Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2012, Article ID 145491,29 pages doi:10.1155/2012/145491

Research Article

Dilation Properties for Weighted Modulation Spaces

Elena Cordero1 and Kasso A. Okoudjou

1 Department of Mathematics, University of Torino, Via Carlo Alberto 10,10123 Torino, Italy

2 Department of Mathematics, University of Maryland, College Park, MD 20742, USA

Correspondence should be addressed to Kasso A. Okoudjou, kasso@math.umd.edu Received 31 January 2011; Accepted 7 March 2011 Academic Editor: Hans G. Feichtinger

Copyright © 2012 E. Cordero and K. A. Okoudjou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We give a sharp estimate on the norm of the scaling operator Uxf (x) = f (Xx) acting on the weighted modulation spaces Mps'q(Rd). In particular, we recover and extend recent results by Sugimoto and Tomita in the unweighted case. As an application of our results, we estimate the growth in time of solutions of the wave and vibrating plate equations, which is of interest when considering the well-posedness of the Cauchy problem for these equations. Finally, we provide new embedding results between modulation and Besov spaces.

1. Introduction

The modulation spaces were introduced by Feichtinger [1], by imposing integrability conditions on the short-time Fourier transform (STFT) of tempered distributions. More specifically, for x,w e rd, we let Mw and Tx denote the operators of modulation and translation. Then, the STFT of f with respect to a nonzero window g in the Schwartz class is

Vgf (x, w) measures the frequency content of f in a neighborhood of x.

For s1, s2 e r, and 1 < p,q <œ, the weighted modulation space Mp1qs2 (r2d) is defined to be the Banach space of all tempered distributions f such that

Here and in the sequel, we use the notation

Vs(x) = (x)s = (l + \x\2)5/2. (1.3)

The definition of modulation space is independent of the choice of the window g, in the sense that different window functions yield equivalent modulation-space norms. Furthermore, the dual of a modulation space is also a modulation space, if p < œ, q < œ, (MPf)' = Mp's^-t, where p',q' denote the dual exponents of p and q, respectively.

When both s = t = 0, we will simply write Mp,q = Mp0'q. The weighted Ll space is

2 2 ' exactly M/2, while an application of Plancherel's identity shows that the Sobolev space H2

coincides with M^s. For further properties and uses of modulation spaces, see Grochenig's book [2], and we refer to [3] for equivalent definitions of the modulation spaces for all 0 <

p,q < œ.

The modulation spaces appeared in recent years in various areas of mathematics and engineering. Their relationship with other function spaces has been investigated and resulted in embedding results of modulation spaces into other function spaces such as the Besov and Sobolev spaces [4-6]. Sugimoto and Tomita [5] proved the optimality of certain of the embeddings of modulation spaces into Besov space obtained in [4, 6]. These results were obtained as consequence to optimal bounds of \\Ux\\MP-q^Mpq [5, Theorem 3.1], where Uxf (•) = f (!•) for X> 0. In the sequel, we adopt the following notation:

fx := Uxf. (1.4)

The operator Ux has been investigated on many other function spaces including the Besov spaces. For purpose of comparison with our results, we include the following results summarizing the behavior of Ux on the Besov spaces [7, Proposition 3].

Theorem 1.1. For X e (0, to), s e r,

C-1X-d/p min{1, Xs}\\f\\BPsq < \\fx\\Bpq < CX-d/p max{1, Xs}||/||bp,,. (1.5)

On the modulation spaces, the boundedness of UX was first investigated by Feichtinger [8, Remark 13] for the Feichtinger algebra, that is, M1,1. In fact, this result is a special case of the boundedness of a general class of automorphisms on M1,1. Recently, the general estimate on the norm of UX on the (unweighted) modulation spaces Mp,q(Rd) was obtained by Sugimoto and Tomita [5]. In this paper, we will derive optimal lower and upper bounds for the operator UX on general modulation spaces Mpt'q(Rd). More specifically, the boundedness of UX on Mpt'q is proved in Theorems 3.1, 3.2, and 3.4, and the optimal bounds on \\UX\\Mpq^jC-q are established by Theorems 4.12 and 4.13. We wish to point out that it is not trivial to prove sharp bounds on the norm of the operator UX, as one has to construct examples of functions in the modulation spaces that achieve the desired optimal estimates. We construct such examples by exploiting the properties of Gabor frames generated by the Gaussian window. It is likely that the functions that we construct can play some role in other areas of analysis where the modulation is used, for example, time-frequency analysis of pseudodifferential operators and PDEs.

Interesting applications concern Strichartz estimates for dispersive equations such as the wave equation and the vibrating plate equation on Wiener amalgam and modulation spaces, where the time parameter of the Fourier multiplier symbol is considered as scaling factor. We plan to investigate such applications in a subsequent paper.

Finally, we prove new embeddings between modulation spaces and Besov spaces, generalizing some of the results of [4]. Although strictly speaking this is not an application of the above dilation results, it is clearly in the spirit of the main topic of the present paper, so that we devote a short subsection to the problem.

Our paper is organized as follows. In Section 2, we set up the notation and prove some preliminary results needed to establish our theorems. In Section 3, we prove the complete scaling of weighted modulation spaces. In Section 4, the sharpness of our results are proved, and in Section 5 we point out the applications of our main results.

Finally, we will use the notations A < B to mean that there exists a constant c > 0 such that A < cB, and A x B means that A < B < A.

2. Preliminary

We will use the set and index terminology of the paper [5]. Namely, for 1 < p < to, let p' be the conjugate exponent of p (1/p + 1/p' = 1). For (1/p, 1/q) e [0,1] x [0,1], we define the subsets

I1 = max |

I2 = max

I3 = max

< -\p p'/ q

(i-) < -

\q' 2) - p''

(I1) < i,

\q' 2j - p'

Ij = min

I2* = min

I3 = min

C-^ > i

\p p'/ q (-,-) > 1

\q' 2) - p'

. (11) 1 in( ^ ) > -•

\q' 2 - p

These sets are displayed in Figure 1. We introduce the indices:

Mp 'q) =

M p'q) =

- - 1 q

2 1 — + -

1 -1 q

2 1 p q

I 1\ r.

- e Ij,

II \ r

-,-) e I

1 1 p'q

1 1 p'q

1 1 p'q

1 1 p'q

e h, e I2 e I3^

Next, we prove a lemma that will be used throughout this paper, and which allows us to investigate the action of Ux only on S(Md).

1/2 0 <1 < 1 (a)

Figure 1: The index sets.

1/2 1 > 1 (b)

Lemma 2.1. Let m be a polynomial growing weight function and A a linear continuous operator from s'(rd) to s'(rd). Assume that

a il- ^ cif IU- vf eSRd)

\\Af\Uq < C\\f Hm" f e Mpmq(Rd). (2.4)

Proof. The conclusion is clear if p,q < œ, because in that case S(rd) is dense in Mmq(rd).

