Sharp embedding relations between Local Hardy and $\alpha ...

arXiv:1609.05280v1 [math.CA] 17 Sep 2016

SHARP EMBEDDING RELATIONS BETWEEN LOCAL HARDY AND α-MODULATION SPACES GUOPING ZHAO, GUILIAN GAO, AND WEICHAO GUO Abstract. We give the optimal embedding relations between local Hardy space and α-modulation spaces, which extend the results for the embedding relations between local Hardy and modulation spaces obtained by Kobayashi, Miyachi and Tomita in [Studia Math. 192 (2009), 79-96].

1. Introduction s Mp,q

The modulation space was first introduced by Feichtinger [1] in 1983. Some motivations and historical remarks can be founded in [2]. As function spaces associated with the uniform decomposition (see [17]), modulation space has a close relationship with the topics of time-frequency analysis (see [6]), and it has been regarded as a appropriate space for the study of partial differential equations s (see [19]). On the other hand, it is well known that the Besov space Bp,q , constructed by the dyadic decomposition on frequency plane, is also a popular function space in the studies of harmonic analysis and partial differential equations. s,α A more general function space, named α-modulation space, denoted by Mp,q , was introduced by Gr¨obner [5] in 1992, in order to link modulation and Besov spaces by the parameter α ∈ [0, 1]. The s are the special α-modulation spaces when α = 0, and the (inhomogeneous) modulation spaces Mp,q s s,α Besov space Bp,q can be regarded as the limit case of Mp,q as α → 1 (see [5]). Sometimes, we use s s,1 for convenience. The interested reader can find to denote the inhomogeneous Besov space Bp,q Mp,q some basic properties of α-modulation spaces in [10]. We note that the α-modulation space is not the intermediate space between modulation and Besov spaces in the sense of complex interpolation (see [8]). Embedding relations among frequency decomposition spaces, including Lebesgue spaces, Sobolev spaces, Besov spaces, modulation spaces and α-modulation spaces, have been concerned by many authors, for example, one can see Gr¨obner [5], Okoudjou [14], Toft [16], Sugimoto-Tomita [15], WangHan [10], Guo-Fan-Wu-Zhao [9] for embedding relation among modulation spaces, Besov spaces and α-modulation spaces. In particular, Kobayashi-Miyachi-Tomita [12] studied the embedding relations s between local Hardy spaces hp and modulation spaces Mp,q , Kobayashi-Sugimoto [13] consider the embedding relations between Sobolev and modulation spaces. We state their results as follows. For (p, q) ∈ (0, ∞]2 , we define A(p, q) = 0 ∧ n(1 − 1/p − 1/q) ∧ n(1/p − 1/q), B(p, q) = 0 ∨ n(1 − 1/p − 1/q) ∨ n(1/p − 1/q), that is,   if (1/p, 1/q) ∈ A1 : 1/q 6 (1 − 1/p) ∧ 1/p, 0, A(p, q) = n(1 − 1/p − 1/q), if (1/p, 1/q) ∈ A2 : 1/p > (1 − 1/q) ∨ 1/2,   n(1/p − 1/q), if (1/p, 1/q) ∈ A3 : 1/p 6 1/q ∧ 1/2;   if (1/p, 1/q) ∈ B1 : 1/q > (1 − 1/p) ∨ 1/p, 0, B(p, q) = n(1 − 1/p − 1/q), if (1/p, 1/q) ∈ B2 : 1/p 6 (1 − 1/q) ∧ 1/2,   n(1/2 − 1/q), if (1/p, 1/q) ∈ B3 : 1/p > 1/q ∨ 1/2. 2000 Mathematics Subject Classification. 42B35, 46E35. Key words and phrases. embedding, local Hardy space, α-modulation space. 1

2

GUOPING ZHAO, GUILIAN GAO, AND WEICHAO GUO 1 q2

1 q1

1 B1

A2 A3 A1 (0, 0)

B3

B2 1

1 p1

Figure 1

(0, 0)

1/2 Figure 2

1 p2

s Theorem A (cf. [12]) Let 0 < p 6 1, 0 < q 6 ∞ and s ∈ R. Then, hp ⊂ Mp,q if and only if s 6 A(p, q) with strict inequality when 1/q > 1/p. s Theorem B (cf. [12]) Let 0 < p 6 1, 0 < q 6 ∞ and s ∈ R. Then, Mp,q ⊂ hp if and only if s > B(p, q) with strict inequality when 1/p > 1/q. s Theorem C (cf. [13]) Let 1 6 p, q 6 ∞ and s ∈ R. Then, Lp ⊂ Mp,q if and only if one of the following conditions holds: (1) p > 1, s 6 A(p, q) with strict inequality when 1/q > 1/p, (2) p = 1, s 6 A(1, q) with strict inequality when q 6= ∞. s Theorem D (cf. [13]) Let 1 6 p, q 6 ∞ and s ∈ R. Then, Mp,q ⊂ Lp if and only if one of the following conditions holds: (1) p < ∞, s > B(p, q) with strict inequality when 1/p > 1/q, (2) p = ∞, s > B(p, q) with strict inequality when q > 1. As mentioned before, α-modulation space is a generalization of modulation space, and that can not be obtained by interpolation between modulation and Besov spaces, so it is interesting to ask what is the sharp conditions for the embedding between α-modulation and local Hardy spaces (Sobolev spaces). The main purpose of this paper is to give the answer. Now, we state our main theorems as follows.

Theorem 1.1. Let α ∈ [0, 1), 0 < p1 < ∞, 0 < p2 , q2 6 ∞, s2 ∈ R. Then ,α hp1 ⊂ Mps22,q 2

(1.1)

s2 6 nα(1/p2 − 1/p1 ) + (1 − α)A(p1 , q2 ),

(1.2)

if and only if 1/p2 6 1/p1 and

with strict inequality in (1.2) when 1/q2 > 1/p1 .

