Box_{k}f_{\ell}\right\|_{p}\right\|_{\ell_{k}^{q}}\geqslant\left\|\left\|\left\|\sum_{\ell}a_{\ell}\Box_{k}f_{\ell}\right\|_{p}\right\|_{\ell_{k\in
A_{j}}^{q}}\right\|_{\ell_{j}^{q}}$
$\displaystyle=\left\|\left\|a_{j}\left\|{\mathscr{F}^{-1}}\sigma_{k}\right\|_{p}\right\|_{\ell_{k\in
A_{j}}^{q}}\right\|_{\ell_{j}^{q}}\approx\left\|a_{j}2^{jd/q}\right\|_{\ell_{j}^{q}};$
$\displaystyle\left\|f\right\|_{B_{p_{0},q_{0}}^{\tau}}$
$\displaystyle=\left\|2^{j\tau}\left\|\sum_{\ell}a_{\ell}\triangle_{j}f_{\ell}\right\|_{p_{0}}\right\|_{\ell_{j}^{q_{0}}}\leqslant\sum_{\left\|\ell\right\|_{\infty}\leqslant
3}\left\|a_{j+\ell}2^{j\tau}\left\|\triangle_{j}f_{\ell+j}\right\|_{p_{0}}\right\|_{\ell_{j}^{q_{0}}}$
$\displaystyle\leqslant\sum_{\left\|\ell\right\|_{\infty}\leqslant
3}\left\|a_{\ell+j}\left\|f_{\ell+j}\right\|_{p_{0}}2^{j\tau}\right\|_{\ell_{j}^{q_{0}}}\lesssim\left\|a_{j}2^{j\tau}2^{jd/p_{0}}\right\|_{\ell_{j}^{q_{0}}}=\left\|a_{j}2^{jd/q}\right\|_{\ell_{j}^{q_{0}}}.$

Take the estimates into (4.1), we have $q_{0}\leqslant q$.

###  4.2 Proof of Theorem 4.2

In this section, we prove the Theorem 4.2. Because we consider the case when
$0<p,q\leqslant\infty$, we can not only use the duality to prove it. Thanks to
the wonderful method in [[2]], also combining the construction in the proof of
Theorem 4.1, we can prove this theorem as well.

####  4.2.1 Sufficiency

By Lemma 2.1, we have $M_{p,q}\hookrightarrow M_{p_{1},q}$; by Proposition
3.4, we have $M_{p_{1},q}\hookrightarrow B_{p_{1},q}^{\sigma(p_{1},q)}$;
finally by Lemma 2.2, we have $B_{p_{1},q}^{\sigma(p_{1},q)}\hookrightarrow
B^{s}_{p_{1},q_{1}}$ for any case in Theorem 4.2.

####  4.2.2 Necessity

If we have $M_{p,q}\hookrightarrow B^{s}_{p_{1},q_{1}}$, then we have

$\displaystyle\left\|f\right\|_{B^{s}_{p_{1},q_{1}}}\lesssim\left\|f\right\|_{M_{p,q}}.$
(4.2)

Just like the proof of Theorem 4.1, we can take some different kinds of $f$
into (4.2) to get some restrictions of indexes, and in some cases, we can take
the same $f$ as in 4.1.2.

(1)

Take $f_{\lambda}$ in the case (1) in 4.1.2 into (4.2), we can get $p\leqslant
p_{1}$.

(2)

Take $f_{\ell}$ in the case (2) in 4.1.2, we have
$\left\|f_{\ell}\right\|_{M_{p,q}}\approx 1$, and
$\left\|f_{\ell}\right\|_{B^{s}_{p_{1},q_{1}}}\geqslant 2^{\ell
s}\left\|\triangle_{\ell}f_{\ell}\right\|_{p_{1}}=2^{\ell
s}\left\|f_{\ell}\right\|_{p_{1}}\approx 2^{\ell s}$. Then we have $s\leqslant
0$.