Consider now the case p = œ or q = œ. For any given f e M^, consider a sequence fn of Schwartz functions, with fn ^ f in S'(rd), and

\\fn\Uq $\\f \\mZ (2.5)

(see the proof of Proposition 11.3.4 of [2]). Since fn tends to f in S'(rd), Afn tends to Af in S'(rd), and VvAfn tends to VvAf pointwise. Hence, by Fatou's Lemma, the assumptions (2.3) and (2.5) hold,

\\Af Wmz? < lin^œf\Afn\Mmq ^ ^Mm? ~ \\f \\Mmq. (2.6)

We will also make use of the following characterization of the modulation spaces by Gabor frames generated by the Gaussian function, which will be denoted through the paper by ^(x) = e-n|x|2, x e rd. Recall that for 0 < a < 1, the family

G(V'ü' 1) = {yKe(■) = MeTaky = e2niey(-- ak), k,e e Zd],

is a Gabor frame for L2(rd) if and only if there exist 0 < A < B < to such that for all f e L2 we have

A\\fIIL2 < X MM)|2 < B\\fII L2 • (2.8)

Moreover, there exists a dual function y e S such that G(y,a, 1) is also a frame for L2 and every f e L2 can be written as

f = X f >fk,e)fk,e = X {f,yk,e)yk,e- (29)

k/eZd k/eZd

It is easy to see from the isometry of the Fourier transform on L2 and the fact that MgTaky = TeM-aky = e2nmkeM-akTey, that G(y, 1, a) is a Gabor frame whenever G(y, a, 1)is one. The characterization of the modulation spaces by Gabor frame is summarized in the following proposition. We refer to [2, Chapter 9] for a detail treatment of Gabor frames in the context of the modulation spaces. In particular, the next result is proved in [2, Theorem 7.5.3] and describes precisely when the Gaussian function generates a Gabor frame on L2.

Proposition 2.2. G(y, a, 1) is a Gabor frame for L2 if and only if 0 < a < 1. In this case, G(y, a, 1) is also a Banach frame for MpS. for all 1 < p,q < to, and s,t e r. Moreover, f e MPS if and only if there exists a sequence (ck/}k/eZd e £pt'q(Zd x Zd) such that f = £k/eZdCk^yk/ with convergence in the modulation space norm. In addition,

Mpq * := ( £ ( E Mvt(k)p) vs(£)q ) • (2.10)

Zd \keZd

3. Dilation Properties of Weighted Modulation Spaces

We first consider the polynomial weights in the time variables vt(x) = (x) = (1 + |x|2)t/2, t r.

Theorem 3.1. Let 1 < p,q <to, t e r. Then the following are true.

uPA Lt,0

(1) There exists a constant C > 0 such that for all f e Mp'0?, 1 > 1,

C-'i^M min{1,X-^If \ „ < UMI^ < Cld^l(p,q) max{l X^Wf ^ (3.1)

(2) There exists a constant C > 0 such that for all f e Mp^, 0 <1 < 1,

C-Wpq) min{1, l-t}||f \\Apq <Wfx\Uq < Cld"2(P'q) max!1, r^lf (3.2)

Proof. We will only prove the upper halves of each of the estimates (3.1) and (3.2). The lower halves will follow from the fact that 0 < 1 < 1 if and only if 1/1 > 1 and f = UxU\/Xf = Ut/xUxf.

We first consider the case X > 1. Recall the definition of the dilation operator UX given by UXf (x) = f (Xx). Since the mapping f ^ (• )f is an homeomorphism from MPq to Mpqts, t , t0, s e r, see, for example, [9, Corollary 2.3], we have

HUxf Ums?

II Ms?

Using (■)tUXf = UX((X 1 ■ ))f and the dilation properties for unweighted modulation spaces in [5, Theorem 3.1], we obtain

< cx^p?

Xd^1(p,q)

^(x-1-) (Of)

Hence, it remains to prove that the pseudodifferential operator with symbol g(t,X) (x) := (x)-t(X-1xf is bounded on Mp,q and that its norm is bounded above by max{1,X-t}. By [2, Theorem 14.5.2], this will follow once we prove that ||g(t/X)(x)||M»-i < max{1,X-t}. To see this, observe first that

g(i'x) (x) I < max{1'X~1}' Vx e rd.

Indeed, let v(t,X)(x) = (X 1x)i. Consider the case t > 0. Since X > 1, we have X 1 \x\ < \x\ and

v(tX)(x) < (x)t.

Analogously, for t < 0, we have v(t,X)(x) < X-t(x)t. Consequently, we get the desired estimates (3.5).

Using the inclusion Cd+1(rd) Mœ'1(rd), we have

g(i'X)(x)H < sup sup I dag(i'x)(x)|.

llM"'1 'a|<d+1 xeRd

By Leibniz' formula, the estimate |d^(x)t| < (x)t-'^', and (3.5), we see that this last expression is estimated by max{1,X-t}.

This concludes the proof of the upper half of (3.1).

We now consider the case 0 < X < 1. Notice that X < 1 if and only if 1/X > 1, and using the upper half of (3.1) we can write

Ux^ = suPKfx'g)1 = x~d supKf gnx)1 < x~dnf ||ms? suP Hg1/xHM-_'i

< c max{1 r^-^'^fH^ sup ||g|l

where the supremum is taken over all g eS and ||gH^q = 1; hence,

P '? -i'0

\fX\Mp0q < Cma^1,^-V+vV^Wf ^m« = Cma^{1 r^1^\\f(3.8) This establishes the upper half of (3.2). □

We now consider the polynomial weights in the frequency variables vs(w) = {w)s,

s e r.

Theorem 3.2. Let 1 < p,q s e r. Then the following are true.

(1) There exists a constant C > 0 such that for all f e Mpqs, X > 1,

C-1xdMp,q) min{1,Xs}IIf IIMP,q < HfxII^P-q < CXd^1(p-q) max{1,Xs}¡¡f I^- (3.9)

(2) There exists a constant C > 0 such that for all f e Mpqs, 0 < 1 < 1, C^M min{1,XS}||f ||jp;, < ||fx||jp;, < CXdf*l(p,q max{1,XS}||f ||jp;,. (3.10)

Proof. Here we use the fact that the mapping f ^ (D)sf is an homeomorphism from Mp'q to MPS-s, t,s,s0 e r (see [9, Corollary 2.3]). The rest of the proof uses similar arguments as those in Theorem 3.1. □

The next result follows immediately by combining the last two theorems.

Corollary 3.3. Let 1 < p,q <to, t,s e r. Then the following are true.

(1) There exists a constant C > 0 such that for all f e MP'/, X > 1,

C-W^ min{1,X-i} min{1,Xs}IIf I^ < If

Mpq - ifXllMPs

< CXd^1(p'q) max{1,X~i} max{1,Xs

(2) There exists a constant C > 0 such that for all f e Mp'q, 0 < X < 1

(3.11)

C-ixd^1(p'q) min{1,X-i} min{1,Xs}If H^q < If

MPs - 11/^1 API

< CXd^2(p'q) max{1,X~i} max{1,Xs

(3.12)

The following result is an analogue of Corollary 3.3 for modulation spaces defined by nonseparable polynomial growing weight function such as vs(x,w) := ((x,o>))s = (1 + |x|2 +

M2)s/2.

Theorem 3.4. Let 1 < p,q <to, s e r. Then the following are true.

