Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 982753,16 pages http://dx.doi.org/10.1155/2013/982753

Research Article

Estimates for Unimodular Multipliers on Modulation Hardy Spaces

Jiecheng Chen,1 Dashan Fan,2 Lijing Sun,2 and Chunjie Zhang3

1 Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

2 Department of Mathematics, University of Wisconsin-Milwaukee, Milwaukee, WI53201, USA

3 School of Science, Hangzhou Dianzi University, Hangzhou 310016, China

Correspondence should be addressed to Chunjie Zhang; purezhang@hdu.edu.cn Received 23 November 2012; Accepted 23 January 2013 Academic Editor: Baoxiang Wang

Copyright © 2013 Jiecheng Chen et al. 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.

It is known that the unimodular Fourier multipliers 0, are bounded on all modulation spaces M^ for 1 < p,q < m. We

extend such boundedness to the case of all 0 < p,q< m and obtain its asymptotic estimate as t goes to infinity. As applications, we give the grow-up rate of the solution for the Cauchyproblemsforthe free Schrodinger equation with the initial data in a modulation

space, as well as some mixed norm estimates. We also study the M*i ^ Msp2 q boundedness for the operator e,t|A| , for the case

.,i.ia/2

0 < a < 2 and a=1. Finally, we investigate the boundedness of the operator e '' for a > 0 and obtain the local well-posedness for the Cauchy problem of some nonlinear partial differential equations with fundamental semigroup e,t|A| 1 .

1. Introduction

A Fourier multiplier is a linear operator H^ whose action on a test function f on R" is formally defined by

H„ f(x) =

The function ^ is called the symbol or multiplier of H^.

In this paper, we will study the unimodular Fourier multipliers with symbol e1^' for t e R+. They arise when one solves the Cauchy problem for dispersive equations. For example, for the solution u(t, x) of the Cauchy problem

idtu + \A\a/2u = 0, (t, x) e R+ x R" u (0, x) = u0 (x),

we have the formula u(t,x) = (e,t|A' uQ)(x). Here A = A x is the Laplacian and e 11 is the multiplier operator with symbol (see [1] for its definition). The cases a = 1,2,3 are of particular interest because they correspond to

the (half-) wave equation, the Schrodinger equation, and (essentially) the Airy equation, respectively.

Unimodular Fourier multipliers generally do not preserve any Lebesgue space Lp, except for p = 2. The L^-spaces are not the appropriate function spaces for the study of these operators and the so-called modulation spaces are good alternative classes for the study of unimodular Fourier multipliers. The modulation spaces M* ?(R") were first introduced by Feichtinger [2-4] to measure smoothness of a function or distribution in a way different from Lp spaces, and they are now recognized as a useful tool for studying pseudodifferential operators [5-7]. We will recall the precise definition of modulation spaces in Section 2 below.

Recently, the boundedness of unimodular Fourier multipliers e '' on the modulation spaces has been investigated in [1, 8-15]. Particularly, one has the following results.

Theorem A (see [11]). Let s e R, 1 < p, q < m, a> 1/2 and a=1. One has, fort > 1,

Jt iAr

«|1/2 -upu

•s+y +

y = n(a-2)

Here (and throughout this paper), we use the notation A< B to mean that there is a positive constant C independent of all essential variables such that A < CB.

Theorem B (see [15]). Let 0 < p < 1, 0 < q < <m, a > n(\/p - 1) and s e R. Then bounded from M^ to

Mp q ifand only if

s > max {0, a - 2}n

1 1 ~P~~2

In this paper, we use a different method from [15] to prove the following theorem, which, in particular, uses the modulation Hardy spaces ^ that will be later defined in Section 2.

Theorem 1. Let se R, 0 < q < œ>, t > 1. For a positive a=1, denote y = n(a -2)\1/p - 1/2\. Leta0 = 2/(2+ n) ifn is even and a0 = 3/(3 + n) ifn is odd.

(i) Assume a > a0.If p > n/(n +a) and n> 2, one has

ILif"|A| jll <tn\1/p-1/2\v.

J IImur») <

IIm^r») +

Particularly, the above inequality holds for all p > 0 if a is a positive even number.

(ii) For any 0 < p < >x>, one has

|Lii|A| j\\ < tK|1/p-1/2|

for any S >0. Here

As+r (Bn ) +

y = n(a - 2)

the fractional Schrodinger semigroup has a growth t"'1^-1/2' when t is growing, but it gains an arbitrary regularity. In the high frequency part, the semigroup can be controlled by

max |1,(t|fc|a-2)

2xk|1/p-1/2|

at each piece of its decomposition with frequency k. This phenomenon was also more precisely observed in [1,15] (see also [11]). Thirdly, the case a = 1 was studied in [8,16].

Since the Lp norm is dominated by the Hp norm and the Riesz transforms are bounded on Hp, by the Riesz transform characterization of the Hp (see Section 2), we easily obtain the following corollary.

Corollary 2. Let s e R, 0 < p, q < >x> and a > 0, a=1. One has for t > 1

eit|A| J\\ <t

J N..s

"|1/p-1/2|||

y = n(a- 2)

Our next result shows that the asymptotic factor /-"I1/?-1/2! in Theorem 1 is the best for all p > 0, at least for a = 2.

Theorem 3. Let a = 2. The asymptotic factor /"!1/i-1/2! in Theorem 1 is the best. Precisely, for t > l,if

||eit|A| f\\ < tS II J IIms„„ <

1 1 p-2

In the next theorem, we state some mixed norm estimates.

Theorem 4. Let 0 < a < 2 and a=1. For 0 < p1 < 1 < p2 < tx>, suppose nr(1/2 - 1/p2) > 1.

(i) Ifr > q, then

eit|A| f\\

Lr(Q,œ)

(iii) Assume n= 1. If a >1/2, then

|1/p-1/2|||

(ii) Ifr < q, then

e't|Ar' f\\

Lr(Q,œ)

for all p > 2/3.