Theorem 1.2. Let α ∈ [0, 1), 0 < p2 < ∞, 0 < p1 , q1 6 ∞, s1 ∈ R. Then ,α ⊂ hp2 Mps11,q 1

(1.3)

s1 > nα(1/p1 − 1/p2 ) + (1 − α)B(p2 , q1 ),

(1.4)

if and only if 1/p2 6 1/p1 and

with strict inequality in (1.4) when 1/p2 > 1/q1 .

Theorem 1.3. Let α ∈ [0, 1), 0 < p, q 6 ∞, s ∈ R. Then if and only if 1/p 6 1 and

s,α L1 ⊂ Mp,q

s 6 nα(1/p − 1) + n(1 − α)A(1, q) with the strict inequality in (1.6) when q < ∞.

(1.5) (1.6)

EMBEDDING BETWEEN LOCAL HARDY AND α-MODULATION SPACES

Theorem 1.4. Let α ∈ [0, 1), 0 < p, q 6 ∞, s ∈ R. Then s,α Mp,q ⊂ L1

if and only if 1/p > 1, and

(1.7)

s > nα(1/p − 1) + (1 − α)B(1, q) with the strict inequality in (1.8) when q > 1.

(1.8)

Theorem 1.5. Let α ∈ [0, 1), 0 < p, q 6 ∞, s ∈ R. Then s,α L∞ ⊂ Mp,q

if and only if p = ∞ and

(1.9)

s 6 n(1 − α)A(∞, q) with the strict inequality in (1.10) when q < ∞.

(1.10)

Theorem 1.6. Let α ∈ [0, 1), 0 < p, q 6 ∞, s ∈ R. Then s,α Mp,q ⊂ L∞

if and only if with strict inequality when q > 1.

3

(1.11)

s > nα/p + n(1 − α)B(∞, q),

Remark 1.7. Obviously, our theorems generalize the main results in [12] and [13]. If we choose p1 = p2 and α = 0 in our theorems, we regain the results obtained in [12, 13]. Furthermore, we use a quite different method to achieve our goals. For the necessity part, we reduce the embedding between local Hardy and α-modulation spaces to the corresponding embedding associated with discrete sequences. For the sufficiency, we estimate the hp atoms under the ”standard” norm of α-modulation spaces. Our method also works well in the special case α = 0. Without using an equivalent norm of modulation space (see Lemma 2.2 in [12]), our method seems more readable and efficient. We also remark that a similar embedding problem (the case p1 = p2 ) is studied by T. Kato in his doctoral thesis [11], in which he use the same method as in [12]. Due to the generality of p1 , p2 and the quite different method, our work has its own interesting. Remark 1.8. Recalling hp ∼ Lp for p ∈ (1, ∞). Except for some endpoints, it is convenient to deal with the case p 6 1 and p > 1 together. Thus, we establish Theorem 1.1 and Theorem 1.2 for all the embedding between α-modulation and local Hardy spaces in the full range. Then, we use Theorem 1.3 to 1.6 to deal with the endpoint cases. Remark 1.9. As mentioned before, Besov space can be regarded as the special α-modulation space with α = 1. However, the embedding relations with α = 1 are the special cases of embedding between Besov and Triebel-Lizorkin spaces, which have been fully studied. So we focus on α ∈ [0, 1) in this paper. Our paper is organized as follows. In Section 2, we give some definitions of function spaces treated in this paper. We also collect some basic properties which is useful in our proof. Section 3 is prepared for some estimates of hp -atoms, which lead to the key embedding relations of our theorems. The proofs of our main results will be given in Section 4 to 7. 2. Preliminaries and Definitions n

Let S := SS(R ) be the Schwartz space and S ′ := S ′ (Rn ) be the space of tempered distributions. Let N0 := Z+ {0} be the collection of all nonnegative integers. We define the Fourier transform F f and the inverse Fourier transform F −1 f of f ∈ S (Rn ) by Z Z −2πix·ξ −1 ˆ ˆ f (ξ)e2πix·ξ dξ. f (x)e dx, F f (x) = f (−x) = F f (ξ) = f (ξ) = Rn

Rn

The notation X . Y denotes the statement that X 6 CY , with a positive constant C that may depend on n, α, pi , qi , si (i = 1, 2), but it might be different from line to line. The notation X ∼ Y

4

GUOPING ZHAO, GUILIAN GAO, AND WEICHAO GUO

means the statement X . Y . X. And the notation X ≃ Y stands for X = CY . For a multi-index k = (k1 , k2 , ..., kn ) ∈ Zn , we denote |k|∞ := max |ki | and hki := (1 + |k|2 )1/2 . i=1,2,...,n

We recall some definitions of the function spaces treated in this paper. First we give the partition of unity on frequency space associated with α ∈ [0, 1). Take two appropriate constants c > 0 and C > 0 and choose a Schwartz function sequence {ηkα }k∈Zn satisfying  α α α  |ηk (ξ)| > 1, if |ξ − hki 1−α k| < chki 1−α ;  α α  suppη α ⊂ {ξ ∈ Rn : |ξ − hki 1−α k| < Chki 1−α }; k P (2.1) α n  η (ξ) ≡ 1, ∀ξ ∈ R ;  k∈Zn k  α|γ|  |∂ γ η α (ξ)| 6 C hki− 1−α , ∀ξ ∈ Rn , γ ∈ (Z+ ∪ {0})n . k

|α|

constitutes a smooth decomposition of Rn . The frequency decomposition The sequence operators associated with the above function sequence are defined by {ηkα }k∈Zn

−1 α α ηk F k := F

(2.2)

n

for k ∈ Z . Let 0 < p, q 6 ∞, s ∈ R, α ∈ [0, 1). Then the α-modulation space associated with above decomposition is defined by !1/q X sq q α s,α n ′ n 1−α s,α kk f kLp Mp,q (R ) = {f ∈ S (R ) : kf kMp,q < ∞} (2.3) hki (Rn ) = k∈Zn

s s,0 0,0 with the usual modifications when q = ∞. For simplicity, we write Mp,q = Mp,q , Mp,q = Mp,q and 0 ηk (ξ) = ηk (ξ).