(1) There exists a constant C > 0 such that for all f e Mi'^, X > 1,

C-1!^ min{X-s,Xs}||f || v,q < ||fX|Uq < CX*1™ max{X-s,Xs}||f |U,. (3.13)

(2) There exists a constant C > 0 such that for all f e MVf, 0 <1 < 1,

C-lXd^1(p'?] min{X-s,Xs

< nfxnjlp.? < CXd^(——) max{X~s, Xs

(3.14)

Proof. We assume s > 0. A duality argument can be used to complete the proof when s < 0 (notice, this duality argument will be given explicitly below in the proof of the sharpness of Theorem 3.1 in the case (1/p, 1/q) e I2, t > 0).

Moreover, since the result has been proved in [5, Theorem 3.1] for s = 0, one can use interpolation arguments along with Lemma 2.1 to reduce the proof when s is an even integer.

The mapping f ^ (x,D)sf is an homeomorphism from to Ms'q, s e r (see [9, Theorem 2.2]). Hence

x" M—

'D)sfxII

= ||Ux((x-1x,XD

Xd^1(P'?)|( X-1x,XD)t

x^mIUX-^'XDY

(3.15)

where in the last inequality we used again the dilation properties for unweighted modulation spaces of [5, Theorem 3.1]. Therefore, writing f = (x,D)-s(x,D)sf, we see that it suffices to prove that the pseudodifferential operator

X-^'XD^ (x,Dys (3.16)

is bounded on Mv'q, and its norm is bounded above by max{1,X~s}max{1,Xs} = max{Xs,X-s}. To this end, we observe that, if s is an even integer, (X-1x,XD)s is a finite sum of operators of the form XkxaDP, with |k| < s and |a| + < s. Now, Shubin's pseudodifferential calculus [10] shows that the operators xaDp(x,D)-s have bounded symbols, together with all their derivatives, so that they are bounded on Mp'q. The proof is completed by taking into account the additional factor Xk. □

Finally, it is relatively straightforward to give optimal estimates for the dilation operator UX on the Wiener amalgam spaces W(¥Lps,Lq). These spaces are images of modulation spaces under Fourier transform, that is, FM^ = W(FLs,Lq). It is also worth noticing that the indices ¡i1 and ¡i2 obey the following relations:

Mp'q) = -1 -Mp?)' p'(?) = -1 -P'q) whenever - + — = ? + — = 1.

p p ? ?

(3.17)

Using the above relations along with the definition of the Wiener amalgam spaces, as well as the behavior of the Fourier transform under dilation, that is, f = X~d(f)1/X, and Corollary 3.3, we obtain the following result.

Proposition 3.5. Let 1 < p,q <<x>, t,s e r. Then the following are true.

(1) There exists a constant C > 0 such that for all f e W (¥Lps,Lqt), X > 1,

min{1,Xi} min{1,X-s}||f ||w(FLs,Lt)

< |fX|W(FLs,Lq) (3.18)

< max{1,xt} max{1,X-s}||f ||w(FLs,Lq)-

(2) There exists a constant C > 0 such that for all f e W (FLp,Lq), X < 1,

rt^1^ min{1,Xt} min{1,X-s}||f ||w(FLp,Lq)

< ||fX||w(FLp,Lq) (3.19)

< CXd^2(p''q,) max{1,Xt} max{1,X-s}||f ||w(FLp,Lq).

Remark 3.6. For L e GL(d,R), the operator UL defined by ULf (■) = f (L-) is clearly bounded on Mp'q(Rd). It is interesting to ask whether versions of Theorems 3.1 and 3.2 can be established for UL.

4. Sharpness of Theorems 3.1 and 3.2

In this section, we prove the sharpness of Theorems 3.1 and 3.2. The sharpness of Theorem 3.4 is proved by modifying the examples constructed in the next subsection. Therefore, we omit it. But we first prove some preliminary lemmas in which we construct functions that achieve the optimal bound.

4.1. Preliminary Estimates

The next two lemmas involve estimates for the modulation space norms of various modifications of the Gaussian. Together with Lemmas 4.3-4.5, they provide examples of functions that achieve the optimal bound under the dilation operator on weighted modulation spaces with weight on the space parameter. Similar constructions for weighted modulation spaces with weight on the frequency parameter are contained in Lemmas 4.6-4.10. Finally, in Lemma 4.11, we investigated the property of the dilation operator on compactly supported functions.

Recall that y(x) = e-n|x|2 for x e rd, and that yX(x) = UXy(x) = y(Xx).

Lemma 4.1. For t,s > 0, one has

MIx X-d/p-t, 0 <X < 1, (4.1)

IWxIIMp* x X-d(1-1/q), X > 1, (4.2)

Mjq x X-d/p, 0 <X < 1, (4.3)

IMIm- x X-d(1-1/q)+s, X > 1- (4.4)

Proof. We will only prove the first two estimates, as the last two, are proved similarly. By some straightforward computations (see, e.g., [2, Lemma 1.5.2]), we get

|V^x(x,^)| = (x2 + 1)-d/V(X2/(X2+1))M2 (1/(X2+1))l-l2 • (4.5)

Hence,

MWjm x WVvVxWj*

= p-2pq-2qX-d/p(X2 + 1)(d/2)((1/p)+(1/q)-1)^|R ^^Ptdx)1/P. (46)

If 0 < X < 1, then

X-| x| t/2 < (^ |x |^t/2 < ( 1 + ^ |x|^t/2 < 2X-(1 + | x 12)/ (4.7)

Thus, we have

• ^ / \"\ \1/P -y \ nv !

X- < e-n| x | ( X x) dx < X-t, 0 < X < 1, (4.8)

\jRd \ X^p / J

and the estimate (4.1) follows.

Now, observe that, if X > 1, then ((VX2 + 1 /X^p)x) x(x), and (4.2) follows. □

Lemma 4.2. For t < 0, X > 1, consider the family of functions

f (x)= X-ty(x - Xet), et = (1,0,0,...,0). (4.9)

Then, there exists a constant C > 0 such that ||f IM? < C, uniformly with respect to X. Moreover,

HfMjfq > X^«1^-^, VX > 1. (4.10)

Journal of Function Spaces and Applications Proof. We have

MP? - \\Vyf (x ,w)(x) = 1 l\ V^(x - lex,w)(x

too II r II LP? II r I1 LP?

(4.11)

t II M / ,„ , \„.\t II <- 1 -t\11| M ,„/,,\-t || < 1

= 1 t||v(?^(x'^)(x + 1e1)t| < 1 t1t||v(?^(x)

The last inequality follows from the fact that the weight (■)t is (■) ^moderate which implies that (x + Xe1)t < Xt(x)-t. This proves the first part of the lemma. Let us now estimate ||fXHm'^ from below. We have

fX(x) = X-tyx(x - ex). (4.12)

Hence, by arguing as above and using (4.5), we have

\\fxIIMsq x X't\\vv^x(x,^)(x +

/r \1/s (4.13)

> X-tXd((/)-1) ( e-^slxl2(x + e^dx^ > x-t+d((/)-1),

which concludes the proof. □

Lemma 4.3. Let 1 < p,q e > 0, t e r, and X > 1. Moreover, assume that (1 /p, 1/q) e Ij. (a) if t > 0, define

f (x) = ^ M d/p-ee2niX -e x^(x) = ^ M-d/s-eMx-1Mx) , in S'(Rd). (4.14)

^ / 0 ^ / 0

Then, there exists a constant C > 0 such that ||f Hm^ < C, uniformly with respect to X. Moreover,