We want to make a few remarks on Theorem 1. First, (iii) in Theorem 1 says that when n = 1,comparedtothecasen > 2 in (i), one obtains a larger range of a and a smaller range of p. We do not know if there is a unified formula regarding a and p for all dimension n > 1. Second, in the proof we will see that, in the low frequency parts of the definition of M^^,

We consider the following linear Cauchy problem with negative power:

idtu+\Aa/1u = 0, (t,x) e R+ x R", a> 0, u (0, x) = uQ (x).

We give the grow-up rate of the solution to the above Cauchy problem in the modulation spaces.

Ms ' P 2 '1

Ms "ta

Theorem 5. Assume a > 0 and y = n(a + 2)\1/2 - 1/p\.

(i) Let 1 < p < 2. One has that for any 0 < r < np/(n + a([n/2] + 1)(2-p))

HwWIuyr») ^(1+t)

([K/2] + 1)|2/p-1||

m°IU:7(r")

+ \\u.

oIIm^(r") •

(ii) For any r <2 < p, one has

IIu(t)IIMs (r") ^ ikii^ (r") + (1 + t)H]llp 1/2|||

^IIm^rt

Now, we study the following Cauchy problem of the nonlinear dispersive equations (NDE):

i^-\A\-a/2u + F(u) = 0, at

u (0, x) = u0 (x),

where F(u) = \u\2ku for some positive integer fc. For T > 0, the space Y = C([0, T], Msp1) is defined by

c([0,r])M:^>1)

i(t,x) : ||m|c([o>t]>m^>1) = supIIu (t,-)IIMsei < œ

We obtain the quantitative forms about the solution to the above Cauchy problem of the nonlinear dispersive equations.

Theorem 6. Let 2 > p > 1, y = n(a + 2)\1/2 - 1/p\, a > 0 and assume

n + a([n/2] + 1)(2- p) Assume u0 e Msr 1r n Ml 1 for any

1 <r <

n + a([n/2] + 1)(2- p)'

There exists T > 0 such that the above Cauchy system (NDE) has a unique solution u e C([0,T], Msp1), where T depends on the norm ||wQ||Ms-i< and ||mq||ms .

According to the inclusions of modulation space (see Proposition 2.5 in [13]), we know the space of initial data

M;7 n m' = Mrs-r if r< p.

Theorem 7. Let p >2,y = n(a + 2)\1/2 - 1/p\. Assume

2k + 1

and u0 e Msr 1 for any

<r <2.

There exists T > 0 such that the above Cauchy system (NDE) has a unique solution u e C([0,T], Msp1), where T depends on the norm \\uQ ||MS .

The rest of the paper is organized as follows. In Section 2, we recall or establish some necessary lemmas and known results. Sections 3 and 4 aredevoted to theproofsofTheorems 1 and 3, respectively. Finally, in Section 5, we give some

1 „ ^ Mf, „ boundedness for

Pl' H r2' H

applications including the M

the operator e,t'A' negative a.

in the case a < 2 and a= 1, including

2. Preliminaries

2.1. The Definitions. The modulation space is originally defined by Feichtinper in 1983 on the locally compact Ablian groups G. When G = R", the modulation space can be equivalently defined by using the unit-cube decomposition to thefrequencyspace (see AppendixAin [13], also [14,17]). The following definition is based on the unit-cube decomposition introduced in [13].

Let a be a fixed nonnegative-valued function in s(R") with support in the cube [-4/5,4/5]n and satisfy a(^) = 1 for any % in the cube [-2/5,2/5]n. By a standard constructive method, we may assume that for all % e R",

(0 = 1,

where ak is the fc-shift of a that is defined by

ak (Z) = a(Z-k), aQ (V = a(Z). (27)

For each k e Z", we use ak(%) as its symbol of a smooth projection nk on the frequency space. Precisely, for any f e S'(R"), we have

nkf = °kf-

Let X be a Banach space of measurable functions on R" with quasi-norm \\ • \\x. We define the modulation space

Ms (X, eq) (R") = {f is measurable :

W/Lw ) = Uk)Shk (ÏÏWxh ' (k) = l + \k\. By definition, we have the inclusion

ms (x, eq) c Mr (x, eq) if r<s. (31)

It is known that the definition of the modulation space MS(X, £q) is independent of the choice of functions a. In this paper, we are particularly interested in the cases X = Lp and X = Hp, where Lp is the Lebesgue space and Hp is the real Hardy space. For all 0 < p,q < rn, we call Ms(Lp,Eq)

the modulation spaces and MS(HP, l) the modulation Hardy space. As a usual notation

Ms (Lp^) = MsM = Ms (R"),

we similarly define

Ms (H,e") = ^M = & (R")

By the definition and known properties of Hp, we have that for all 1 < p < >x>,

^ (R")=M^ (R"),

and for all 0 < p < 1,

(rn)cM] For simplicity in notation, we denote

(R") = ^ (R"). MM (R) = M

")■ (36)

The following imbedding relation can be found in Proposition 5.1.5 of [18]. Let s1, s2 e R, 0 < p1, p2, q1, q2 < œ. if

s2 < s1>

P1 < p2>

q1 <q2

MSl c MS2 . Mp1 'h C Mp2 «2 .

2.2. Hp Spaces. It is well known that the Hardy space Hp(rn) coincides with the Lebesgue space Lp(R") when 1 < p < >x>. For 0 < p < 1, the space Hp(r") has many characterizations. We will use its Riesz transform characterization in this paper. For an integer L > 0 and multi-index J = (j1,...,jN) e {1,2,..., n}L, let Rj denote the generalized Riesz transform

R (f) = RjRjRjL (f),

where each R(f) is the jth Riesz transform of f if j = 0 and R0(f) = f .It isknownthat for p > (n - 1)/(n -1 + L) and all f e Hp n L2,

L» -IR (f)L, /

where is a sum of finite terms.

The operator rcfce,t|A| is a convolution. We have

R/ (f)\

Also it is well known that Rj is bounded on Hp spaces for any 0 < p < <m.

2.3. Some Lemmas and Known Results

Lemma 8. Let 0 < p < >x> and t > 1. Suppose that there is an integer N > 0, such that for all test functions f

'f\l <tb11

for m < N and

H,/ce't|A| f\\ <t°2 m

for m > N. Here b > ^2 >0 and d is a real number. rThen for

f e Hp n L2,

•, has

lle*t|A fll < tbi

where v is an arbitrary positive number.