(3)

Denote $B_{j}=\left\\{k\in{\mathbb{Z}}^{d}:k+[-3/4,3/4]^{d}\cap
D_{j}\neq\emptyset\right\\}$, choose $g\in\mathscr{S},$ such that supp
$g\subseteq D_{0}$, then denote $g_{\ell}(\cdot)=g(2^{-\ell}\cdot)$. Take
$f_{\ell}={\mathscr{F}^{-1}}g_{\ell}$, so supp ${\mathscr{F}}f_{\ell}\subseteq
D_{\ell}$. So by Corollary 3.3, we have

$\displaystyle\left\|f_{\ell}\right\|_{M_{p,q}}=\left\|\left\|\Box_{k}f_{\ell}\right\|_{p}\right\|_{\ell_{k\in
B_{\ell}}^{q}}\leqslant\left\|\left\|{\mathscr{F}^{-1}}\sigma_{k}\right\|_{p}\right\|_{\ell_{k\in
B_{\ell}}^{q}}\lesssim 2^{\ell d/q};$
$\displaystyle\left\|f_{\ell}\right\|_{B^{s}_{p_{1},q_{1}}}\geqslant 2^{\ell
s}\left\|\triangle_{\ell}f_{\ell}\right\|_{p_{1}}=2^{\ell
s}\left\|f_{\ell}\right\|_{p_{1}}\approx 2^{\ell(s+d(1-1/p_{1}))}.$

Take the estimate into (4.2), let $\ell\rightarrow\infty$, we can get
$s\leqslant d(1/p_{1}+1/q-1)$.

(4)

Take $f_{\ell}(x)=\sum_{k\in A_{\ell}}e^{ikx}\eta(\frac{x-k}{a})$, we can
choose $a<<1$ such that $\left\|f_{\ell}\right\|_{p_{1}}\geqslant C2^{\ell
d/p_{1}}$(see [2]), then by the same calculation of case (4) in 4.1.2, we have

$\displaystyle\left\|f_{\ell}\right\|_{B^{s}_{p_{1},q_{1}}}\geqslant
C2^{\ell(s+d/p_{1})};\ \ \left\|f_{\ell}\right\|_{M_{p,q}}\lesssim 2^{\ell
d/q}.$

Take the estimates above into (4.2) and put $\ell\rightarrow\infty$, we have
$s\leqslant d(1/q-1/p_{1})$.

(5)

Combine case(2)-(4), we have $s\leqslant\sigma(p_{1},q)$. In contrast with the
necessary condition we need in Theorem 4.2, we only need to prove that if
$M_{p,q}\hookrightarrow B_{p_{1},q_{1}}^{\sigma(p_{1},q)}$, then we have
$q_{1}\geqslant q$.

By the same way in the proof of Theorem 4.1, take $f=\sum a_{\ell}f_{\ell}$,
$f_{\ell}$ in case (2)-(4), we can get
$\left\|a_{\ell}\right\|_{q_{1}}\lesssim\left\|a_{\ell}\right\|_{q}$, which
means that $q_{1}\geqslant q$.

##  5 Embedding between Sobolev spaces and modulation spaces

In this section, we can get the sufficient and necessary conditions of the
embedding between $L^{r}$-Sobolev spaces and modulation spaces. We consider
the most general case of this kind of embedding with the form like
$W^{s,r}\hookrightarrow M_{p,q}$, which is the general case of Proposition 3.5
where $r=p$. Also, we consider the case when $0<q\leqslant\infty$. Unlike the
proof in that proposition, we mostly use the uniform decomposition of
frequency space and the Theorem 4.1 we get in last section.

Our main result is:

######  Theorem 5.1.

Let $1\leqslant p,r\leqslant\infty,0<q\leqslant\infty,s\in{\mathbb{R}}$, then
$W^{s,r}\hookrightarrow M_{p,q}$ if and only if $r\leqslant p$ and one of the
following conditions is satisfied:

(1)

$r>q,s>\tau(r,q)$;

(2)

$1<r\leqslant q,s\geqslant\tau(r,q)$;

(3)

$r=1,q=\infty,s\geqslant\tau(r,q)$;

(4)

$r=1,0<q<\infty,s>\tau(r,q)$;

(1)(2)(3)(4)$\frac{1}{r}$$\frac{1}{q}$011 Figure 1: the four cases of
$(1/r,1/q)$ in Theorem 5.1

######  Remark 5.2.

As the can see, the sufficient and necessary conditions we got in the theorem
above is the same as the result in Proposition 3.4. The index $p$ here only
make sense in $r\leqslant p$. The reason of this may be that the modulation
space have good embedding property of the index $p$ like in Lemma 2.1.