HfiHjf? Z 1-d^p-e' V1> 1. (4.15)

(b) If t < 0 define

f (x) = £ |fc|-d/p-e>(x - k) = £\k\-d/p-e-tTky(x) ,, in S'(Rd). (4.16)

k / 0 k / 0

Then, there exists a constant C > 0 such that ||f ||ms0' < C, uniformly with respect to X. Moreover,

\\fx\\jf'q > rd/s-e-t ' VX> 1. (4.17)

Proof. We only prove part (a) as part (b) is obtained similarly. We use Proposition 2.2 to prove that f defined in the lemma belongs to M'^. Indeed, Q(y, 1,X-1) is a Gabor frame, and the coefficients of f in this frame are given by ck= 6k,o|^|-d/s-e if £ / 0 and c0,0 = 0. It is clear that

Hck/HC = |ck/|s(k)stY/s^ = |£|q(-d/s-eA < gO, (4.18)

' \£ezAkeZd / / \£ = 0 /

because q/p > 1. Thus, f e MP'0 with uniform norm (with respect to X). Given X> 1, we have

M«- sup |\.M,5/|^imi y-qwj^r/1- (419)

1,0 iky ^=1 -tfi (419)

WAWm? = suP KfA'gI * II^OMrP'?' Kfx,WI

Using relation (4.5),

<fx,?> = X|erd/p-%<M0,e) = £|Cd/p-e(1 + X2V 7 e-^2^. (4.20)

e / o e / o

Therefore, if X> 1,

IIMIm- * c£l e | -d/p-( 1 + x2)^

d/2 ~|e|2/(x2+1)

> CX-d£ |£|-d/s-ee-п|£|2/(x2+1)

£ / 0

> CX-d ^ |£|-d/'-ee-n|£|2/(X2+1) (4.21)

0<|£|<X

> CX-dX-d/s-e 2 e~n

0<|£|<X

> CX-dX-d/s-ee-nXd = CX-d/s+e,

from which the proof follows. □

The next results extend [5, Lemmas 3.9 and 3.10].

Lemma 4.4. Let 1 < p, q <», t > 0, e> 0. Suppose that f e S(rd) satisfy supp f c [-1/2,1/2]d and f = 1 on [-1/4,1/4]d.

(a) If 1 < q < <x>, define

f (y) = E |fc|-(d/q)-e-iMfcTfc^(y), in 5'(Rd).

keZd\{0}

(4.22)

Journal of Function Spaces and Applications Then, f e MC'0q(Rd) and

WfXWjtpq > X-d((2/P)-(1/q))+e-t, V0 < X < 1.

(4.23)

(b) If q = to, let

f (y) = E MkTkty(y), in S'(rd).

(4.24)

Then, f e MP'0TO and

XII mto > X-(2d/p)-t, V0 <X < 1.

(4.25)

Proof. We only prove part (a), that is, the case 1 < q < to as the case q = to is proved in a similar fashion.

Let g e S(rd) satisfy supp g c [-1/8,1/8]d, and |g| > 1 on [-2,2]d. The proof of each part of the Lemma is based on the appropriate estimate for Vgf.

Let us first show that f e Ap'q(Rd). We have

( e-W^m^y - k)g(y - x)dy

f(y - k)g(y - x)j (1 + |w - k|2)-d (I - Ay)V2ni(w-k)y }dy

(1 + |w - k|2)' C

+ |w - k|2) |?1 +^21<2d Hence,

E C^f d?1 (Tkf )(y) (d?2g) (x - y)e-2ni("-k)ydy |?1+?2|<2d jRd

E (|Tk (d?1 <^)|*|d?2 g|)(x).

(4.26)

/tip-q Mt,0

\Vgf W

Tp,q Lt,0

MM E |k|-d/q-e-if e-2ni(--k)y^(y - k)g(y - x)dy

V JRd \ J Rd k = 0 •'Rd

P \ q/p \ 1 /q

(x)tpdx ! dw

Rd k=0

q \ 1/q

-d/q-e-t

k=0 (1 + |w - k|2) |?1+?2|<2d

E HK^ <^)|*|d?2g|||Lp ) dw

(4.27)

Using Young's inequality: |||Xk(dP f)| * |dP2g|||Ls < HTkdP1 f||L1 ||dP2g||Ls, and the

estimate HTkdP f ||L1 < (k)t|dP1 f ||L1, we can control (4.27) by

«M k = 0

-d/q-e

k=0 (1 + |w - k|2)'

q \ 1/q

-d/q-e

.¿eZd^+[-l/2'l/2]d \k = 0 (1 + |W - k|2)

q\ 1/q

q \ 1/? dw

< C( X( £|k|

-d/q-e

eeZd\k = 0 + |£ - k|2^

-d/q-e

(1 + |k|2)d

since {|k|-d/q-e}kf 0 e £q.

Next, we prove (4.23). Since VgfX(x,w) = X-dVgi-1 f (Xx,X-1o>), we obtain

(4.28)

Vgfxhp? = 1-d(1+1/P-1/?YJ» |v&-1 f(x'W)\P(l-1x)Ptd^j'?/Pd^\ /. (4.29)

Observe that

(i-1(x)) > rV| if |x|x|e|

(4.30)

and suppg((■ - x)/X) c £ + [-1/4,1/4]d, for all 0 < X < 1, x e £ + [-1/8,1/8]d. Since supp f (■ - k) c k + [-1/2,1/2]d and f (t - k) = 1 if t e k + [-1/4,1/4]d,the inner integral can be estimated as follows:

pt \ 1/P

^j" |V&-1 f (x, w) dx^j

/ MJ e

e=^ e+[-1/8,1/8]d

2 |k|-d / q-e-t{ e-2ni(w-k)yf(y -k=0 J»1

\ 1/p x^ dx J

/ \1/P Z(\erd/q-e-txd|g(-x(^-e))|x-t|e|t)v )

> (E(w^X^^^-e))|)p) •

(4.31)

Consequently,

q/p \ 1/q

VgfxW^q = x~d(1+1/p-1/q) M Q |Vgi-1 f(x^f^x^dx) dw

> x-d(1+1/P-1/q) (I NE(er^q^x^^-x^-e))^p) dw

= xd-t-d/q x-d(1+1/p-1/q) N ( £ (|e|-d/q-e|g(^ + xe^)p j dw

q/p \ 1/q

> x-t-d/p U (E (|e|-d/q-e|g(w + Xe)|)p ) dw

\ \ rn< 1/1. /

^e / 0

\ q/p \ 1/q

je|<1/x

\ q/p \ 1/q

'|e|-d/q-^p )

| w| <1 \ | e |<1/x '

> x-t-d/p (| ( E (|e|-d/q-e)

= X-i-d/p M E (|^|-d/q-e)P) > X-t-d/p Xd/q+e M E )

> X-t-2(d/p)+d/q+e

(4.32)

which completes the proof. □

Lemma 4.5. Let 1 < p,q <to be such that (1 /p, 1/q) e I3. Let e > 0, t < 0, and 0 < X < 1. (a) If t < -d define

f (x) = Xd/q-2d/p+2d£ kr^^Mx), in S' (Rd) •

(4.33)

Then, there exists a constant C > 0 such that \\f H^ < C, uniformly with respect to 1. Moreover,

Wfibpq £ 1dfi2ipq)+e , V0 <1< 1. (4.34)

(b) If -d < t < 0, choose a positive integer N large enough such that 1/N < (p-1)/2-pt/2d. Define

f (x) = 1d/qXifc|d(2/Np-1)-e/NTiNfc^(x), in S'(rd). (4.35)

Then, the conclusions of part (a) still hold.