Proof. The case p > 1 is proved in [11]. It suffices to show the lemma for 0 < p < 1. By the Riesz transform characterization of Hp, for \k\ > N, we have

fil -HW^Rr (f)\

<Itb2|fc|d||R/f||I, -t*2|k|d||

IIh? ■

By checking the Fourier transform, we have the identity

nk = X nk+ink> (46)

lil^s1

j = Ü1>...>j«)> \i\

So for \k\ > N - 1, one has

= max {\;1\)...)\;„\}. (47)

¡t|A|"

1 ^k+j^k (Sa (t)f)

< I |k+/|A|"/2 (nkf)\\ <tfc2|k|d|M|№■

A similar argument shows that for \k\ < N - 1,

|pfce'| | f||HP <t1hJ\\Hp

foranyf e Hp nL2. The rest ofthe lemma easily follows from the definition of the modulation spaces. □

Lemma 9 (see [18,19]). Let E c R" denote an open set and f e C™(E). If< e Cm(E) and the rank ofthe matrix

{Dxpxt^(x))ni =1 is at least m > 0 for all x e supp(f), then

eiA0(*V (x) dx

-m/2||

lie2» ■

Lemma 10. Let 2 > a > 0 and a=1. Suppose that a is a Cm function with support in [-1,1]n. Then

IIF-1 (a(i;) e,t|?r )|| * min |1,f-"/2}. (52) II \ /IIl~(r") l j

Proof. The case a = 2 is known [20]. It then suffices to show that for 0 < a <2,

if-1 (o(t) e,t|?r )\\ <t-n/2

ii v /iil~(r")

for large t. Let r be a standard bump radial function supported in the set

and satisfying, for all % = 0,

Ir(2} |*|) = 1.

Noting the support condition of a, we write

|F-1 (a(Ç) eit|?r)(x)\

= |f a(Ç)eim" e'^dl, 1 J r"

f | T(2 k|)5«)e't|?re(*'?)#

,= ot j r"

il A il A jr"

jeAiUA2UA3

where the sets Afc, k = 1,2,3 are defined by

A1 = {i>0: 2-j(a-1)-5 > M

1 ]J at

A 2 = {j > 0 : n2

. 9-j(«-1)+^ m

25 \x\ ;(1-a) 2 5 \x\

j > 0 : —— > 2;(1 a) > -—

at atn

For j e A 1,we use polar coordinates to write

2-X f Tm)a(2-jZ)e'tr"'^e'^^#

= - L If "'-Kli

xe e ls lsl"

where d^' is the induced Lebesgue measure on the unit sphere S"-1. When n is even, taking integration by parts for n/2 times on the inside integral, we obtain

j€A1 Jr

^ ^-«/2 I 2-i"(1-"/2) ^ ^-"/2 jeAi

When n is odd, we use integration by parts for n/2+1/2 times on the inside integral,

(54) <\[ r(K\)ä(2-^Wi2-'"|i|" e'^^

\\ r " \ \

n/(n+1)

* ^-n/22j«w/2

Again we obtain that for odd n,

I 2-- [

j€A1 Jr

For j e A2, without loss of generality, we assume \x1\ > (\x\/n). Perform integration by parts on the ^ variable for suitable amount of times. We similarly obtain

I 2-jn [ T(\t\)a(2-jt) e't2-J"|?re'2" jeA2 Jr"

For j e A 3, invoking Lemma 9, we obtain

T(\i;\)a(2-ii;)e'{tz' m

< min -

Noting that A3 contains no more than (10/\1 - a\) + log2n numbers of j, it is easy to check

£2"" min

2 jaw/2

The lemma is proved.

Lemma 11 (see [21, pages 163-171]). Let 0 < p < 2 and

(64) □

Suppose that Tm is a Fourier multiplier with symbol m. Ifm is a bounded function which is of class CN in R" \ {0} and if

< (^\^\-1)|J| for \J\<N (66)

with A> 1, then Tm is a bounded operator on Hp and IIT H * ,n(1/p-1/2)

Lemma 12. Let \k\ > 1 and a=1. For all 1 < p < >x>, one has

* \\nkf\\LP if m»-2 < i,

JfjAj"

< (t\k\ ) hkf\\Le if t\k\ ^ I-

This lemma can be found in Section 4.2 of [11].

Lemma 13. Let F be a compact subset in R", and let 0 < p < q < rn. There exists a constant C depending only on the diameter ofF and p, such that

llfL <CMlP (69)

for all f e Lp satisfying supp f c F.

This lemma is the Nikolskij-Triebel inequality, see Proposition 1.3.2 in [20] (also Lemma 2.5 in [22]).

Lemma 14. Let 0 < p < 1 and F, F' be compact subsets of R". Then there exists a constante depending only on the diameters ofF, F' and p, such that

lllfl*ML <CllfllL. MIl- (70)

for all f,geLp satisfying supp f c F and supp g c F'.

This is Lemma 2.6 in [22] (see also Proposition 1.5.3 in [20]).

Lemma 15 (Pitt's theorem). If 1 < p < 2 and 0 < a < 1/p', then

(f lf(x)f\xrnp'dx)UP lf(x)lp\xrpdx]llp.

VJr" l l / \Jr" /

Lemma 16. Let se R and p,p¡ > l (j = l,2,... ,m) satisfy

P Pi P2 Pm.

Then one has

* n\\u

n\kj"lMs„ ,

j=i P>-1

This result is a particular case of Lemma 2.5 in [8].

3. Proof of Theorem 1

The operator rcfce,t|A| is a convolution operator with the symbol a(£-k)eit|?r. This symbol is a Cm function on R" \{0} with compact support. Clearly for any N > 0 and % = 0, we have that for \k\ < 10,

(|)Wk)e'^ *{(1 + t)r1}l/l for \J\<N.

So Lemma 11 implies the following estimate.

Proposition 17. Let 0 < p < œ>. For any k with \k\ < 10, one has

Ih^fL < max {W^Wf^ ■ (75)

By the proof of Lemma 8 and Proposition 17, we have that for all k < 9,

Ih^fL < max{Lt^-^IMU. (76)

The following proposition extends Lemma 12 to all p e (0,1].