Remark 2.1. We recall that the above definition is independent of the choice of exact ηkα (see [10]). Also, for sufficiently small δ > 0, one can construct a function sequence {ηkα }k∈Zn such that ηkα (ξ) = 1, α α and ηkα (ξ)ηlα (ξ) = 0 if k 6= l and ξ lies in the ball B(hki 1−α k, hki 1−α δ) (see [3, 7]). Lemma 2.2 (Embedding of α-modulation spaces). Let 0 < p1 , p2 , q1 , q2 6 ∞, s1 , s2 ∈ R, α ∈ [0, 1). ,α ,α if and only if ⊂ Mps22,q Then Mps11,q 2 1 1 1 1 1    p2 6 p1  p2 6 p1 1 1 1 1 (2.4) or q 6 q1 q > q1    s22  s22 s1 s1 α α α α 1−α 1−α − − − − 6 + < + . n p2 n p1 n p2 q2 n p1 q1

The above embedding lemma can be verified by the same method used in the proof of embedding between modulation spaces. We omit the details here. One can see [9] for a more general case.. Next, we give some definitions and properties of sequences associated with α.

Definition 2.3. Let 0 < p, q 6 ∞, s ∈ R, α ∈ [0, 1). Let ~a := {ak }k∈Zn be a sequence, we denote its lps,α (quasi-)norm  !1/p  X sp  p  1−α  |ak | hki , p1

sup

.

eα hkin(1−1/p) k k aχ(Q∗ )c kLp

n 1

p 1 ( n |Q|hki 1−α )n(1−1/p) 6 1,

|Q|hki 1−α >1

p 1 n |Q|hki 1−α < 1

EMBEDDING BETWEEN LOCAL HARDY AND α-MODULATION SPACES

9

where we use n(1 − 1/p) 6 0. Thus, we deduce that eα sup hkin(1−1/p) k k aχ(Q∗ )c kLp

k∈Zn

.

sup

√ n

k∈Zn

1

eα hkin(1−1/p) k k aχ(Q∗ )c kLp +

|Q|hki 1−α 1

 Proposition 3.3. Let 0 < p < 1, we have kakM n(1−α)(1−2/p),α . 1 p,p

p

for all h -atoms. Proof. Take a to be an hp atom, and without loss of generality we may assume that √ a is supported in a cube Q centered at the origin. We denote Q∗ the cube with side length 2 nl(Q) having the same center as Q, where the l(Q) is the side length of Q. By the same argument as in the proof of Proposition 3.2, we only need to verify eα kak fn(1−α)(1−2/p),α := k{hkin(1−2/p) k k akLp }k∈Zn klp . 1. Mp,p

By the (quasi-)triangle inequality, we have

n(1−2/p) e α kakM kk akLp }k∈Zn klp fn(1−α)(1−2/p),α =k{hki p,p

n(1−2/p) e α eα kk aχ(Q∗ )c kLp }k∈Zn klp .k{hkin(1−2/p) k k aχQ∗ kLp }k∈Zn klp + k{hki

e α aχQ∗ kLp . 1 obtained in the proof of Proposition 3.2, we deduce that Recalling k k n(1−2/p) eα }k∈Zn klp . 1, k{hkin(1−2/p) k k aχQ∗ kLp }k∈Zn klp . k{hki

(3.7)

where we use the fact p < 1. On the other hand, by the same method as in the proof of Proposition 3.2, we obtain that eα 1 k lp k{hkin(1−2/p) k k aχ(Q∗ )c kLp } √ n

|Q|hki 1−α 1

1/p

p  hki−n (hki 1−α )n(1−1/p) 

(3.9) . 1,

>1

where we use the fact p < 1. Thus, n(1−2/p) e α eα kk aχ(Q∗ )c kLp } √ k{hkin(1−2/p) k k aχ(Q∗ )c kLp }k∈Zn klp .k{hki n

1

|Q|hki 1−α 1

klpk klpk . 1.

(3.10) 

10

GUOPING ZHAO, GUILIAN GAO, AND WEICHAO GUO

4. Proof of Theorem 1.1 ,α . In this section, we give the proof of Theorem 1.1, i.e. the sharp conditions of hp1 ⊂ Mps22,q 2

4.1. Necessity of Theorem 1.1. To verify the necessity, we first show that the embedding hp1 ⊂ ,α actually implies the embedding about corresponding weighted sequences. Mps22,q 2 ,α Proposition 4.1. Let 0 < p1 < ∞, 0 < p2 , q2 6 ∞, s ∈ R. If hp1 ⊂ Mps22,q , then we have 2 (1) 1/p2 6 1/p1 ,

(2) 2 +nα(1−1/p2 )+s2 ,1 1 ),1 , ⊂ lqn(1−α)/q lpn(1−1/p 2 1

(3)

2 )+s2 ,α 1 ),α . ⊂ lqnα(1−1/p lpnα(1−1/p 2 1

Proof. We only state the proof for q2 < ∞, since the case of q2 = ∞ can be treated by the same method with a slight modification. By the Littlewood-Paley characterization of local Hardy space , we actually have ,α . kf k 0 (4.1) kf kMps2,q Fp ,2 . 2