Proof. (a) For the range of p,q being considered, d/q + 2d - 2d/p = d^2(p,q) + 2d > 0, and so if 1< 1, then id/q+2d-2d/p < 1.

Next, notice that Q(y,12,1) is a Gabor frame. So, to check that f e M^, we only need to verify that the sequence c = {ck&} = {|k|-e/26^o,k / 0}k&eZd e &vt 0q. But, the condition t <-d guarantees this, since

\\c\\c = 1d/q+2d-2d/p ik\-pe/2 (1 + |k|2)pt/2^ < C. (4.36)

Next, as in the proof of Lemma 4.3, we have

M«- suP WJX,&/\ ^ uru 2>4\\)X,YIV (437)

1,0 iky ^=i M-t0 (437)

\\fx\\Mpq = suP \{h>g)\ > \M\>m\{h>v)\

lgll J M

(4.38)

In this case,

(f1,y) = 12d+dMp,q) ^ |k|-e/2 VvV1(1k, 0)

= 12d+d^2(p,q) ^ |k|-e/V1 + 12)-d/2g-^14|k|2/(12+1). k / 0

Therefore, if 1 < 1,

WfxWjp* > C^+^M £ |k|-e/2(1 + 12)-d/2e-n14|k|2/(12 + l) ',0 k / 0

> C12d+d^2(p,q) ^ |k|-e/2 > C12d+d^i(p,q) ^ |k|-e/2 (4.39)

k/ 0 0<m<1/12

> C12d+dp2(p,q) 1e 1~2d = C1dMp,q)+e

which completes the proof of part (a).

(b) If p > 1, the assumptions -d < t < 0 and 1/N < (p - 1)/2 - pt/2d are sufficient to prove that f e MP? In addition, the main estimate is that

WfxWjr* > CXd/qE|k|d(2/NP-1)-e/N(1 + X2)-d/2e-пx2Nlkl2/(x2+1) ',0 k / 0

k/ (4.40)

> CXd/q E |k|d(2/Np-1)-e/N > CXd^i(P,q)+e.

0<|k|<t/XN q

We now state results similar to the above lemmas when the weight is in the frequency variable.

Lemma 4.6. For s < 0, 0 < X < 1, consider the family of functions

f (x) = XsMx-1e1 y(x), e1 = (1,0,0,...,0). (4.41)

Then, there exists a constant C > 0 such that ||f ||Mp,q < C, uniformly with respect to X. Moreover,

||fx||Mpq > Xs-d/p, V0 <X < 1. (4.42)

Proof. We have

^ x ||Vf (x,w)(w)s|Lp,q = XsWVfV(x,w - X_te0 (W)s||Lpq

s , (4.43)

= Xs||v^(x,w)(w + X-^)sWipq < XsX-s||V^^(w)-s||ip,q < 1,

where we have used again the fact that the weight (-)s is (■) s-moderate. Thus, the functions f have norms in Mp'? uniformly bounded with respect to X. Let us now estimate ||fX||Mp,q from below. We have

fx(x) = Xs Me, yx(x). (4.44)

By using (4.5), we obtain

WW fxW= XsWVvyx(x,w - e^(w)sWLp?

/c x 1/q (4.45)

> Xs-d/p( e-nqW (w + e\)qsdwj > Xs-d/p,

as desired.

Lemma 4.7. Let 1 < p,q <<be such that (1/p, 1/q) e I£. Assume that s < 0, e> 0 and X > 1. (a) If q > 2 and s < 0, or 1 < q < 2 and s < -d, de^'ne

f(x) = E |e|d(1/q-1)-ee2ntX-1e>(x) = E e^^M^x), in S'(RdV (4.4g)

e / 0 e / 0

Then, there exists a constant C > 0 such that ||f || jp,q < C, uniformly with respect to X. Moreover,

11 fx 11 Mp q > Xd(1/q-1)-e, VX> 1 (4.47)

(b) If 1 < q < 2 and -d < s < 0, choose a positive integer N such that 1/N < -sq/d, and define

f (x) = E |g|d(t/Nq-tУe/Ne2пiX~N£• >(x) = E ^^^^Mx-Ne^x), in S' (rd).

e / 0 e / 0

(4.48)

Then, the conclusions ofpart (a) still hold.

Proof. (a) First of all, notice that Q(y, 1,X-1) is a frame. In addition, q > 2 is equivalent to 1/q - 1 < -1/?. Thus, for all s < 0, {|e|d( 1/q-1)-e, e / 0} e £?, which ensures that the function f defined above belongs to This is also true when 1 < q < 2 and s < -d.

To prove (4.47), we follow the proof of Lemma 4.3. In particular, we have

||fx||Mp,q > C^er^(1 + xiy^e-^2^ 0,s e / 0

/ (4.49)

> CX-d E |e|d(t/q-t)-ee-п|e|2/(X2+t),

0<|e|<x

from which (4.47) follows.

(b) In this case, Q(y, 1,X-N) is a frame. Moreover, the choice of N insures that d(1/(Nq) - 1) + s < -d which is enough to prove that f e MPs and that ||f Hjpq < C. Relation (4.47) now follows from

||fx||Mp,q > CE |e|d(t/Nq-t)-e/N (1 + x2)-d/2e_пX-2N+2lel2/(x2+t) 0,s e / 0

/ (4.50)

> CX-d E |g|d(t/Nq-í)-e/Ne-пX-2N+2|ei1/(X2+t) > cxd(t/q-t)-e. 0<|e|<xN

The next lemma is proved similarly to Lemma 4.4, so we omit its proof.

Lemma 4.8. Let 1 < p,q < to, s < 0, e > 0. Suppose that f e S(rd) satisfies supp f c [-1/2,1/2]d and f = 1 on [-1/4,1/4]d.

(a) 1/ 1 < q < to, define f (y) = EkeZd\{0)|k|-d/q-e-sMfcTfcf (y), in S'(rd). Then, f e MpS(rd) and

||fi||Mp;q > rd(2/p-1/s)+e+s, V0 <1 < 1. (4.51)

(b) I/ q = to, let

f (y) = X lkl-se2n'kyTkf (y), in S'(rd). (4.52)

Then, f e MpJ'TO and

||fi^p,to > \-2d/p+s, V0 <1 < 1. (4.53)

Lemma 4.9. Lei 1 < p,q <to be such that (1 /p, 1/q) e I3. Lei e > 0, s > 0 and 0 < 1 < 1. Assume that p > 1, and choose a positive integer N such that 1/N < (p - 1)/2. Define

f (x) = 1d/q^|fc|d(2/Np-1)-e/NTANfc^(x), in S'(rd). (4.54)

Then, there exists a constant C > 0 such that ||f Hj^q < C, uniformly with respect to 1. Moreover,

llfi|lMpS £ ldM+e. (4.55)

Proof. In this case, Q(y,1N, 1) is a frame. The condition 1/N < (p - 1)/2 is equivalent to 2/Np-1 < -1/p which is enough to show that {|k|d(2/Np-1)-e/N}k f 0 e ¿p. Therefore, f e A™ with |f |mp'' < C, where C is a universal constant. The rest of the proof is an adaptation of the proof of Lemma 4.5. □

Notice that the previous lemma excludes the case p = 1. We prove this last case by considering the dual case. Observe that the case (1/to, 1/to) e I*1n I3 was already considered in dealing with the region I^.