Proposition 18. Let 0 < p < 1. Foranyk with \k\ > 1,forany a e R \ {0,1}, one has

\\ 't|A|"/2 11 hke 1 1 f

* max {l,(t|k|a-2)

a-n«(i/i-i/2)

T ]\\nk+jf}\LP ■

Proof. The proof uses the same idea used in proving the case p > 1 which was represented in [11]. For the convenience of the reader, we present its proof.

Let Qfc be the kernel of rcfce,t|A| . Then

Qk = f-i (a(t-k)etm°). By Lemma 14 and (46), we have

|ke't|A| ' fL * \\Qk\\LP T \K+jf|

Thus to prove the proposition, it suffices to show

\\F-i (a (Ç - k) e't|?|")\\Lp * max {l, (i|k|"-2)"(i/p-l/2)} .

For simplicity, we prove the case n = 2. The proof for n > 3, is tedious but shares the same idea as that for n = 2.

First we study the case t\k\"-2 > 1. For i = 1,2, and k = (k1, k2), if a > 2 we denote

C, (k)=ai(|k,| + l){(|ki| + l)2 + (\k2 | + l)2}W2)-i,

D, (k) = ai(|k,|-l){(|ki|-l)2 + (\k2\-l)2 }(»/2)-i. If a < 2, we denote

C, (k) = Mt(\k, | + l){(|ki|-l)2 + (|k2| - l)2}(»/2)-i,

D, (k) = |a|t(\kí |-l){(|ki| + l)2 + (\k2\ + l)2}(a/2yi.

Also, for i = 1,2 and j e N, we define sets Fj = {xj e R : Dj (k) - t\k\a-2 < \x;\ < C; (k) + t\k\a-2), Gjj = {Xj e R :Ci (k) + t\k\a-2 + j-1

< \xi\<Ci (k) + t\k\a-2 + j), Huj = {*, e R :D, (k)-t\k\a-2 -j

< \x,\<D, (k)-t\k\a-2 -;+!}.

It is easy to check

Length (F) * t\k\a-2, Length (Ghj) = Length (HUj) = 1.

Kt,j = Gt,j u Ht,j.

We have for i = 1,2,

XFi (xi) + I%KiJ (xd = 1.

>t|Ç+fc|°

iR2 \f-1 (ak (Ç) e't|?r )(x)\Pdx = [r2 \f-1 (a^e^)(x)\Pdx *[r2 xf> (*1)xf2 (^2)\f-1 (a(Ç)

œ /•

+ 1 \,Xklt (x1)XF2 (x2)

;-=1 Jr2

x \ f-1 (a(0 eit|?+fcr)(x)\Pdx

œ /•

+ 1 \R2 *Km (X2)^Fi (x1)

m=1 J r

x \f-1 (atf) eit|?+fcr)(x)\Pdx

œ œ /•

+ 1 1 L ^ (X1)^K2,m (x2)

j=1 m=1 Jr

x \f-1 (a(^)eit|?+fc|")(x)\Pdx

= II1 + II2 + II3 + II4,

f-1 (a (Ç) e't|?+fcr ) (x) = ir2 (Ï) (88)

It is easy to check that if i\fc\" 2 >1 and £ e supp(a), the phase function

®(t,k,x,$) = + ^ + (x•!■),

satisfies

|detD?.D?.> t\k\a-2. So by Lemma 9, we have

II1 * (t-1 \k\2-a)P \ Xf, W Xf2 (x2) dx

v 7 Jr2 1 2

* (t-1\k\2-a)p(t\k\a-2)2 = t2-P\k\(a-2)(2-p).

Observe the easy fact that if x e K1j and % e supp (a), for any integer L,

0Çi (t,k,x,Ç)J \j + t\k\

, i, I = 1,2.

Perform integration by parts on ^ and £2 variables both for L times such that Lp > 1. An easy computation shows that

iJ4 I 2 XKhl (x1)XK2,m (x2)

i=1 m=1 JR

(j + mr2)Lp (m+mr2)

* I Ii

j>t|fc|" 2 m>t|fc|

„-2 jLP mLP

\r2 XKhi (xù XK2tm (x2) dx * 1

The estimates for II2 and II3 are exactly the same. We only estimate II2. Take integration by parts on ^ variable for L times with pL> 1. Again, a simple computation shows that

j>t|fc|'

X2-fp [R2 XKi,i (x2)dx

* (t\kr-2)-Lp+2 * 1,

if we chose a suitably large L. These estimates on IIj, j = 1,2,3,4, indicate

f-1 (a(t-k)em")||ip * t2/P-1\k\(«-2)(2/P-1)

= (t\k\a-2)

2x2(1/p-1/2)

provided i\fc\" 2 > 1.

Wenowturntoshowthecaset\k\"-2 < 1.For«= 1,2,and k = (k1,k2), let C(k) Dt(k) be the numbers defined above. For i = 1,2 and j e N, we define sets

F = {x, :D, (k)-1<\x,\\<Cl (k) + 1}, G y = {x,:C, (k) + j<\x,\<C, (k) + 1 + j}, H, 'j = {x, :D, (k)-1-j<\x,\<D, (k)-j}. It is easy to check

Length (F,) < 1, Length (G hj) = Length (%, J = 1.

K¡'j = Gij u Hi,j.

|r2 \f-1 (ak (!) e't|?r )(x)\Pdx

< |r2 XSi (x1) Xg2 (x2) \f-1 (a (!) e't|?+fcr ) (xfdx

œ /•

+ I I 2^Ki,i (x1)^S2 (x2) j=1 J r

x\f-1 (a (!) eit|^+fc|")(x)\Pdx

œ /•

+ 1 I 2^K2,m (x2)XFi (x1)

m=1 J r

x\f-1 (a (!) eit|^+fc|")(x)\Pdx

œ œ /•

+ I 1 |ra2XKlJ (x1)X«2,„ (x2)

j=1 m=1 Jr

x\f-1 (a(!) eit|?+fc|")(x)\Pdx.