As for the other side of this kind of embedding, we also have:

######  Theorem 5.3.

Let $1\leqslant p,r\leqslant\infty,0<q\leqslant\infty,s\in{\mathbb{R}}$, then
$M_{p,q}\hookrightarrow W^{s,r}$ if and only if $p\leqslant r$ and one of the
following conditions is satisfied:

(1)

$r<q,s<\sigma(r,q)$;

(2)

$q\leqslant r<\infty,s\leqslant\sigma(r,q)$;

(3)

$r=\infty,0<q=1,s\leqslant\sigma(r,q)$;

(4)

$r=\infty,1<q\leqslant\infty,s<\sigma(r,q)$;

###  5.1 Proof of Theorem 5.1

In this subsection, we prove Theorem 5.1. By the Proposition 3.5, we can
easily prove the sufficiency. As for necessity, we mainly construct some
funcions to the embedding estimates to get some restrictions of the indexes.

####  5.1.1 Sufficiency

When $1\leqslant q\leqslant\infty$, by Proposition 3.5, in any case of
(1)-(4), we have $W^{s,r}\hookrightarrow M_{r,q}$; so, if $r\leqslant p$, by
Lemma 2.1, we have $M_{r,q}\hookrightarrow M_{p,q}$.

When $0<q<1$, which could only happen in case (2) or (4), then we have
$s>\tau(r,q)=\tau(r,1)-d+d/q$. Take $\varepsilon>0$ small enough, such that
$s>\tau(r,1)-d+d/q+\varepsilon$, then by Proposition 3.5, we have
$W^{s,r}\hookrightarrow M_{r,1}^{d/q-d+\varepsilon}$. Then by (4) in Lemma
2.1, we have $M_{r,1}^{d/q-d+\varepsilon}\hookrightarrow M_{p,q}$, as desired.

####  5.1.2 Necessity

If we have $W^{s,r}\hookrightarrow M_{p,q}$, then we have

$\displaystyle\left\|f\right\|_{M_{p,q}}\lesssim\left\|f\right\|_{W^{s,r}}.$
(5.1)

(i)

Choose $\eta\in\mathscr{S}$, such that supp $\eta\subset[-1/8,1/8]^{d}$,
denote $f={\mathscr{F}^{-1}}\eta$, for $\forall\lambda\leqslant 1$, take
$f_{\lambda}(x)=f(\lambda x)$ into (5.1), we have
$\left\|f_{\lambda}\right\|_{r}\approx\left\|f_{\lambda}\right\|_{W^{s,r}}\lesssim\left\|f_{\lambda}\right\|_{M_{p,q}}\approx\left\|f_{\lambda}\right\|_{p}$,
so we have $r\leqslant p$. Therefore, when $r=\infty$, we know that
$p=\infty$, this is the case in Proposition 3.5.

(ii)

When $1\leqslant r<\infty$, by Lemma 2.2, we have $B_{r,r\wedge
2}^{s}\hookrightarrow F_{r,2}^{s}\hookrightarrow W^{s,r}\hookrightarrow
M_{p,q}$, so when $r\leqslant 2$, we have $B_{r,r}^{s}\hookrightarrow
M_{p,q}$, by Theorem 4.1, we know that if $r\leqslant q$, then
$s\geqslant\tau(r,q)$; if $r>q$, then $s>\tau(r,q)$.

When $r>2$, we have $B_{r,2}^{s}\hookrightarrow M_{p,q}$, also by Theorem 4.1,
we know that if $2\leqslant q,$ then $s\geqslant\tau(r,q)$; if $2>q$, then
$s>\tau(r,q)$.

(iii)

We prove that $W^{\tau(r,q),r}\hookrightarrow M_{p,q}$ is not true for any
$2\leqslant q<r\leqslant p$, in which case $\tau(r,q)=d(1/q-1/r)$, for
convenience we also denote it by $\tau$.