Lemma 4.10. Let 1 < q <to be such that (1/to, 1/q) e I3. Let e > 0, s < 0 and 1 > 1. (a) If 1 < q < 2, choose a positive integer N such that 3/N < q - 1. Define

f (x) = 1d(1-2/q) X ¿|d(3/Nq-1)-e/NMx-N^(x), in S (rd). (4.56)

Then, there exists a constant C > 0 such that ||f || jp,q < C, uniformly with respect to X. Moreover,

llfxllj- £ ^^ (4.57)

(4.58)

(b) If 2 < q< to, choose a positive integer N such that N > 2 + q. Define

f(x)= xd+d(2-N)/q£|e|d((N-t)/Nq-t)-e/NMx-Ne<Kx), in S'(rd). e =/ 0

Then, the conclusions ofpart (a) still hold.

(c) If q = 1 and s < -d, define

f (x) = £ |e|-e/2Mx-2e^(x), in S' (rd).

e =/ 0

Then, there exists a constant C > 0 such that ||f ||mto,1 < C, uniformly with respect to X. Moreover,

WfxWM- £ ^ (4.60)

(4.59)

(d) If q = 1 and -d < s < 0, choose a positive integer N such that 1/N < -s/2d. Define

f (x) = £ er^^^Mx-Ne^x), in S'(rd). (4.61)

e =/ 0

Then, the conclusions ofpart (c) still hold.

Proof. (a) In this case, Q(y, 1,X-N) is a frame. The hypotheses 1 < q < 2 and X> 1 imply that Xd(1-2/q) < 1. In addition, the condition 3/N < q -1 is equivalent to 3/Nq -1 < -1 /q which is enough to show that {|e|d(3/Nq-1)-^/N}e10 e iqs. Therefore, f e with ||f ||m-;? < C, where C is a universal constant. The rest of the proof is an adaptation of the proof of Lemma 4.5.

(b) Assume that 2 < q < to. The proof is similar to the above with the following differences: N > q + 2 and X > 1 imply that Xd(1+(2-N)/q) < 1. In addition, the condition q > 2 implies that (N-1)/Nq-1 < -1/q. This is enough to show that {|e|d((N-1)/Nq-1)-^/N}e f 0 e eq. Therefore, f e MTOs? with ||f Hm^? < C, where C is a universal constant.

(c) In this case, Q(y, 1,X-2) is a frame. The fact that s < -d implies that (|e|-e/2}e=0 e £\. Therefore, f e MTOs1 with ||f ||^to,1 < C, where C is a universal constant. The rest of the proof is an adaptation of the proof of Lemma 4.5.

(d) In this case, Q(y, 1,X-N) is a frame. The fact that -d < s < 0 and the choice of N imply that d(2/N - 1)+ s < -d. Therefore, {|e|d(2/N-:1)-e/2}e f 0 e i\. Therefore, f e M^ with

|f |^to,1 < C, where C is a universal constant. The rest of the proof is an adaptation of the proof of Lemma 4.5. □

We finish this subsection by proving lower bound estimates for the dilation of functions that are compactly supported either in the time or in the frequency variables.

Lemma 4.11. Let u e S(rd), 1 e (0, to), and 1 < p,q < to.

(i) If u is supported in a compact set K c rd, then, for every t e r and 1 > 1,

||ui|Up0q > 1-d(1-1/q) min{1,1-t}. (4.62)

(ii) If U is supported in a compact set K c rd, then, for every s e r and 1 < 1,

||u1||Mp;q > Crd/p min{1,1s}. (4.63)

Proof. We use the dilation properties for the Sobolev spaces (Bessel potential spaces) Hvs (rd) (see, e.g., [7, Proposition 3]):

C-11-d^p min{1,1s}||u||Hp <||u1||h < C1~d/p max{1,1s}||u||Hp, 1 < p <to, s> 0. (4.64) (i) Let u be supported in a compact set K c rd, we have u e Mp,q ^ u e FLq, and

c^mUM < ||u|FLq < CKHUHA?*, (4.65)

where CK> 0 depends only on K (see, e.g., [11,12]). Hence, if 1 > 1,

"u^ - ||('>HL - ||('>HL - ||F-1(u1)||Htq = 1-d|K^H-h,Hq

t (4.66)

> rd(r!) -d/q min{1,1-t}.

(ii) Now let u be supported in a compact set K c rd. We have u e Mp,q ^ u e Lp, and

CKM^ < |u|lp < Ck||u||Mp,q, (4.67)

where CK > 0 depends only on K (again, see, e.g., [11]). Arguing as in part (i) above with 0 < 1 < 1,

HUMa* - ||<D>su1||^q - ||(d)su1||lp - uHm > C1-d/p min{1,1s}||u||HP (4.68)

and the proof is completed.

4.2. Sharpness of Theorems 3.1 and 3.2

We are now in position to state and prove the sharpness of the results obtained in Section 3. In particular, Theorem 3.1 is optimal in the following sense.

Theorem 4.12. Let 1 < p,q < to.

(A) If t > 0 then the following statements hold.

Assume that there exist constants C > 0 and a, 5 e r such that

C-1 1^11/||Xoq < \\Uxf ||Xoq < cn/IIm-' Vf e Mq 1 > 1 (4.69)

then, a > (p , q), and ¡5 < d^2 (p , q) - t.

Assume that there exist constants C > 0 and a, ¡5 e r such that

cia\\f llMp0q < IIU1/ILp0 < Cl5\\f lUq Vf eMpq 0 < 1 < 1 (4.70)

then, a > d^1 (p, q), and 5 < d^2 (p, q) - t.

(B) If t < 0 then the following statements hold.

Assume that there exist constants C > 0 and a, ¡5 e r such that

C— 15llf lU; < llUif IIm- < Ciallf llMi,- Vf e Mq 1 > 1, (4.71)

then, a > d^i(p,q) - t, and ¡5 < d^2(p,q).

Assume that there exist constants C > 0 and a,p e r such that

C~! ia|lf IIms? < llUif llMpq < Cl5llf llмp0q, Vf eM^ 0 < 1 < 1 (4.72)

then, a > d^i(p,q) - t, and 5 < d^2(p,q).

Proof. It will be enough to prove the upper half of each of the estimates, as the lower halves will follow from the fact that f = U1U1/1f. Moreover, the proof relies on analyzing the examples provided by the previous lemmas and by considering several cases.

Case 1 ((1 /p, 1/q) e t > 0). In this case, we have 1 > 1 and ¡i\(p,q) = 1/q - 1. Substitute f (x) = ^(x) = in the upper half estimates (4.69) and use Lemma 4.1 to obtain

1-d(1-1/q) SMm? < C1a|MUp0q, (4.73)

for all 1 > 1. This immediately implies that a > -d(1 - 1/q) = d^\(p,q).