2

1

Let f ∈ S be a nonzero function whose Fourier transform has compact support in B(0, 1). Set c fλ (ξ) = fb(ξ/λ), λ > 0. Using (4.1) and the local property of α-modulation and local hardy space, we obtain kfλ kLp2 . kfλ kLp1 for sufficiently small λ and consequently λn(1−1/p2 ) . λn(1−1/p1 ) . Letting λ → 0+ , we deduce 1/p2 6 1/p1 , (4.2) which is the first desired condition. Let g be a nonzero Schwartz function whose Fourier transform has compact support in {ξ : 3/4 6 |ξ| 6 4/3}, satisfying g(ξ) = 1 on {ξ : 7/8 6 |ξ| 6 8/7} . Set gbj (ξ) := b g(ξ/2j ). By the definition of ∆j , we have ∆j gj = gj for j > 0, and ∆l gj = 0 if l 6= j. Denote  fj = k ∈ Zn : suppηkα ⊂ {ξ : (7/8) · 2j 6 |ξ| 6 (8/7) · 2j } , Γ (4.3) fj | ∼ 2jn(1−α) for j > N , where N is a sufficiently large number. For a truncated (only we have |Γ finite nonzero items) nonnegative sequence ~a = {aj }j∈N , we define X GhN := aj gjh , gjh (x) := gj (x + jhe0 ), (4.4) j>N

where h ∈ R, e0 = (1, 0, · · · , 0) is the unit vector of Rn . By the definition of α-modulation space, we obtain that 1/q2 !1/q2  X X X s2 q2 s2 q2 h q2 h q2  ,α = > hki 1−α kα kGhN kMps2,q hki 1−α kα k GN kLp2 k GN kLp2 2



=

2

∞ X X

j=N k∈Γ fj



∼

∞ X

X

j=N k∈Γ fj



=

∞ X

j=N

k∈Zn

aqj 2 kF −1 ηkα kqL2p2 hki

j∈N0 k∈Γ fj

s2 q2 1−α

1/q2 

1/q2

aqj 2 2jq2 (nα(1−1/p2 )+s2 ) 



∼ 

∼

1/q2

aqj 2 2jq2 (nα(1−1/p2 )+s2 ) 2jn(1−α) 

∞ X X

j=N k∈Γ fj

∞ X

j=N

aqj 2 hki

nαq2 1−α

1/q2

(1−1/p2 ) js2 q2 

2

1/q2

fj |aq2 2jq2 (nα(1−1/p2 )+s2 )  |Γ j

= k{aj }j>N kln(1−α)/q2 +nα(1−1/p2 )+s2 ,1 . q2

(4.5)

EMBEDDING BETWEEN LOCAL HARDY AND α-MODULATION SPACES

On the other hand, using the orthogonality of {gjh } as h → ∞, we deduce that



 1/2 1/2

∞

X





 X

h h 2 h 2 kGN kFp0 ,2 = |∆j GN | = |aj gj |

1



j∈N0

p

j=N

p L 1 L 1  p1 /2 1/p1   1/p1 Z Z X ∞ ∞ X h→∞   h 2 p  = |aj gj |  dx −−−−→  |aj gj | 1 dx Rn



≃

∞ X

j=N

11

(4.6)

Rn j=N

j=N

1/p1

apj 1 2jn(1−1/p1 )p1 

= k{aj }j>N kln(1−1/p1 ),1 . p1

Combining (4.6) and (4.5), we have

k{aj }j>N kln(1−α)/q2 +nα(1−1/p2 )+s2 ,1 . k{aj }j>N kln(1−1/p1 ),1 , p1

q2

n(1−α)/q2 +nα(1−1/p2 )+s2 ,1 n(1−1/p1 ),1 . ⊂ lq2 which implies the desired embedding lp1 nα(1−1/p2 )+s2 ,α nα(1−1/p1 ),α . Let f ∈ S be a nonzero smooth ⊂ lq2 Next, we turn to the proof of lp1 α α function whose Fourier transform has small  support, such that k fk = fk and l fk = 0 if k 6= l,  α

where we denote fbk (x) = fb

ξ−hki 1−α k α hki 1−α

sequence ~b = {bk }k∈Zn , we define

F h (x) =

X

. For a truncated (only finite nonzero items) nonnegative

bk fkh ,

fkh (x) = fk (x − kh),

k∈Zn

where h ∈ R. By a direct computation, we have X

h

,α = kF kMps2,q 2

2

k∈Zn

X

∼

k∈Zn

s2 q2 bqk2 hki 1−α

s2 q2 bqk2 hki 1−α

kfk kqL2p2 hki

(4.7)

!1/q2

nα 1−α (1−1/p2 )q2

!1/q2

(4.8) ∼ k~bklnα(1−1/p2 )+s2 ,α . q2

On the other hand, for p1 > 1, using the orthogonality of {fkh }k∈Zn as h → ∞, we obtain p1 ! p1 ! p1 Z X Z X 1 1 h→∞ p1 h h h p bk fk dx kF kh 1 ∼ kF kLp1 = −−−−→ |bk fk | dx Rn k∈Zn Rn k∈Zn !1/p1 X p nα ∼ k~bk nα(1−1/p1 ),α . ∼ b 1 hki 1−α (1−1/p1 )p1 k

lp 1

k∈Zn

For p1 6 1, we use quasi-triangle inequality to deduce that X X X kF h kph1p1 = k bk fkh kph1p1 6 kbk fkh kph1p1 = kbk fk kpL1p1 k∈Zn

∼

X

k∈Zn

k∈Zn

nα bpk1 hki 1−α (1−1/p1 )p1

k∈Zn

∼ k~bk nα(1−1/p1 ),α , p1

lp 1

where we use the fact kfk kLp1 ∼ kfk kLp1 (see Proposition 2.6). So we have 1 . lim kF h khp1 . k~bkpnα(1−1/p 1 ),α

h→∞

lp 1

(4.9)

12

GUOPING ZHAO, GUILIAN GAO, AND WEICHAO GUO

Combining (4.8) and (4.9), we obtain k~bklnα(1−1/p2 )+s2 ,α . k~bklnα(1−1/p1 ),α q2

(4.10)

p1

nα(1−1/p1 ),α

which implies the desired embedding lp1

nα(1−1/p2 )+s2 ,α

⊂ lq2

.