If not, we have $W^{\tau,r}\hookrightarrow M_{p,q}$ which is equivalent to
$L^{r}\hookrightarrow M_{p,q}^{-\tau}$. So we have

$\displaystyle\left\|f\right\|_{M_{p,q}^{-\tau}}\lesssim\left\|f\right\|_{r}.$
(5.2)

Choose $\eta\in\mathscr{S}$, such that supp $\hat{\eta}\subset[-1/8,1/8]^{d}$,
take $f(x)=\sum_{k\in{\mathbb{Z}}^{d}}a_{k}e^{ikx}\eta(x-k)$, then
$\hat{f}(\xi)=\sum_{k\in{\mathbb{Z}}^{d}}a_{k}e^{ik(\xi-k)}\hat{\eta}(\xi-k)$.
Then by the orthogonality of $\Box_{k}$, we have

$\displaystyle\Box_{k^{\prime}}f=\begin{cases*}a_{k}e^{ikx}\eta(x-k);&$k=k^{\prime}$;\\\
0;&$k\neq k^{\prime}$.\end{cases*}$

So, we have $\left\|f\right\|_{M_{p,q}^{-\tau}}\approx\left\|\left\langle
k\right\rangle^{-\tau}a_{k}\right\|_{\ell_{k}^{q}}$.

By the orthogonality of $L^{2}$ space, we know that

$\displaystyle\left\|f\right\|_{2}=\left(\sum_{k\in{\mathbb{Z}}^{d}}\left\|a_{k}e^{ikx}\eta(x-k)\right\|_{2}^{2}\right)^{1/2}\approx\left\|a_{k}\right\|_{\ell^{2}}.$

By the rapidly decrease of $\eta$, we know that

$\displaystyle\left|f(x)\right|\leqslant\sum_{k\in{\mathbb{Z}}^{d}}\left|a_{k}\right|\left|\eta(x-k)\right|\leqslant\left\|a_{k}\right\|_{\ell_{k}^{\infty}}\sum_{k\in{\mathbb{Z}}^{d}}(1+\left|x-k\right|)^{-N}\lesssim\left\|a_{k}\right\|_{\ell_{k}^{\infty}}.$

So we have
$\left\|f\right\|_{\infty}\lesssim\left\|a_{k}\right\|_{\ell_{k}^{\infty}}$,
then by interpolation, we have
$\left\|f\right\|_{r}\lesssim\left\|a_{k}\right\|_{\ell_{k}^{r}}$.

Take the two estimates into (5.2), we have $\left\|\left\langle
k\right\rangle^{-\tau}a_{k}\right\|\lesssim\left\|a_{k}\right\|_{\ell_{k}^{r}}$,
which is equivalent to

$\displaystyle\sum_{k\in{\mathbb{Z}}^{d}}\left\langle k\right\rangle^{-\tau
q}b_{k}\lesssim\left(\sum_{k\in{\mathbb{Z}}^{d}}b_{k}^{r/q}\right)^{q/r}.$

So we know that $\left\\{\left\langle k\right\rangle^{-\tau
q}\right\\}\in\ell^{(r/q)^{\prime}}$, which is equivalent to $\tau
q(r/q)^{\prime}>d$, so we have $\tau>d(1/q-1/r)$, which is a contradiction.

(iv)

We prove that when $r=1\leqslant p,0<q<\infty,s=\tau(1,q)=d/q$,
$W^{s,1}\hookrightarrow M_{p,q}$ is not true.

If not, we have $W^{d/q,1}\hookrightarrow M_{p,q}\hookrightarrow M_{\infty,q}$
which is equivalent to $L^{1}\hookrightarrow M_{\infty,q}^{-d/q}$, i.e.

$\displaystyle\left\|f\right\|_{M_{\infty,q}^{-d/q}}\lesssim\left\|f\right\|_{1}.$
(5.3)

Choose $\eta\in\mathscr{S}$ such that $\hat{\eta}=1$ on $[-1,1]^{d}$, for
$0<t<1$, take $f(x)=t^{-d}\eta(x/t)$, then $\hat{f}=1$ on
$\frac{1}{t}[-1,1]^{d}$, denote
$K_{t}=\left\\{k\in{\mathbb{Z}}^{d}:k+[-3/4,3/4]^{d}\subseteq\frac{1}{t}[-1,1]^{d}\right\\}$,
then we have

$\displaystyle\left\|f\right\|_{M_{\infty,q}^{-d/q}}=\left\|\left\langle
k\right\rangle^{-d/q}\left\|\Box_{k}f\right\|_{\infty}\right\|_{\ell_{k}^{q}}\geqslant\left\|\left\langle
k\right\rangle^{-d/q}\left\|\Box_{k}f\right\|_{\infty}\right\|_{\ell_{k\in
K_{t}}^{q}}\approx\left\|\left\langle
k\right\rangle^{-d/q}\right\|_{\ell_{k\in K_{t}}^{q}}.$

Also, $\left\|f\right\|_{1}=\left\|\eta\right\|_{1}\approx 1$. So, take the
two estimates into (5.3), we have $\left\|\left\langle
k\right\rangle^{-d/q}\right\|_{\ell_{k\in K_{t}}^{q}}\lesssim 1$, take
$t\rightarrow 0$, we have $\left\|\left\langle
k\right\rangle^{-d/q}\right\|_{\ell_{k\in{\mathbb{Z}}^{d}}^{q}}\lesssim 1$,
which is a contraction.