Case 2 ((1/p, 1/q) e I2, t < 0). This is the dual case to the previous case and can be handled as follows. In this case, we have 1 < 1 and ¡2(p,q) = 1/q - 1. Assume that the upper-half

estimate in (4.72) holds. Notice that (1/p, 1/q) e I2 if and only if (1/p', 1/q') e I* and that 1 < 1 if and only if 1/1 > 1.

yu^A^ = sup|(f1/1,gI = 1d supKf^)I < 1d|f ^ sup < 1d+lf|Lp'-q' sup||glU^

(4.74)

where the supremum is taken over all g eS and ||g||Mpoq = 1; hence,

||f1/1||MV< 1d+PUf . (4.75)

Thus, from Case 1 above, -p - d > d^1(p',q') = d/q' - d. Hence, ¡5 < d^2(p,q).

Case 3 ((1/p, 1/q) e I3, t > 0). In this case, we have 1 < 1 and ¡2(p,q) = -2/p + 1/q. First assume that 1 < q < to and that the upper-half estimate in (4.70) holds for all f e Mp0 and 0 < 1 < 1 but that ¡5 > d^2(p, q) - t. Then there is e > 0 such that ¡5 > d^2(p, q) - t + e. For this choice of e > 0, we construct a function f as in (4.22) of Lemma 4.4 such that

1dMp,q)-t+e <||f1||x? < ^HfH^ (4.76)

for some C > 0 and all 0 <1 < 1. This leads to a contradiction on the choice of e.

When q = to, the function given by (4.24) of Lemma 4.4 gives the optimal bound.

Case 4 ((1/p, 1/q) e I*, t < 0). In this case, 1 > 1, and ¡1i(p,q) = -2/p + 1/q. This is the dual of Case 3, and a duality argument similar to the used in Case 2 above gives the result.

Case 5 ((1/p, 1/q) e I*, t < 0). In this case, 1 > 1, and ¡1(p,q) = -1/p. Assume that the upper-half estimate in (4.71) holds and that a < d^1(p,q) - t. Then, choose e > 0 and construct a function f as in part (b) of Lemma 4.3. A contradiction immediately follows.

Case 6 ((1/p, 1/q) e I1, t > 0). In this case, 1 < 1, and ¡2(p,q) = -1/p. This is the dual of Case 5.

Case 7 ((1/p, 1/q) e I*, t > 0). In this case, 1 > 1, and ¡1(p,q) = -1/p. Assume that the upper-half estimate in (4.69) holds for all f e and 1 > 1, but that a < d^1(p,q). Then, there is e > 0 such that a < d^1(p,q) - e. For this choice of e > 0, we can now construct a function f as in Lemma 4.3, part (a), such that

<^1^ < C1af Hm« (4.77)

for some C > 0 and all 1 > 1. This leads to a contradiction on the choice of e.

Case 8 ((1/p, 1/q) e I1, t < 0). In this case, 1 < 1, and ¡2(p,q) = -1/p. This is the dual of Case 7.

Case 9 ((1 /p, 1/q) e I2,, t < 0). In this case, 1 > 1, and ¡1(p,q) = 1/q - 1. The function constructed in Lemma 4.2 leads to the result.

Case 10 ((1/p, 1/q) e I2, t > 0). In this case, 1 < 1, and ¡2(p,q) = 1/q - 1. This is the dual of Case 9.

Case 11 ((1/p, 1/q) e I3, t < 0). In this case, 1 < 1, and ¡1(p,q) = -2/p + 1/q, and Lemma 4.5 can be used to conclude.

Case 12 ((1/p, 1/q) e I3, t > 0). In this case, 1 > 1, and ¡2(p,q) = -2/p + 1/q. This is the dual of Case 11. □

We next consider the sharpness of Theorem 3.2. Theorem 4.13. Let 1 < p,q < to.

(A) If s > 0 then the following statements hold.

Assume that there exist constants C > 0 and a,p e r such that

^llf IIm- < llU1f lU- < Cla|lfllмP0:, Vf e Mpq 1 > 1 (4.78)

then, a > d^i(p,q) + s, and ¡5 < dp2(p,q).

Assume that there exist constants C > 0 and a, 5 e r such that

C-1 1a|lf IIm- < llU1f IImpS < C15llf IIm^ Vf e 0 < 1 < 1 (4.79)

then, a > d^1 (p,q) + s, and ¡5 < dp2(p,q).

(B) If s < 0 then the following statements hold.

Assume that there exist constants C > 0 and a, 5 e r such that

C- 15llf IIm- < lUf llXi < C1allf llVf e Mq 1 > 1, (4.80)

then, a > d^1 (p, q), and ¡5 < d^2 (p, q) + s.

Assume that there exist constants C > 0 and a, 5 e r such that

C- 1allf\U:! < llU1f IIm- < C15llf lU^ Vf eMq 0 <1 < 1 (4.81)

then, a > d^1 (p, q), and ¡5 < d^2 (p, q) + s.

Proof. As for the time weights, it is enough to prove the upper half of each estimates. Moreover, in what follows we consider only 6 of the 12 cases to be proved, since the others are obtained by the same duality argument used in the previous theorem.

Case 1 ((1/p, 1/q) e I1, s > 0). In this case, 0 <1 < 1 and ¡2(p,q) = -1/p. Assume there exist constants C > 0 and ¡5 e r such that the upper-half estimate (4.79) holds. Taking the Gaussian f = y as in Lemma 4.1 and using (4.3), we have

^ < IMImp;1 < 15HMaZ' (4.82)

for all 0 < 1 < 1. This gives p < -d/p.

Case 2 ((1/p, 1/q) e I1, s < 0). Here, 1 < 1, and we test the upper-half estimate (4.81) on the family of functions (4.41). Using (4.42), we obtain p < s - d/p.

Case 3 ((1/p, 1/q) e I|, s > 0). Here 1 > 1, ¡1(p,q) = 1/q - 1. We assume the upper-half estimate (4.78) and test it on the dilated Gaussian function in (4.4), obtaining a > d(1/q-1)+s.

Case 4 ((1/p, 1/q) e I*, s < 0). Here, 1 > 1, ¡1(p,q) = 1/q-1. We use a contradiction argument based on Lemma 4.7.

Case 5 ((1/p, 1/q) e I3,s > 0). Here, 1 < 1, ¡2(p,q) = -2/p + 1/q. The sharpness is obtained by testing the upper-half estimate (4.79) on the family of functions f1, defined in Lemma 4.9 when p > 1.

If p = 1, we consider the dual case, that is (1/to, 1/q) e I*, s < 0. Here 1 > 1, ¡1(TO,q) = 1/q. We use a contradiction argument based on Lemma 4.10.

Case 6 ((1/p, 1/q) e I3, s < 0). Here, 1 < 1, ¡2(p,q) = -2/p+1/q. The sharpness is obtained by testing the upper-half estimate (4.81) on the family of functions f1, defined in Lemma 4.8. □

5. Applications

5.1. Applications to Dispersive Equations

5.1.1. Wave Equation-

Let us first recall the Cauchy problem for the wave equation:

d2u - Axu = 0 u(0,x) = uo(x), dtu(0,x) = u1 (x),

with t > 0, x e Rd, d > 1, Ax = + ••• d2xd. The formal solution u(t,x) is given by

M(t,x)=f e2nix cos(2nW)u0(Z)d£ +\ e2nx sin(2nt|| ^) ui(№, Jr JR 2n ^ 1

= H00u0 (x) + H01 (x)

with, a0(i) = cos(2nt | £| ) and ox(¿) = sin(2nt | I| )/2n| I| .