Using the same argument as we used before, we can show

||f-1 (a (! - k) eit|?r)}[lî<1- (100)

We complete the proof of Proposition 18. □

We are now in a position to prove Theorem 1.

Proof. By an argument involving interpolation and duality, it suffices to show the case p < 1. Using Proposition 18, the inequality in (76) and the definition of the modulation spaces, we easily obtain (ii) in Theorem 1.

To show (i) and (iii) in Theorem 1, by Proposition 18 and the definition of the modulation spaces, it suffices to show

IM« (t)f\\L? < max {1, t"(1/^-1/2)} I |kf|

I? ■ (101)

Again, by Lemma 14, the proof of the inequality in (101) can be reduced to show that for t > 1,

(flf o(!)em"e'^d!Pdx)1/P < t"(1/^1/2). (102) V Jr" I Jr" J

We show (iii) first. The proof of n = 1 may illustrate the method. When n = 1

(f \f a(!)em"e'(^x)d!Pdx) r \\\\ r \\

<(f \ f a(!) e'^V^ d!Pdx) \J|x|<t\J r /

+ (f \f a(!)em''e'(t-x)di;PdxX'P. \J|x|>t\J r j

P \ 1/^ >t|?| e'(?-*>dr (103)

By Holder's inequality and the Plancherel theorem, the first term above

P \1/^

(f \f a(!)e'm"e'(^x)d!Pdx) \J|x|<t\J r /

» \ 2 \ 1/2

(|r \a(!)e't|?r\2d!) t(1/^-1/2) <t(1/^-1/2).

For the second term, performing integration by parts, we obtain

(f W o(!)e>m\'^x)d!Pdx)lF \J|x|>tU r /

i f I fm a P \

= 2(1 II a(!)e't^ cos !xd! dx) \J|x|>t|J 0 /

= 2(\ -Hp |f+M [iataWr1 (%)}

\J|x|>t \xY I J0 1

x sin !xd! , (2-p)/2p

P \1/Î dx

l|x|>t |x|2^/(2-^)

ç \ ç +œ

{iata (!) !"-1 + a' (!)}

J|x|>t\JQ

|x|2^/(2-^)

xe,t|^| sin !xd! , (2-p)/2p

2 X 1/2

Î9 \ 1/2

R1 (\a(!)\\!\'-1 + \a' (!)\) d!) <t^-1/2

Now we return to show (i) of Theorem 1. We will prove only the case a < 1. Write

([ \[ a(Z) e't|?r e,(^x)dtPdx)1/P

r" \\ r" \\

*([ \[ °(i;)e'm'' e,(^x)dtPdx)1/P (107) \J|x|<t\J r" /

([ \[ a«)e't|?r # V J|x|>t \ JR"

P \1/P dx) = J1 + J2.

Using Holder's inequality and the Plancherel theorem, we obtain

h *(f |a(Z)e,m'X fdt)1/2i"(1/P-1/2) * i»(1/i-1/2). (108) V Jr» | | )

For i,j e {1,2,..., n}, we denote sets Et = [x e R" : \x\ > t}, Et,t = {x e Et: |xt | > | Xj | yj = i}. We now write

I ([ \[ a(0e't|?r ¿«^

z~i V Je,, \ Jr"

P \1/p dx ) . (110)

To show (102), it now suffices to show that for each i,

[ \[ a(Ç)eit|Ç|" e'(^dt je,, \ jr"

p \1/P dx) *i"(1/P-1/2).

Using the Leibniz rule, for any positive integer L, we have

a(L) (a (x) ) = I cfo(L-fc) (a (x)) 3(fc) (e'^) . (112)

Here, an easy induction argument shows that, for k> 1,

By the definition, it is easy to see that each O a is an LOT and COT(Rn - {0}) function with support in the cube [-112,1/2]n.

Let L = [(n/2) + a] + 1. Performing integration by parts on variables for L times, we have

([ \[ ct(Ç) eit|Ç|"e'^d^dx) * ( [ \x\-Lp\[ 9(L) (a^e'^ e'^di;

JEt i \ J

!,J (Lw-" \L

x (Ç) eit|?r ei(*'?) d!;

P \1/P

We first estimate each Ij, 0 < j < L - 1. Recall that we assume p e (0,2). Let q = 2/p, so pq = 2p/(2 - p). By the choice of L and the assumption

it is easy to see Lpq' > n. Therefore, by Holder's inequality, we obtain

( [ \x\-LM'dx)

, \1/W)

„<«(J*)-,»*!«V«W, №3) x([ \[ ä,0^„^v^^Î" (118)

j-1 v j^" \ j J

where e COT(Rn - {0}) is a homogeneous function of

degree ja - k for each j. We now write

df] (oWe1^ )

= e (a(L) (a (x)) + IiVr-LOj>L>a (x)

* i-L+«(1/i-1/2)||g(L)a|| * pL+KU/p-1/2)

For each j = 1,2, ...,L- 1, by the choice of L, the assumption on p, and an easy computation, it is not difficult to see that we may obtain a number a^ in the interval [0, n/2) satisfying

Oj^ (x) = I C^-k) (x)) Ä ( £ ) \x\L-k. (115)

L- a.: > n\---),

' \p 2''

a.• + ja - L > —,

a: + j - L< 0.

By Holder's inequality and Pitt's theorem, for each j, we obtain

I} < *

-(L-a.-)p

f ÜrV« (!) e't|?r

xei(x<)d!

P \1/i d x)

< ( f M-1-^'dx)

1/(^')

(f |x|-"J f

V JR" h

xr> ! \!\ j"-L^j'L'« (!)e't|?|V(^d!

2 1/2 dx)

< t-L+«J+j+»(1/P-1/2) (j \|x|«j ^j«-1 ®.Ua(x)\2dx)

< t-L+«j+;+n(1/p-1/2)

Combining all the estimates, we have

f \f a(!)eit|Ç|V(*-°d!

JE», \ JR"

< I^ + I t-i+«j+;'+n(1/p-1/2)

< IL + t»(1/P-1/2).

It remains to estimate Il.