In fact, we already get the result as desired. To be specific,from (i,ii) we
can get the necessity of the condition (2)(3); the necessity of condition (4)
can be got by (ii,iv); as for condition (1), when $q<r\leqslant 2$ or $r>2>q$,
the necessity is proved in (ii), when $2\leqslant q<r$, the necessity is
proved in (iii).

###  5.2 Proof of Theorem 5.3

When $1\leqslant q\leqslant\infty$, we can get the result as desired just by
duality of Theorem 5.1, so we only need to consider the case $0<q<1$, which
could only happen in case (2) or (3). Also, we know that $\sigma(r,q)=0$ in
these cases.

Sufficiency: by Lemma 2.1 and dual of Proposition 3.5, we can get
$M_{p,q}\hookrightarrow M_{r,1}\hookrightarrow W^{s,r}$ when $s\leqslant 0$.

Necissity: if we have $M_{p,q}\hookrightarrow W^{s,r}$ which is equivalent to
$M_{p,q}^{-s}\hookrightarrow L^{r}$. So, we have

$\displaystyle\left\|f\right\|_{r}\lesssim\left\|f\right\|_{M_{p,q}^{-s}}.$

For any $k\in{\mathbb{Z}}^{d}$, take $f(x)=e^{ikx}\eta(x)$ into the equation
above, where $\eta\in\mathscr{S}$ and supp
$\hat{\eta}\subseteq[-1/8,1/8]^{d}$. Then we have $1\lesssim\left\langle
k\right\rangle^{-s}$, which means that $s\leqslant 0$.

##  6 Embedding between Triebel spaces and modulation spaces

In this section, we mainly study the embedding $F_{p_{0},q}^{s}\hookrightarrow
M_{p,q}$. We first consider a special case like $F_{p,2}^{s}\hookrightarrow
M_{p,q}$, which is just the same as Proposition 3.5 in some cases. And using
this embedding and interpolation method we can get the result we need.

As for special case, we have:

######  Theorem 6.1.

Let $0<p,q\leqslant\infty$, then $F_{p,2}^{s}\hookrightarrow M_{p,q}$ if and
only if one of the following conditions is satisfied:

(1)

$q\geqslant p,s\geqslant\tau(p,q)$;

(2)

$q<p,s>\tau(p,q)$.

As for the other side of this kind of embedding, we also have:

######  Theorem 6.2.

Let $0<p,q\leqslant\infty$, then $M_{p,q}\hookrightarrow F_{p,2}^{s}$ if and
only if one of the following conditions is satisfied :

(1)

$q\leqslant p,s\leqslant\sigma(p,q)$;

(2)

$q>p,s<\sigma(p,q)$.

Then, our main result is:

######  Theorem 6.3.

Let $0<p,p_{0},q\leqslant\infty$, then $F_{p_{0},q}^{s}\hookrightarrow
M_{p,q}$ if and only if $p_{0}\leqslant p$ and one of the following conditions
is satisfied:

(1)

$p_{0}\leqslant q,s\geqslant\tau(p_{0},q)$;

(2)

$p_{0}>q,s>\tau(p_{0},q)$.

(1)(2)$\frac{1}{p_{0}}$$\frac{1}{q}$01$\tau=d(\frac{1}{q}-\frac{1}{p_{0}})$$\tau=0$$\tau=d(\frac{1}{p_{0}}+\frac{1}{q}-1)$
Figure 2: the two cases of $(1/p_{0},1/q)$ in Theorem 6.3

As for the other side of this kind of embedding, we also have:

######  Theorem 6.4.

Let $0<p,p_{1},q\leqslant\infty$, then $M_{p,q}\hookrightarrow
F_{p_{1},q}^{s}$ if and only if $p_{1}\geqslant p$ and one of the following
conditions is satisfied:

(1)

$p_{1}\geqslant q,s\leqslant\sigma(p_{1},q)$;

(2)

$p_{1}<q,s<\sigma(p_{1},q)$.