,α . By Proposition 4.1, we obtain Now, we are in the position to verify the necessity for hp1 ⊂ Mps22,q 2 1/p2 6 1/p1 and 2 )+s2 ,α 1 ),α . ⊂ lqnα(1−1/p lpnα(1−1/p 2 1

2 +nα(1−1/p2 )+s2 ,1 1 ),1 , ⊂ lqn(1−α)/q lpn(1−1/p 2 1

n(1−α)/q +nα(1−1/p )+s ,1

n(1−1/p ),1

2 2 2 1 implies s2 6 nα(1/p2 − 1/p1 ) + n(1 − α)(1 − ⊂ lq2 By Lemma 2.4, lp1 1/p1 − 1/q2 ) and the inequality is strict if 1/q2 > 1/p1 . On the other hand, by Lemma 2.4 and nα(1−1/p2 )+s2 ,α nα(1−1/p1 ),α , we obtain s2 6 nα(1/p2 − 1/p1 ) for 1/q2 6 1/p1 , and s2 < n(1 − ⊂ lq2 lp1 α)(1/p1 − 1/q2 ) + nα(1/p2 − 1/p1 ) for 1/q2 > 1/p1 . Combining with the above estimates, we obtain s2 6 nα(1/p2 − 1/p1 ) + (1 − α)A(p1 , q2 ), with strict inequality for 1/q2 > 1/p1 .

(1−α)A(p ,q ),α

1 2 for 1/q2 6 1/p1 , 4.2. Sufficiency of Theorem 1.1. We only need to verify hp1 ⊂ Mp1 ,q2 (1−α)A(p ,q )−ǫ,α 1 2 p1 and h ⊂ Mp1 ,q2 for 1/q2 > 1/p1 , where ǫ is any positive number. Then the fi(1−α)A(p1 ,q2 ),α nα(1/p −1/p1 )+(1−α)A(p1 ,q2 ),α (1−α)A(p1 ,q2 )+ǫ,α nal conclusion follows by Mp1 ,q2 ⊂ Mp2 ,q2 2 or Mp1 ,q2 ⊂ nα(1/p2 −1/p1 )+(1−α)A(p1 ,q2 )+ǫ,α Mp2 ,q2 . For 1/q2 6 1/p1 . We want to verify that 1 ,q2 ),α . hp1 ⊂ Mp(1−α)A(p 1 ,q2

Taking a fixed f ∈ hp1 (p1 6 1), we can find a collection of hp1 -atoms {aj }∞ j=1 and a sequence of P∞ ∞ complex numbers {λj }j=1 which is depend on f , such that f = j=1 λj aj and  1/p ∞ X  |λ|p  (4.11) 6 Ckf khp , j=1

where the constant C is independent of f . For p1 6 1, we use Proposition 3.2 to deduce that kf kM (1−α)A(p1 ,∞),α = kf kM n(1−α)(1−1/p1 ),α =k p1 ,∞

p1 ,∞

∞ X j=1



. 

Similarly, we use Proposition 3.3 to deduce that

.

λj aj kM n(1−α)(1−1/p1 ),α

∞ X j=1

∞ X j=1

p1 ,∞

1/p1

kλj aj kp1 n(1−α)(1−1/p1 ),α  Mp1 ,∞

1/p1

|λj |p1 

. kf khp1 .

kf kM (1−α)A(p1 ,p1 ),α = kf kM n(1−α)(1−2/p1 ),α . kf khp1 p1 ,p1

p1 ,p1

(4.12)

(4.13)

for p1 < 1. By a direct calculation, we also have 0,α kf kM (1−α)A(∞,∞),α = kf kM∞,∞ = sup kα k f kL∞ . kf kL∞ . ∞,∞

(4.14)

k∈Zn

Recalling kf kM (1−α)A(2,2),α = kf kM 0,α ∼ kf kL2 , we obtain the following inequality: 2,2

2,2

kf kM (1−α)A(2,2),α . kf kL2 . 2,2

By an interpolation among (4.12),(4.13),(4.14) and (4.15), we obtain the desired conclusion.

(4.15)

EMBEDDING BETWEEN LOCAL HARDY AND α-MODULATION SPACES

13

For 1/q2 > 1/p1 . We want to verify that 1 ,q2 )−ǫ,α , hp1 ⊂ Mp(1−α)A(p 1 ,q2

where ǫ is any fixed positive number. In fact, observing that ((1 − α)A(p1 , q2 ) − ǫ)/n + (1 − α)/q2 < ((1 − α)A(p1 , p1 )/n + (1 − α)/p1 for 1/q2 > 1/p1 , we use Lemma 2.2 to deduce 1 ,q2 )−ǫ,α 1 ,p1 ),α . ⊂ Mp(1−α)A(p Mp(1−α)A(p 1 ,q2 1 ,p1

(1−α)A(p1 ,p1 ),α

Recalling hp1 ⊂ Mp1 ,p1

, we obtain the desired conclusion:

1 ,q2 )−ǫ,α 1 ,p1 ),α ⊂ Mp(1−α)A(p hp1 ⊂ Mp(1−α)A(p 1 ,q2 1 ,p1

for any fixed positive number. 5. Proof of Theorem 1.2 5.1. Necessity of Theorem 1.2. As in the proof of Theorem 1.1, we use following proposition ,α ⊂ hp1 can be reduced to the necessity of embedding between to show that the necessity of Mps22,q 2 corresponding weighted sequences. ,α ⊂ hp2 , then we have Proposition 5.1. Let 0 < p2 < ∞, 0 < p1 , q1 6 ∞, s1 ∈ R. If Mps11,q 1

(1)

1/p2 6 1/p1 , (2) 2 ),1 1 +nα(1−1/p1 )+s1 ,1 , ⊂ lpn(1−1/p lqn(1−α)/q 2 1

(3) 2 ),α 1 )+s1 ,α . ⊂ lpnα(1−1/p lqnα(1−1/p 2 1

Proof. Take f ∈ S to be a nonzero smooth function whose Fourier transform has compact support b in B(0, 1). Denote fc λ (ξ) = f (ξ/λ), λ > 0. By the assumption, we have ,α kfλ kFp0 ,2 . kfλ kMps1,q 2

1

1

(5.1)

for λ 6 1. By the local property of α-modulation and Triebel space, we actually have kfλ kLp2 . kfλ kLp1 , and consequently λn(1−1/p2 ) . λn(1−1/p1 ) . Letting λ → 0+ , we conclude 1/p2 6 1/p1 .