It is easy to see that the choice of L and the condition

- it n is even

a > aQ =

n + 2 3

if n is even if n is odd

in the theorem imply La-L > -n/2.So,byHolder's inequality and Pitt's theorem again, we obtain

Î\ |x|if

\JEW \JR"

h =tL(l x^l \!\La-L0L'La Me^'e^d!

tL( f |x|-L^dx)

\ 1/W)

f \f \!\La-L^L'L'« (!)e't|?|V(^d! JR" \ JR"

< t"(1/i-1/2)||\!\La-LoL'La (!)||2 <t"(1/^-1/2).

This completes the proof of (102).

2 1/2 dx

When a = 2,4,.. .,we have

(f \f a(!)e'm"e'(t-x)d!Pdx)

\ jem \ j r" j

p \1/i

'Üe,^'""«W

for any integer L, where «(!) is a Cœ function. Thus it is trivial to see that

(f \f a(!)em"e'(^d!Pdx)1/P < t"(1/^-1/2) (125)

v jem \ j r" j

for all p > 0. This proves (i) in Theorem 1.

4. Proof of Theorem 3

Recall that the function a chosen in the definition of the modulation space is flexible. We may choose

where each dj is a nonnegative valued function in S(R) with support in [-4/5,415] and satisfies Qj(!j) = 1 if e [-2/5,2/5] and

Xe*, fe ) = 1, e^ fe ) = 6j fo + kj). (127)

For simplicity, we work on the case 0 < p < 2. Let 0 e Cm(R) be a nonnegative function with support in the set [1/16,1/8]. For

' = (x1 ,x2,..., xn) ,

define a function O on R" by

®(x) = n$(xj). i=1

and an f e S(R") by f = F-1(0). Let k e Z". It is easy to see that

nk (f)(x) = e(k)f(x), (130)

where e(k) = 1 if k = 0 and e(k) = 0 if k = 0. Similarly we have

nk (e't|A|f)(x) = e(k)(e't|A|f)(x). (131)

Suppose that we have some S such that

„i(|A| -

¥ f IIm^(R") < t llf Hm^(RT

By the choice of f, we have

e»t|A| j\\ = II e>t|A|

j iim'(r") II Q

<ts\\f\\ (r, ) =ts||

On the other hand,

II ii|A| rI

lhe | 71

Jr" j=1 Jr

where the phase function Xj) is defined by V(ïrxj) = t?2 +ÏJ xr

f(^j,xj)=2t^j + xj

The critical point of Xj) is at

-X; p* =_¿

= 2t .

Thus, by the stationary phase method (see [19, Proposition 3, page 334]), an easy computation gives that, as t ^ œ>,

[ n [ ^(Qe^d^

Jr" j={ Jr

>(n [ \[

\j=1 J^- |<2t\Jr

n [|^i|<2t

+ O (i"(1/P-1/2)-1/2) _ ^«(1/^-1/2)

Thus the inequality

Jf|A| .

We ' (r") * t ll/L^(r")

implies

i"(1/P-1/2) *ts, as i —> œ. This shows the conclusion.

5. Applications

5.1. Operators Sa^(t). Let a=1 and ß e R. In [23], to investigate the absolute convergence for multiple Fourier series, Wainger studied the oscillating multipliers Sa>^(t) with symbol

In [24], Miyachi proved that in the case a > 0 and a = 1,for 0 < p < œ,

11^ (DAI

(134) if and only if

By Theorem 1 and its proof, we not only obtain the bound-edness of Sa^(t) on for any fie R, but also gain a

regularity of fi if fi > 0.

Theorem 19. Let 0 < p < œ and a=1. One has for t > 1

,n|1/p-1/2|

11^ (t)fL

(137) where

y = n(a- 2)

Particularly, if ß = n(a - 2)\1/p - 1/2\, we have

vfL **

n|1/p-1/2||

Proof. The proof of Theorem 19 is the same as the proof of Theorem 1. We skip it. □

M* Boundedness. We will study the

M* ^ Mi „ boundedness for the operator e,t|A| in the case 0 < a < 2 and a = 1. The other cases of a will be addressed in another paper. First we estimate rcfce,t|A| for \k\ < 10.

Proposition 20. Let a= 1, 0 < a < 2 and \k\ < 9. One hasfor all 0<r<1<p<rn,

|vt|A| 4 *(1+t)-n(1/2-1/p)\\nk(f)\\Lr.

Proof. By Lemma 9, we know

*(1+t)

III1 ^L™

On the other hand, it is known from [11]

Jf|A|"

III? ^L?

*(1 +tf1/P-1/2|

for all 1 < p < m. Interpolating these two inequalities with the energy inequality

,it|A|"

III2 ^L2

we find that for any 1 < p < œ,

¡t|A|"

*(1+t)-n(1/P-1/2). (151)

Let 1 < p < <x>. For f e L1,we have

ttfce't|A| ' f||T , = sup \< nkSa (t)f,g>\

< f,^2 y0>

,it|A|"

Now for any 0 < r < 1,we obtain that for all 1 < p < œ,

n^2 f\\ <I\\nk+A (t)nk |l|<1

<(1+ma-2) <(1+t\kr2)n(1/P-m

^-«(1/^-1/2). (160) R (f)\\L1

k (j)wl'

<(1+tf(1/^-1/2)llfllL1 ■ This shows that for | k| < 10,

nfce't|A|"f|L , <(1+t)

-«(1/2-1/p)||

Now if 0 < r < 1,we obtain that for all 1 < p < œ,

k^'fL - Xk+A (t)nk (f)le

As a consequence the proposition, we have the following.

Corollary 23. Let 0 < a < 2, a= 1, \k\ > 9 and 0 < p1 < 1 < p2 < œ, one has

ll^'^viL - Kf iu if t\k\'-2 < 1,

Jf|A|"

< (1+t)-"(1/2-1/i)||nfc (f)||L1

< (1+t)-n(1/2-1/p)\\nk (f)\\Lr■ The last inequality is from Lemma 13.

- \k\»(«-2)(1/p2-1/2)t»(1/P2-1/2)||^kf||Hpi ift\k\a-2 > 1.

By Corollaries 21 and 23 and the definition of the

modulation spaces, we now obtain the ^

As these discussions, we obtain the following corollary. boundedness of the modulation spaces.