We recall that H0i i = 0,1, are examples of Fourier multipliers which are defined by

f (x) = f e2niMl)f(№l, (5.3)

where a is called the symbol.

The boundedness of HGi,i = 0,1 on modulation spaces was proved in [13,14] and in [15]. Moreover, some related local-in-time well-posedness results for certain nonlinear PDEs were also obtained in [14,15] for initial data in modulation spaces.

Proposition 5.1. Let s e m, and 1 < p,q < <x>. Then, the solution u(t,x) of (5.1) with initial data

(u0,u1) e M'l x 1 satisfies

l|u(t,Ollx* < C0(1 + O^INH^ + C1t(1 + t)d+1\\u1\\Mqs_i, (5.4)

where C0 and C1 are only functions of the dimension d.

Proof. It was proved in [13] that 00(g) e W(FL1,L°°) and in [15] that 01(g) e W(FL1,^). In addition, it was shown in [15] that the solution satisfies

llu(t, OiU^ < iiH00 u0\ Mp,q + iiho1 u1\ xj

< I|00\w(fl1,lto) w^wm^ + w^ww(FL^W^U*;- (5.5)

< C0(OIN||Mpq + Cl(t)WulWмp;q_1 •

We can now use the results proved in Section 3 to estimate C0 (t) and C1 (t). More specifically, setting 00(g) = cos |g|, for t > 0, we can write o0(g) = (00)2nt. Using (3.10) with ^1(to, 1) = 1, [a2(to, 1) = 0, we have, for every R> 0,

LC0,Rtd+1WooWW(FL1,L~), t > R-

\\(a0)2nt\\w (FL1,L~) < J " - (5.6)

Hence,

C0R, 0 < t < R Co(t) < ^ ;' _ (5.7)

'0 ,Rt

CO Rtd+1, t > R.

Setting o1(g) = sin |g|/|g|, for t> 0, we can write o1(g) = t(o1)2ni and, for every R> 0,

_ fQRWSTWw(FLU?), t < R

imhntllw(FLl,Lr) 1 ' (5.8)

, ^ IdRt-lloillw(FL1,lto), t > R•

Journal of Function Spaces and Applications Hence,

(C1Rt, 0 < t < R Ci(i) <j( / _ (5.9)

C1 Rtd+2, t > R,

and the estimate (5.4) becomes

||u(t,Ollxq < Co(1 + t)d+ 1\\uo\\X;i + Cit(1 + t)d+ 1\\uit> 0. (5.10)

and satisfies the following estimate.

(5.11)

5.1.2. Vibrating Plate Equation

Consider now the following Cauchy problem for the vibrating plate equation

d2tu + A2xu = 0, u(0, x) = u0(x), dtu(0, x) = u1(x),

with t > 0, x e rd, d > 1. The formal solution u(t, x) is given by

f , x f sin(4n 2t|£|2)

u(t,x) = e2nixt cos(4n2t|||2)S5(£)d£ + e2nixI > ui(£)d£, (5.12)

JR V ' 4п2|¿,Г

Proposition 5.2. Let s e m, and 1 < <œ. Then, the solution u(t,x) of (5.11) with initial data

(u0, u1) e Mp'l x Mil-2 satisfies

||u(t,ohm« <C0(1 + O^iNl^ + C1t(1 + t^u ||x*, (5.13)

where C0 and C1 are only functions of the dimension d.

Proof. Here the solution is the sum of two Fourier multipliers u = H0u0 + H1u1 having symbols 00(I) = cos(4n2t|||2) e W(FL1,Lco) (see [13]) and 01 (¿) = sin(4n2t|||2)/4n2|||2 e W(FL1,L|f) (see [16]).

Since 00(I) = cos(|^|2)2nvt and 01(1) = t(sin(|||2)/|^|2)2nVt, using the same arguments as for the wave equation, we obtain

||u(t, ■) hm^ < C0(1 + t)d/2 ||u0 H^ + C1t(1 + t)d/2+1 ||u1 lu^, t > 0. (5.14)

5.2. Embedding ofBesov Spaces into Modulation Spaces

We generalize some results of [4]. But first, we recall the inclusion relations between Besov spaces and modulation spaces (see [5,17]). Consider the following indices, where ¡i, i = 1,2 were defined in Section 2:

11 Vl{pq) = ¡l^q) + p, V2(p,q) = ¡ifaq) + p• (5.15)

The following result was proved in [6, Theorem 3.1] and in [17, Theorem 1.1]. Theorem 5.3. Let 1 < p,q <to and s e r.

(i) If s > dvl(p,q), then B^R) Mpq(Rd).

(ii) If s < dv2(p,q), then Apq(Rd) ^ BPp,q(Rd).

The next results improve those in [4, Theorem 3.1]. Theorem 5.4. Let 1 < p < 2.

(i) If s > d(1/p - 1/p') and 1 < q < p, then B™ ^ Mp.

(ii) If s > d(1/p - 1/p') and 1 < q <to, then BPpq ^ Mp.

Proof. (i) For s > d(1/p - 1/p') = dv1(p,p), Theorem 5.3 says that Bp'p Mp,p. However, the inclusion relations for Besov spaces give Hence the result follows.

(ii) If s > d(1/p - 1/p') > 0, and q < p, then this is exactly as (i) above. If p < q, then there exists an e> 0 such that s = d(1/p - 1/p') + e and

Bpq = Bp,q , ^ Bpp , ^ mp, (516)

s d(l/p-1/p')+e d(l/p-1/p') (5.16)

where the last inclusion follows from (i). □

The next results improve those in [4, Theorem 3.2].

Theorem 5.5. (i) Let 1 < p < 2, s > 0. Then, BPp,q ^ Mpp',for all 1 < q <to.

(ii) If 2 < p < to, s> d(1/p' - 1/p), then Bp* ^ Mpp,for all 1 < q <to.

Proof. (i) For 1 < p < 2, v1(p,p') = 0 and using Theorem 5.3, we obtain Bp,p ^ Mp'P. Since Bpq ^ Bpp<, for all 1 < q <to, s> 0, the result follows. (ii) If 2 < p < to,

vl(p,p') = - --<- • (5.17)

p p p'

Hence, if s > d(1/p'- 1/p), Theorem 5.3 gives Bppp ^ Mpp'. If s > d(1/p'- 1/p), the inclusion relations for Besov spaces give B^ ^ Bd(p1/p,-1/p). This is easy to see if q < p'. On the other hand, if q > p', it follows by an application of Holder's inequality for & spaces. In any case, this concludes the proof. □

Acknowledgments

The authors would like to thank Fabio Nicola for helpful discussions. They are grateful to the anonymous referees for their valuable comments. K. A. Okoudjou would also like to acknowledge the partial support of the Alexander von Humboldt foundation. K. A. Okoudjou partially is supported by ONR Grant N000140910324 and by RASA from the Graduate School of UMCP.

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