(5.2)

Let g be a nonzero Schwartz function whose Fourier transform has compact support in {ξ : 3/4 6 |ξ| 6 4/3}. Set gbj (ξ) := b g(ξ/2j ). By the definition of ∆j , we have ∆j gj = gj for j > 0, and ∆l gj = 0 if l 6= j. Denote  Γj = k ∈ Zn : suppηkα ∩ {ξ : (3/4) · 2j 6 |ξ| 6 (4/3) · 2j } 6= ∅ . (5.3)

We have |Γj | ∼ 2jn(1−α) . For a truncated (only finite nonzero items) nonnegative sequence ~a = {aj }j∈N0 , we define X Gh := aj gjh , gjh (x) := gj (x + jhe0 ), (5.4) j∈N0

14

GUOPING ZHAO, GUILIAN GAO, AND WEICHAO GUO

where h ∈ R, e0 = (1, 0, · · · , 0) is the unit vector in Rn . By the definition of α-modulation space, we obtain that  q1  ! q1 1 1 X X X s1 q1 s1 q1 q q α h α h h 1 1   1−α 1−α ,α = kk G kLp1 kaj k gj kLp1 kG kMps1,q . hki hki 1



.

1

X X

j∈N0 k∈Γj



∼

X X

j∈N0 k∈Γj



=

X

j∈N0

k∈Zn

j∈N0 k∈Γj

 q1



1

s1 q1 aqj 1 kF −1 ηkα kqL1p1 hki 1−α



 q1

1

aqj 1 2jq1 (nα(1−1/p1 )+s1 ) 

∼ 

∼

 q1

X X

j∈N0 k∈Γj

X

j∈N0

1

aqj 1 2jq1 (nα(1−1/p1 )+s1 ) 2jn(1−α) 

 q1

1

nαq1 aqj 1 hki 1−α (1−1/p1 ) 2js1 q1 

(5.5)

 q1

1

|Γj |aqj 1 2jq1 (nα(1−1/p1 )+s1 ) 

= k~akln(1−α)/q1 +nα(1−1/p1 )+s1 ,1 . q1

On the other hand, by the same argument as in the proof of Proposition 4.1, we deduce that lim kGh kLp2 = k~akln(1−1/p2 ),1 .

h→∞

(5.6)

p2

Since Gh is a Schwartz function, by the definition of local Hardy space, we have kGh kLp2 . kGh khp2 . Thus, (5.7) k~akln(1−1/p2 ),1 = lim kGh kLp2 . lim kGh khp2 . h→∞

h→∞

p2

Combining (5.5) and (5.7), we have k~akln(1−1/p2 ),1 . k~akln(1−α)/q1 +nα(1−1/p1 )+s1 ,1 , p2

q1

n(1−1/p2 ),1 n(1−α)/q1 +nα(1−1/p1 )+s1 ,1 . ⊂ lp2 which implies the desired embedding lq1 nα(1−1/p2 ),α nα(1−1/p1 )+s1 ,α . Let f ∈ S be a nonzero ⊂ lp2 Next, we turn to verify lq1 α α whose Fourier transform has small support, such that  k fk = fk and l fk = 0 if  

denote fbk (x) = fb

α

ξ−hki 1−α k α hki 1−α

~b = {bk }k∈Zn , we define

smooth function k 6= l, where we

. For a truncated (only finite nonzero items) nonnegative sequence

F h (x) =

X

k∈Zn

bk fkh ,

fkh (x) = fk (x − kh),

(5.8)

where h ∈ R. Observing that F h is a Schwartz function and kF h kLp2 6 kF h khp2 , we use the assumption to deduce ,α . (5.9) kF h kLp2 . kF h kMps1,q 1

1

By the same argument as in the proof of Proposition 4.1, we obtain kF h k s1 ,α ∼ k~bk nα(1−1/p1 )+s1 ,α , lim kF h kLp2 ∼ k~bk nα(1−1/p2 ),α , h→∞

Mp1 ,q1

lp 2

lq1

which implies k~bklnα(1−1/p2 ),α . k~bklnα(1−1/p1 )+s1 ,α . p2

q1

By the arbitrary of ~b, we obtain the desired conclusion.

(5.10) 

,α ⊂ hp2 . Using Proposition 5.1, we obtain 1/p2 6 1/p1 Now, we turn to verify the necessity of Mps11,q 1 and the embedding relations: 2 ),1 1 +nα(1−1/p1 )+s1 ,1 , ⊂ lpn(1−1/p lqn(1−α)/q 2 1

n(1−α)/q +nα(1−1/p )+s ,1

n(1−1/p ),1

2 ),α 1 )+s1 ,α . ⊂ lpnα(1−1/p lqnα(1−1/p 2 1

2 1 1 1 implies s1 > nα(1/p1 − 1/p2 ) + n(1 − α)(1 − ⊂ lp2 By Lemma 2.4, lq1 1/p2 − 1/q1 ) and the inequality is strict if 1/p2 > 1/q1 . On the other hand, by Lemma 2.4 and

EMBEDDING BETWEEN LOCAL HARDY AND α-MODULATION SPACES

15

nα(1−1/p ),α

nα(1−1/p )+s ,α

2 1 1 , we obtain s1 > nα(1/p1 − 1/p2) for 1/p2 6 1/q1 , and s1 > nα(1/p1 − ⊂ lp2 lq1 1/p2 ) + n(1 − α)(1/p2 − 1/q1 ) for 1/p2 > 1/q1 . Combining with the above estimates, we obtain s1 > nα(1/p1 − 1/p2) + (1 − α)B(p2 , q1 ), with strict inequality for 1/p2 > 1/q1 .