Corollary 21. Let 0 < a<2, a =1, \k\ < 10 and 0 < p1 < 1 < p2 < œ, one has

Theorem 24. Let 0 < a < 2, a = 1, and 0 < p1 < 1 < p2 < >x>, one has that

|Vt|A|" f||№2 <|M|№1 ift<1,

it|A|"

4, <t*(1/ft-1/2)iMi„, ift>1,

||e,t|A| f|Ms <(1+t)n(1/p2-1/2) ||

where for convenience, one denotes Lœ = Hm.

Proposition 22. Let 0 < a < 2 anda = 1. One has for \k\ > 9 and all 0<r<1<p<œ

||^Ar/2f||Lf, <(1+t\k\^-2)-n{1/2-1/P)llnk (f)^. (156) Proof. By Lemma 9, it is trivial to see that

H^'lL - min |1> t-"/2\k\-"(a-2)/2|. (157)

Interpolating this inequality with the energy inequality, we find that for any p <2,

y = n(a-2)(±-1).

Next, we give the proof of Theorem 4.

Proof. By the definition of the modulation space, we know

e'^'fl

MJ,2 '„(r")

Lr(Q,œ)

a/2 II

llL ^L™

So by duality, for \ k\ > 9,

< (1+t|k|"-2)

:-2x-«(1/p-1/2)

<(1+mr2)

2xK(1/p-1/2).

I(1 + |k|Hke'i|A|"/2fll<? fc

If r/q > 1, write

(1 + ^^ (t)f\\l2 =9fc (t). (165)

By the Minkowski inequality, we have

If r/q < 1, then

Lr(0,m)

I? (i (,)|

j? (JT ^

[zo+iwtf

sr|| it|A|" \\nke 1

f\\ dt

J N T D-,

r \q/r

II >t|A|'- jrii j. Ilnke 1 f\\ dt

U k ^ 11 T D^

When \k\ < 10,

J«/2 "r

\\Lf2'

||nke,t|A|" f\\ dt

im \ q/r

0 (1+i)-r»(1/2-1/P2)dij — |M|^i .

— I ll^i

When \k\ > 9,

f\\ dt

J 11 T D->

,0 N il^2

■|k|2-a

J ' ll»ke"|4| f|L„d<

J|k |k

+ ')|k|2-a \he'm° f\Ld<

|k|2-a xq/r

— ( Ik^IIld, I di

r q/ r

+ ll^i

Nrn(1/2-1/p2)

fllLi J

■■ -f"LD, (|k|2— (i\k\"-2)....... 'r2'dt

— Il^kf IIlD, \k\(2-")q/r + ||^kf IIld, (\k\(a-2)r"(1/P2 -1/2) |~2 a i-r»(1/2-1/^2)di)

— |Kf||L, \k\(2-a)q/r

This shows that if r > q, then

m;2 >,(r")

Lr(0,m)

?(1 + \k\)Sq(k) (2-")q/r||^kf||

eit|A| f II

Lr(0,m)

/•to

?(1 + \k\)

srll it|A|a'^|| J.

"nke 1 f II di k J IIlp2

As in the previous case,

|0 ||^ke't|A|a/2f||LD2di — ll^kf||HDl

¡t|A|a

f |l dt

J M T D->

for \k\ < 10, and

/•TO .

(1 + \k\)sr I L

J0 ll * IIlD2

— (1 + \k\)Sr||^kf||LDi \k\(a-2)r"(1/2-1/P2)

/•TO

X J i-r"(1/2-1/P2)di

J|k|2-a

/•TO

(1 + \k\)sr||^kf||LDi | j|k|

-rn(1/2-1/p2)

— (1 + \k\)sr||^kf||LDi (k)(a-2).

This shows that

m;2 >,(r")

Lr(0,m)

?(1 + \k\)sr||^kf||L,i (k)(a-2) k

f ll^(,2_a)/r(r").

The theorem is proved.

5.3. Schrodinger Equation. Consider the Cauchy problem of the linear free Schrodinger equation

idtu = Axu, (t,x) e R+ x R" u (0, x) = uQ (x). The formal solution to this equation is given by

" (t'x) = T^ J e't|?| "0 ($) e2n'^xd<; = (e ,iAU0) (x).

(2n) Jr»

By Theorem 1, we obtain the growth rate as t ^ m for the solution to the linear free Schroodinger equation.

Theorem 25. Let u(t, x) be the solution of the above Cauchy problem of the Schrodinger equation. For 0 < p < m, one has

\\u ft 0H,(R») *(1 + i"(1/i-1/2)) kl^r^ (176) where the asymptotic factor ^"(1/i-1/2) is sharp ast ^ m.

ml ,„(r»)

ml >„(r")

5.4. Linear Cauchy Problem with Negative Power. We start with the following linear Cauchy problem with negative power:

idtu + iA^^ a/1u = 0, (t, x)e R+ x R", a > 0,

u (0, x) = uQ (x). The formal solution to this equation is given by

u(t,x) = -^ f (!) e2^*d! = (e-^^k) (x).

(2n) Jr» v /

Proposition 26. Let a > 0 and 1 < p < 2. One has

-it|A|-

<(1+ t)(["/21+1)|2/^-1| I ¡nf

|i|»<1

hr(r")'

1 n + a([n/2] + 1)(2- p) r np

Proof. We only prove the case of odd n, since the proof for even n is similar. For any fixed

we write

„-¡t|A|-

1—n f ad^fe'^\!\-ßf(!)e2^xd! (182)

n) Jr»

where Iß is the Riesz potential of order ß. The kernel of

nQ|A|ß/2e-'t|A|-a/2 is

Q„'ß (x) = f \!\ße't|?r"a(!)e2m?-d!. (183)

We first show

¡n^ *f||Ll <(1 + t)[2]+1||f||Ll.

To this end, by Young's inequality, we need to show

Ml1(r») <(1+t)

[n/21+1

It suffices to show the case t > 1. As the same argument in the proof of Theorem 1, with the Schwarz inequality we have

"-A« <-tn"+L\Îr. \^"|ir a^2"'t*d!

d x. (186)

Performing integration by parts (n+1)/2 times on the second term, without loss of generality, we may write

\f \!\ße,t|?ra(!)e2m?'*d!