###  6.1 Proof of Theorem 6.1

For $1<p<\infty$, we know that $F_{p,2}^{s}\approx W^{s,p}$, so by Theorem
5.1, we get the result as desired.

For $0<p\leqslant 1$, we know that $F_{p,2}^{s}=h_{p}$ the local Hardy space,
so by the result in [20], we get the result as desired.

For $p=\infty$, if we have $F_{\infty,2}^{s}\hookrightarrow M_{p,q}$, then by
Lemma 2.2, we have $W^{\infty,s}\hookrightarrow
F_{\infty,2}^{s}\hookrightarrow M_{p,q}$, then by Theorem 5.1, we have the
condition as desired. As for sufficiency, in condition (1), by Lemma 2.2, we
have $F_{\infty,2}^{s}\hookrightarrow B_{\infty,\infty}^{0}$, then by
Proposition 3.4, we have $B_{\infty,\infty}^{0}\hookrightarrow
M_{\infty,\infty}$; in condition (2), by Lemma 2.2, we have
$F_{\infty,2}^{s}\hookrightarrow B_{\infty,\infty}^{s}\hookrightarrow
B_{\infty,q}^{d/q}$, then by Proposition 3.4, we have
$B_{\infty,q}^{d/q}\hookrightarrow M_{\infty,q}$.

###  6.2 Proof of Theorem 6.2

By the same discussion in the proof of Theorem 6.1, we only need to consider
the case when $p=\infty$. Also, when $1\leqslant q\leqslant\infty$, by
duality, we can get the result as desired. When $0<q<1,p=\infty$, we have
$\sigma(p,q)=0$.

Sufficiency: by Lemma 2.1, we have $M_{\infty,q}\hookrightarrow M_{\infty,1}$.
By Theorem 6.1, we have $F_{1,2}\hookrightarrow M_{1,\infty}$, by duality, we
have $M_{\infty,1}\hookrightarrow F_{\infty,2}$, so we have
$M_{\infty,q}\hookrightarrow F_{\infty,2}\hookrightarrow F_{\infty,2}^{s}$ for
$s\leqslant 0$.

Necessity: if we have $M_{\infty,q}\hookrightarrow F_{\infty,2}^{s}$, then by
Lemma 2.2, we have $F_{\infty,2}^{s}\hookrightarrow B_{\infty,\infty}^{s}$. By
Theorem 4.2, we have $s\leqslant\sigma(\infty,q)$.

###  6.3 Proof of Theorem 6.3

####  6.3.1 Sufficiency

For condition (1): by Lemma 2.2, we have $F_{p_{0},q}^{s}\hookrightarrow
B_{p_{0},q}^{s}$, then by Theorem 4.1, we have $B_{p_{0},q}^{s}\hookrightarrow
M_{p,q}$.

For condition (2): by Lemma 2.2, we have $F_{p_{0},q}^{s}\hookrightarrow
B_{p_{0},p_{0}}^{s}\hookrightarrow B_{p_{0},q}^{\tau(p_{0},q)+\varepsilon}$
for some $0<\varepsilon<<1$, then by Theorem 4.1, we have
$B_{p_{0},q}^{\tau(p_{0},q)+\varepsilon}\hookrightarrow M_{p,q}$.

####  6.3.2 Necessity

If we have $F_{p_{0},q}^{s}\hookrightarrow M_{p,q}$, then we have

$\displaystyle\left\|f\right\|_{M_{p,q}}\lesssim\left\|f\right\|_{F_{p_{0},q}^{s}}.$
(6.1)

(i)

By the same method as in subsection 5.1.2, we can get $p_{0}\leqslant p$.

(ii)

By Lemma 2.2, we have $B_{p_{0},p_{0}\wedge q}^{s}\hookrightarrow
F_{p_{0},q}^{s}\hookrightarrow M_{p,q}$, then by Theorem 4.1, we have
$s\geqslant\tau(p_{0},q)$.

(iii)

When $p_{0}>q\geqslant 2$, by Lemma 2.2, we have $W^{s,p_{0}}\approx
F_{p_{0},2}^{s}\hookrightarrow F_{p_{0},q}^{s}\hookrightarrow M_{p,q}$, then
by Theorem 5.1, we have $s>\tau(p_{0},q)$.