5.2. Sufficiency of Theorem 1.2. As in the proof of Theorem 1.1, we actually only need to verify (1−α)B(p2 ,q1 )+ǫ,α (1−α)B(p2 ,q1 ),α ⊂ hp2 for 1/p2 > 1/q1 , where ǫ is any ⊂ hp2 for 1/p2 6 1/q1 , and Mp2 ,q1 Mp2 ,q1 positive number. For 1/p2 6 1/q1 . We want to verify 2 ,q1 ),α ⊂ hp2 . Mp(1−α)B(p 2 ,q1

(5.11)

for 1/q1 > (1 − 1/p2 ) ∨ 1/p2 , where 1/qe1 = ⊂ Mp0,α Observing that B(p2 , q1 ) = 0 and Mp0,α 2 ,q1 2 ,qe1 (1 − 1/p2 ) ∨ 1/p2 , we only need to prove (5.11) for 1/p2 6 1/q1 6 1 − 1/p2 and p2 = q1 6 2. By a dual argument, in 1/p2 6 1/q1 6 1 − 1/p2, (5.11) can be verified by the sufficiency of Theorem 1.1 proved before. Thus, we only need to verify (5.11) for p2 = q1 6 2. Observing that (5.11) has been verified for p2 = q1 = 2, by an interpolation argument, we only need to show (5.11) for p2 = q1 6 1. In fact, when p2 = q1 6 1, we use the quasi-triangle inequality to deduce that !1/p2 !1/p2 X X X p p 2 2 kf khp2 = k α ∼ = kf kMp0,α,q , (5.12) kα kα k f k hp 2 6 k f k hp 2 k f k L p2 k∈Zn

k∈Zn

2

k∈Zn

1

which is just the embedding (5.11) for p2 = q1 6 1. For 1/p2 > 1/q1 . We want to verify

2 ,q1 )+ǫ,α ⊂ hp2 Mp(1−α)B(p 2 ,q1

(1−α)B(p2 ,p2 ),α

for any fixed positive number ǫ. In fact, by the embedding Mp2 ,p2 observing that

⊂ hp2 proved before,

n(1 − α)/p2 + (1 − α)B(p2 , p2 ) < n(1 − α)/q1 + (1 − α)B(p2 , q1 ) + ǫ, we use Lemma 2.2 to deduce 2 ,p2 ),α 2 ,q1 )+ǫ,α ⊂ hp2 , ⊂ Mp(1−α)B(p Mp(1−α)B(p 2 ,p2 2 ,q1

which is the desired conclusion. 6. Embedding between L1 and α-modulation spaces 6.1. Proof of Theorem 1.3. We first give the proof for necessity. It is known that hp = Lp when p > 1, but h1 $ L1 , due to the different structure between h1 and L1 , we have the a additional s,α restriction associated with L1 ⊂ Mp,q . s,α Proposition 6.1. Let 0 < p 6 ∞ 0 < q < ∞, s ∈ R, then L1 ⊂ Mp,q implies

s < nα(1/p − 1) + n(1 − α)A(1, q) Proof. Take f to be a smooth function satisfying that suppfb ⊂ B(0, 2) and fb(ξ) = 1 on B(0, 1). Denote fbj (ξ) = f (ξ/2j ) for j > 1. By the scaling of L1 , we obtain kfj kL1 ∼ 1 for all j. However, by the assumption of fj , for any finite subset of Zn , denoted by A, we can find a sufficiently large J such that ! 1q ! q1 X X sq sq nα ∼ hki 1−α kF −1 ηkα kqLp hki 1−α hki 1−α (1−1/p)q k∈A

k∈A

.

X

k∈Zn

hki

sq 1−α

q kα k fJ kLp

! q1

s,α . kf k 1 . 1. ∼ kfJ kMp,q J L

16

GUOPING ZHAO, GUILIAN GAO, AND WEICHAO GUO

By the arbitrary of A, we actually obtain X

k∈Zn

hki

sq 1−α

hki

nα 1−α (1−1/p)q

! q1

. 1.

which yields that s < nα(1/p − 1) + n(1 − α)(−1/q) = nα(1/p − 1) + n(1 − α)A(1, q).



By the same method used to prove Proposition 4.1, we obtain s 6 nα(1/p − 1) + n(1 − α)A(1, q).

(6.1)

Then, we use Proposition 6.1 to conclude that inequality (6.1) must be strict when q 6= ∞. Next, we turn to the sufficiency part. When q = ∞, we have p > 1 and s 6 nα(1/p − 1). Using Young’s inequality, we deduce that s

s

−1 α α s,α = sup k f k p hki 1−α . sup kF ηk kLp hki 1−α kf kL1 kf kMp,∞ L k k∈Zn

k∈Zn

nα

(6.2)

s

. sup hki 1−α (1−1/p) hki 1−α kf kL1 . kf kL1 . k∈Zn

For q < ∞, taking ǫ to be a fixed positive number, observing that n(1 − α)/q + nα(1/p − 1) + n(1 − α)A(1, q) − ǫ < nα(1/p − 1) , we use Lemma 2.2 to deduce kf kM nα(1/p−1)+n(1−α)A(1,q))−ǫ,α . kf kM nα(1/p−1),α . kf kL1 . p,q

(6.3)

p,∞

6.2. Proof of Theorem 1.4. By the same method used in the proof of Proposition 5.1, we can verify the necessity of Theorem 1.4. On the other hand, using Theorem 1.2 and the fact that h1 ⊂ L1 , we get the sufficiency of Theorem 1.4. 7. Embedding between L∞ and α-modulation spaces 7.1. Proof of Theorem 1.5. First, we give the proof for necessity. By the same method used in the proof Proposition 4.1, we deduce 1/p 6 1/∞, which yields p = ∞. We also have s 6 n(1 − α)A(∞, q) with strict inequality when q 6= ∞. Next, we turn to the sufficiency part. For q = ∞, we have s 6 0. Thus, we have s,α 0,α = sup kα 6 kf kM∞,∞ kf kM∞,∞ k f k∞ 6 sup kf k∞ 6 kf kL∞ .

k∈Zn

k∈Zn

When q 6= ∞ and s < −n(1 − α)/q, by Young’s inequality, we have ! 1q ! q1 X X sq sq q 1−α s,α = hki 1−α . kf kL∞ . kf kL∞ , kα kf kM∞,q k f kL∞ hki

(7.1)

k∈Zn

k∈Zn

where we use sq/(1 − α) < −n in the last inequality.