J|x|>t \ JR"

(n+1)/2

■>|x|>t ^

(n+e)/2

* \r„ V (! e't|^|-"a (!) e2ni^'Xd!

d x, (187)

where a(!) is a Cœ function supported in [-1,1]n and f(!) is a function satisfying

\f(!)\<\!\

ß-((n+1)«/2)-((«+1)/2)

Choose a small e > 0 such

2ß-a(n+1)-e>0.

Thus by Schwarz's inequality and the Pitt's theorem, we obtain

\\\\\\ d x

\!\ße,t|?ra(!)e2m?'*d!

J|x|>t \ JR"

(«+1)/2 I

xe2n^xd!

t("+1)/2( f \a(!)\2\!\-"-£\!\2ß-a("+1)d!)1/2 < t

v Jr» J

(n+1)/2

Combining these estimates, we have

-¡t|A|-

fll <t("+1)/2||Ißf||L1 <t

(n+1)/2||

,1 ^ (191)

where L-^ denotes the L1 Sobolev space of order -p. On the other hand, we have the easy energy estimate

-it|A|-

f |L2 < Wf\\L2 ■

An interpolation yields that for all 1 < p < 2,

||rcQe-it|A f || < t("+1)(1/^-1/2)||

y=ß(p -1 '■

We now use the Sobolev imbedding theorem and the almost orthogonality of {nk} to obtain

-it|A|-

f\\ < t("+1)(1/i-1/2) I lln.fl J llri(n») ¿-I \\ JJ \

|i|»<1

hr(r")

1=1 + ß(2/p-1) = n + ß(2-p) (196) r p n np

Since ß is an arbitrary number larger than a([n/2] + 1), the proposition now follows from Lemma 13. □

Lemma 27. Let a > 0 and 2 < p < >x>. One has

-;t|A|-

f\\ * I h\f\ lll2(r") |,6i ' '

hr(r")'

for any r <2.

Proof. The almost orthogonality (Identity (46)) and the energy estimate give

„-¡t|A|-

l2(r")

* I lb/1

l2(r")'

Thus the lemma follows from Lemma 13. □

Now we are ready to give the proof of Theorem 5.

Proof. The proof of (i) can be obtained from Propositions 18 and 26, and the definition and the modulation spaces. Similarly, the proof of (ii) follows by Proposition 18, Lemma 27 and the definition of the modulation spaces. □

5.5. Nonlinear Cauchy Problem with Negative Power. Now, we study the following Cauchy problem of the nonlinear dispersive equations (NDE):

i^-\A\-a/2u + F(u) = 0, at

u (0, x) = u0 (x),

where F(u) = \u\2ku for some positive integer fc. For T > 0, the space Y = C([0, T], Msp1) is defined by

C([0,T],M:^>1)

u(t,x) : ||m|c([o,T]>ms l) = sup IIu(t,-

w 0<t<T

Our proof will follow the same method used in [8], or, more precisely, the idea introduced in an earlier paper [13]. Now we give the proof of Theorem 6.

Proof. In the proof, the letters Cj, j = 1,2,3 denote some positive constants that are independent of all essential variables. We write the Cauchy problem in the equivalent form

u (t, ■) = eit|Ar'2MQ - i f e'^I^F (u (r, ■)) dr, (201)

and consider the mapping

i [ e,(t-r')|Ar/2F(u(т,■))dт. (202) Jo

Iu = e u0 - i I e

We want to show that I is a contraction. By Theorem 5 and Lemma 12,

II it|A|-"/2 ||

II UÜ||y

<C (1 + T)(["/21+1)I2/P-1I||U II +C IIU II < + i) IIuü|IM^7(r») + ^IFOHM^rt

By Lemma 16, there is a constant A 2fc+1 > 0 for which we have

Thus, by Theorem 5 and Lemma 12,

III e'(t-T)|Ar"/2F(u(r,-))dr

IIIIII ü III

\\u(t,-)\2ku(t,-)\\MS < A2k+1lIu(t,-)Cl

C([0,T]>M^i)

* sup [ \\e

0<t<T Jo

[ l^w^F (u(t,-))\\ dt

Jo II Kl

C1 sup f(l + (í-т))(["/2]+1)|2/P-1|

0<t<T Jo

x ||\M(T,-)\2fcM(T,-)||MS dT

< C2MkT(1 + T)«"^1^-1 sup IIu(t,-)

,2fc+1

* C2MkT(1 + D(["/21+1)|2/P-11 sup \\u(tr)

Q< t<T

The last inequality is because

r(2k+1)>p

and the imbedding relation Msp 1 c M*(2fc+1) 1. As a consequence,

\\1u \c([Q,T1>M^>1)

= sup \\3w(f,-)\\MSi

*c3(1+ T)([n/21+1)I2/p-11

2fc+1 •s

,2fc+1

x ( KIIm^r") + T sup IIu(t,-)IIu:

+ c1|IMo|Ims,1(R").

Fix an L > 0 such that

c3lhLs7(r") + c1IImoIIm^>1(r") < 4.

MS(2t+l),

Let BL be the closed ball of radial L centered at the origin in the space C([0, T],Msp1). Nowwe choose T0 such that for any u e BL and T e (0,T0],

ll1u^C([Q'T1'M^,1)

<C3(1+ T)([n/21+1)|2/p-1|

r 2fc+1

q|im^,1 (r")

By this choice, I is a mapping form BL into BL. Furthermore, an easy computation gives that

W1u - 1vyc([o>T]>M^,i)

<C3 (1 + T)(["/2]+1)|2/^-11

(2k+1)L

x T sup llu (t, ■) - V (t,

< V^C([0>T],M^>1)'

for all T e (0, T0] if T0 is suitably chosen. Finally, using the Banach contraction mapping theorem, we obtain a fixed point in BL. This is the solution of the Cauchy problem. We prove the theorem. □

Using a similar argument with the help of (ii) in Theorem 5, we can prove Theorem 7.

Acknowledgment

This work is partially supported by the NSF of China (Grants nos. 10931001,10871173,11201103).

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