(iv)

When $2\geqslant p_{0}>q$, by a similar proof of Proposition 3.2 in [21], we
have
$\left\|2^{jd/q}a_{j}\right\|_{\ell_{j}^{q}}\lesssim\left\|2^{j(s+d(1-1/p_{0}))}a_{j}\right\|_{p_{0}}$,
then we have $s>d(1/q+1/p_{0}-1)$.

(v)

We prove that $F_{p_{0},q}^{\tau(p_{0},q)}\hookrightarrow M_{p,q}$ is not true
for any $p_{0}>q,q<2,p_{0}>2$, in which case $\tau(p_{0},q)=d(1/q-1/p_{0})$.

If not, we have $F_{p_{0},q}^{d(1/q-1/p_{0})}\hookrightarrow M_{p,q}$, also by
sufficiency, we have $F_{p_{0},p_{0}}^{0}\hookrightarrow
M_{p_{0},p_{0}}\hookrightarrow M_{p,p_{0}}$, then by interpolation in Lemma
2.2 and 2.1, we have $F_{p_{0},2}^{d(1/2-1/p_{0})}\hookrightarrow M_{p,2}$
which is contradiction with case (iii).

Combine the case(i),(ii), we get the condition (1); combine the case
(i),(iii),(iv),(v), we get the condition (2) as desired.

######  Remark 6.5.

  1. 1.

As for case (v) in the proof above, we prove it by contradiction using the
method of interpolation. Also, we can prove it directly by take some good
function into the norm inequality. One can modify the proof of Proposition 3.1
in [21] to get the result.

  2. 2.

The proof of Theorem 6.4 is similar to the proof above. For the reader’s
convenience, we give the outline of the proof here. The sufficiency can get by
the relation of Triebel spaces and Besov spaces, then use Theorem 4.2 can get
the result. As for necessity, we can use a revised version of Proposition
3.1,3.2 in [21] directly to get the result as desired.

  3. 3.

As we can see, the embedding we consider above is not the most general case
like $F_{p_{0},q_{0}}^{s}\hookrightarrow M_{p,q}$. I believe that for the
general case the sufficient and necessary is the similar to Theorem 6.3. But I
have not proven it yet. The most difficult case left is the same as the open
problem in [21].

##  7 Embedding between Fourier $L^{p}$ spaces and modulation spaces

In this section, we consider the embedding
$M_{p,q}^{s}\hookrightarrow{\mathscr{F}}L^{r}$. Because of the orthogonality
of $\Box_{k}(\sigma_{k})$, we can calculate the ${\mathscr{F}}L^{r}$ norm
easily. Then, we can get the result as follow. Recall that when $0<p<1$, we
denote $p^{\prime}=\infty$.

######  Theorem 7.1.

Let $0<p,q,r\leqslant\infty,s\in{\mathbb{R}}$, then we have
$M_{p,q}^{s}\hookrightarrow{\mathscr{F}}L^{r}$ if and only if one of the
following conditions is satisfied:

(1)

$p\leqslant 2,q\leqslant r\leqslant p^{\prime},s\geqslant 0$;

(2)

$p\leqslant 2,r\leqslant p^{\prime},r<q,s>d(1/r-1/q)$.

As for the other side of this kind of embedding, we also have:

######  Theorem 7.2.

Let Let $0<p,q,r\leqslant\infty,s\in{\mathbb{R}}$, then we have
${\mathscr{F}}L^{r}\hookrightarrow M_{p,q}^{s}$ if and only if one of the
following conditions is satisfied:

(1)

$p\geqslant 2,p^{\prime}\leqslant r\leqslant q,s\leqslant 0$;

(2)

$p\geqslant 2,p^{\prime}\leqslant r,r>q,s<d(1/r-1/q)$.