7.2. Proof of Theorem 1.6. First, we give the proof for necessity. By the structure of L∞ , we s,α establish the following proposition for restriction of Mp,q ⊂ L∞ .

s,α Proposition 7.1. Let 0 < p 6 ∞, q > 1, then Mp,q ⊂ L∞ implies

lqs+nα(1−1/p),α ⊂ l1nα,α .

Proof. Take f ∈ S to be a nonzero smooth function whose Fourier transform hassmall support,  such α 1−α k ξ−hki that f (0) = 1, α fk = fk and α fk = 0 if k 6= l, where we denote fbk (x) = fb . For a α k

l

hki 1−α

truncated (only finite nonzero items) nonnegative sequence ~a = {ak }k∈Zn , we define X F (x) = ak f k . k∈Zn

By a direct computation, we have

,α ∼ k~ aklnα(1−1/p2 )+s2 ,α . kF kMps2,q 2

2

q2

(7.2)

EMBEDDING BETWEEN LOCAL HARDY AND α-MODULATION SPACES

On the other hand, observing that F (x) =

P

αn

k∈Zn

kF kL∞ > F (0) =

X

k∈Zn

ak hki 1−α e2πihki

αn

ak hki 1−α f (0) =

X

k∈Zn

α 1−α

k·x

17

α

f (hki 1−α x), we have

αn

. ak hki 1−α = k~aklnα,α 1

Thus, we have s,α ∼ k~ aklnα(1−1/p)+s,α . . kF kL∞ . kF kMp,q k~aklnα,α 1 q

By the arbitrary of ~a, we obtain the desired conclusion.



Using Proposition 7.1 and Lemma 2.4, we deduce that s > nα/p for 1 6 1/q, and s > nα/p + n(1 − α)(1 − 1/q) for 1 > 1/q, which is just the conclusion. Now, we turn to the proof of sufficiency. In fact, by Lemma 2.2, we have that X s,α (7.3) kα kf kL∞ 6 k f kL∞ = kf kM 0,α . kf kMp,q . ∞,1

k∈Rn

where s > nα/p for 1 6 1/q, or s > nα/p + n(1 − α)(1 − 1/q) for 1 > 1/q. Acknowledgements. This work was supported by the National Natural Foundation of China (No. 11601456).

References [1] H. G. Feichtinger, Modulation spaces on locally compact Abelian group, Technical Report, University of Vienna, 1983. Published in: “Proc. Internat. Conf. on Wavelet and Applications”, 99-140. New Delhi Allied Publishers, India, 2003. [2] H.G. Feichtinger, Modulation spaces: looking back and ahead, Sampling Theory in Signal and Image Processing, 5(2) (2006), 109-140. [3] M. Fornasier, Banach frames for α-modulation spaces, Appl. Comput. Harmon. Anal. 22(2) (2007), 157-175. [4] D. Goldberg, A local version of real Hardy spaces, Duke Mathematical Journal, 46(1)(1979), 27-42. [5] P. Gr¨ obner, Banachr¨ aume Glatter Funktionen and Zerlegungsmethoden, Doctoral Thesis, University of Vienna, 1992. [6] K. Gr¨ ochenig, Foundations of Time-Frequency Analysis, Birkh¨ auser, Boston, MA, 2001. [7] W. Guo, J. Chen, Strichartz estimates on α-modulation spaces, Electronic J. Differential Equations 118 (2013), 1-13. [8] W. Guo, D. Fan, H. Wu, G. Zhao, Sharpness of Complex Interpolation on α-Modulation Spaces, Journal of Fourier Analysis and Applications (2015) 1-35. [9] W. Guo, D. Fan, H. Wu, G. Zhao, Full characterization of inclusion relations between α-modulation spaces, arXiv:1606.01386v2. [10] J. Han, B. Wang, α-modulation spaces (I) embedding, interpolation and algebra properties, J. Math. Soc. Japan, 66 (2014), 1315-1373. [11] T. Kato, On modulation spaces and their applications to dispersive equations, Doctoral Thesis, (2016). [12] M. Kobayashi, A. Miyachi, N. Tomita, Embedding relations between local Hardy and modulation spaces, Studia Math, 192(2009), 79-96. [13] M. Kobayashi, M. Sugimoto, The inclusion relation between Sobolev and modulation spaces, J. Funct. Anal., 260 (2011) 11, 3189-3208. [14] K. Okoudjou, Embeddings of some classical Banach spaces into modulation spaces, Proceedings of the American Mathematical Society, 132(6)(2004), 1639-1647. [15] M. Sugimoto, N. Tomita, The dilation property of modulation space and their inclusion relation with Besov spaces, J. Funct. Anal., 248 (2007), 79-106. [16] J. Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus, I, J. Funct. Anal., 207 (2004), 399-429. [17] H. Triebel, Modulation spaces on the euclidean n-space, Z. Anal. Anwend., 2 (5) (1983), 443-457. [18] H. Triebel, Theory of function spaces II, Birkhaser. Basel (1992). [19] M. Ruzhansky, M. Sugimoto, B. Wang, Modulation spaces and nonlinear evolution equations, Progress in Mathematics 301 (2012), 267-283.

18

GUOPING ZHAO, GUILIAN GAO, AND WEICHAO GUO

School of Applied Mathematics, Xiamen University of Technology, Xiamen, 361024, P.R.China E-mail address: [email protected] School of Science, Hangzhou Dianzi University, Hangzhou, 310018, P.R. China E-mail address: [email protected] School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006, P.R.China E-mail address: [email protected]