###  7.1 Proof of Theorem 7.1

####  7.1.1 Sufficiency

If we have condition (1), when $1\leqslant p\leqslant\infty$, by the
orthogonality of $\sigma_{k}$, H$\ddot{o}$lder’s inequality of $L^{p}$ spaces
and Hausdorff-Young’s inequality of Fourier transform, we have

$\displaystyle\left\|\hat{f}\right\|_{r}\approx\left\|\left\|\sigma_{k}\hat{f}\right\|_{r}\right\|_{\ell_{k}^{r}}\leqslant\left\|\left\|\sigma_{k}\hat{f}\right\|_{p^{\prime}}\right\|_{\ell_{k}^{r}}\leqslant\left\|\left\|\Box_{k}f\right\|_{p}\right\|_{\ell_{k}^{r}}\leqslant\left\|\left\|\Box_{k}f\right\|_{p}\right\|_{\ell_{k}^{q}}=\left\|f\right\|_{M_{p,q}}\leqslant\left\|f\right\|_{M_{p,q}^{s}}.$

When $0<p<1$, by Lemma 3.1, we have
$\left\|\sigma_{k}\hat{f}\right\|_{p^{\prime}}\leqslant\left\|\Box_{k}f\right\|_{1}\leqslant
C\left\|\Box_{k}f\right\|_{p}$, then by the same estimates we can get the
result as desired.

If we have condition (2), by Lemma 2.1, we have $M_{p,q}^{s}\hookrightarrow
M_{p,r}$, then by the sufficiency of condition(1), we have
$M_{p,r}\hookrightarrow{\mathscr{F}}L^{r}$.

####  7.1.2 Necessity

If we have $M_{p,q}^{s}\hookrightarrow{\mathscr{F}}L^{r}$, then we have

$\displaystyle\left\|\hat{f}\right\|_{r}\lesssim\left\|f\right\|_{M_{p,q}^{s}}.$
(7.1)

The conditions we need are contained the following cases.

(i)

For any $f\in\mathscr{S}$ with supp $\hat{f}\subseteq[-1/8,1/8]^{d}$, in this
case we have $\left\|f\right\|_{M_{p,q}^{s}}\approx\left\|f\right\|_{p}$. So,
(7.1) means that

$\displaystyle\left\|\hat{f}\right\|_{r}\lesssim\left\|f\right\|_{p}.$

Then by Hausdorff-Young’s inequality on compact set (see Tao’s note of Math
254B, one can extend the result to the case of $0<p<1$ with the same proof),
we have $r\leqslant p^{\prime},p\leqslant 2$.

(ii)

For any $k\in{\mathbb{Z}}^{d}$, take $f(x)=e^{ikx}\eta(x)$, where
$\eta\in\mathscr{S},$supp $\hat{\eta}\subseteq[-1/8,1/8]^{d}$, then we know
that supp $\hat{f}\subseteq k+[-1/8,1/8]^{d}$. So, we have

$\displaystyle\left\|f\right\|_{M_{p,q}^{s}}=\left\langle
k\right\rangle^{s}\left\|\eta\right\|_{p}\approx\left\langle
k\right\rangle^{s};\ \
\left\|f\right\|_{{\mathscr{F}}L^{r}}=\left\|\hat{\eta}\right\|_{r}\approx 1.$

Take the estimates into (7.1), we have $1\lesssim\left\langle
k\right\rangle^{s}$, which means that $s\geqslant 0$, which is the result in
condition (1).

(iii)

When $r<q$, take $f(x)=\sum_{k\in{\mathbb{Z}}^{d}}a_{k}e^{ikx}\eta(x)$, $\eta$
is the same as case (ii). Then we have
$\hat{f}(\xi)=\sum_{k\in{\mathbb{Z}}^{d}}a_{k}\hat{\eta}(\xi-k)$. So, we have

$\displaystyle\left\|f\right\|_{M_{p,q}^{s}}\approx\left\|\left\langle
k\right\rangle^{s}a_{k}\right\|_{\ell_{k}^{q}};\ \
\left\|\hat{f}\right\|_{r}\approx\left\|a_{k}\right\|_{\ell_{k}^{r}}.$

Take the estimates above into (7.1), we have
$\left\|a_{k}\right\|_{\ell_{k}^{r}}\lesssim\left\|\left\langle
k\right\rangle^{s}a_{k}\right\|_{\ell_{k}^{q}}$, then we have
$\left\|\left\langle
k\right\rangle^{-sr}b_{k}\right\|_{\ell_{k}^{1}}\lesssim\left\|b_{k}\right\|_{\ell_{k}^{q/r}}$,
which means that $\left\\{\left\langle
k\right\rangle^{-sr}\right\\}\in\ell^{(q/r)^{\prime}}$. So, we have
$sr(q/r)^{\prime}>d$, i.e. $s>d(1/r-1/q)$, which is the result in condition
(2).

######  Remark 7.3.

The proof of Theorem 7.2 is similar to the proof above, we omit it here.

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