Christ M., Kenig C.E., Sadosky C. Harmonic Analysis and Partial Differential Equations: Essays in Honor of Alberto P. Calderon

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Chapter 4: Transference Couples and Weighted Maximal Estimates

have (in the notation of (4.27))

{Wkj ,y}j>1 C Mp,_,(R), and

'P,",

({WkJ,,y}j>1) < Kp,w

IIWIIMp.,(R) NP

({kj}j>1)

Theorem 4.20 can be extended so as to transfer maximal multiplier esti-

mates from LP(T) to L7(w). Let Mp(Z) be the space of Fourier multipliers for

17(T). Given a sequence of Fourier multipliers {Oj}j>1 C Mp(Z), we denote

by 91p ({lbj}j>1) the strong type (p, p) norm of the maximal multiplier trans-

form on LP(T) corresponding to {l6j}j>1. In particular, for {kj}j>1 C L'(T),

we have Np

({k3}3>1)

= 'tp

({}). The extended version of Theorem

4.20 takes the following form.

Theorem 4.21. Suppose that w E Wp, %Y E Mp,,,,(]R), and the support of WY

is contained in [-1/2,1/2]. Then for each sequence {0j}j>1 C Mp(Z), we

have {Wjj,v}j>1 C MP,u,(R), and

9lp,.

({Wws,',}j>1) C Kp,w II41IIMp,u(R)% ({(bj}j>1)

. (4.28)

Proof. It is enough to establish (4.28) in the case of a finite sequence

Mp(Z). Thus we wish to show that

91p'.

1) :5 Kp,, IIWIIMp,,(R)`1p ({Oj} 1)

. (4.29)

In the special case where each (bj, 1 < j < N, is finitely supported, there is

a trigonometric polynomial kj such that kj = Oj, and it follows by Theorem

4.20 that (4.29) holds in the case where each cj has finite support.

In order to treat the general case of a sequence {Oj}N j=1 C Mp(Z), let

{jctt}'

denote the Fejer kernel for T, and for n > 0, 1 < j < N, put

latt j = l 5,.. Since O,,,j is finitely supported, we have for each n > 0:

Kp,.

and consequently

91".

({Won

}j_1) < Kp pp IIMp.W(R) °.p ({4 } 1) (4.30)

Temporarily fix j in the range 1 < j < N.

It is easy to see that

the sequence IWOn,j,*}0

is uniformly bounded on R and tends point-

wise on R to WO" 'F as n -* oo.

It follows that for f E S(R), we have

f) - ,p

f)vIIL2(R)

-+ 0, as n -4 oo. Hence as n -+ oo,

II1

sup

I (Wmn.j,k

1)V

1sup

I (Wm,, f JV III

-3 0.

L (R)

Berkson, Paluszyziski, and Weiss

This furnishes a subsequence {sups<j<N

I) vI}

which conver-

ges

v_1

a.e. on R to supl<j<N

I(Wij,. f )vl. Using this together with (4.30),

can apply Fatou's Lemma to deduce that for each f E S(R),

sup

(W0j,,p

I<j<N

Kp,w

II LQ(w)

LP(w)

(4.31)

It follows, in particular, that { Wo 4, } j

C Mr,. (R). Since S(R) is dense

in L1(w), we can now use (4.31) to obtain (4.29), and thereby complete the

proof of Theorem 4.21.

A straightforward "surgical" procedure will enable us to pass from Theorem

4.21 to the case where %F is assumed to be an arbitrary compactly supported

element of Mp,w(R). The specific outcome is formulated as follows.

Theorem 4.22. Suppose that w E Wp, P E Mp,w(R), and T has compact

support. Let N be a nonnegative integer such that the support of ' is con-

tained in [-N - (1/2), N+ (1/2)]. Then for each sequence

C Mp(Z),

we have

}j>1 S M,,,w(R), and

9tp ,

({Wwj,'p}j>1 ) (2N + 1)Kpw N'IIM,,.(R) "p ({oj}j>1)

Proof. We can assume without loss of generality that T vanishes outside

[-N - (1/2), N + (1/2)). For -N < k < N, let Wk = X[k-(1/2),k+(1/2))T,

and let Tk be the translate of 'Pk by k. Thus, 'P = Lk

T k, and, for

-N < k < N, Tk has support contained in [-1/2,1/2]. Given a sequence

{¢j}j>1 C Mp(Z), we apply Theorem 4.21 to Tk. This gives us:

` tp,w (IWo"Tk }j>1) < Kp,w II'`IIM,,W(R) `nP ({(pj}j>1) ,

for - N < k < N.

Since 91p,w ({WOj,rk}j>1) = 9Zp,w ({w,,,vk}j>1), for -N < k < N, we

have

`npw

,yk}j>1)

< Kp,w (I1`IIM,,,(R) 97p

for - N < k < N.

(4.32)

The remainder of the proof is evident from (4.32) and the identity

W4,,,y=

for all j> 1.

k=-N

Chapter 4: Transference Couples and Weighted Maximal Estimates

We close the discussion with some observations about the relationship

between the existing literature and our results in this section. Theorem 4.22

constitutes a weighted generalization of the unweighted result in [2, Theorem

(1.1)]; that is, Theorem 4.22 essentially reduces to [2, Theorem (1.1)] when

w E Wp is specialized to be the function identically one on R. In the special

case of Theorem 4.20 where w = 1, %F = X(_1/2,1/2), and {k3},>1 is the

Dirichlet kernel for T, we have the unweighted Carleson-Hunt Theorem for

LI(R), 1 < p < oo, which was shown by other methods in [14]. The situation

regarding the weighted Carleson-Hunt Theorem in the setting of R is more

complicated. It is known in the literature that the demonstration in [12]

of the weighted Carleson-Hunt Theorem for T can be adapted to provide a

weighted Carleson-Hunt Theorem for L1(w), w E Ap(R), 1 < p < oo (see the

comments in [10, p. 466]). Theorem 4.20 falls short of this much generality,

since it only provides the weighted Carleson-Hunt Theorem for L1(w) when

wEWW.

References

[1] N. Asmar, E. Berkson, and T. A. Gillespie, Transferred bounds for square

functions, Houston J. Math. (memorial issue dedicated to Domingo Her-

rero), 17 (1991), 525-550.

[2] ,

On Jodeit's multiplier extension theorems, Journal d'Analyse

Math. 64 (1994), 337-345.

[3] P. Auscher and M. J. Carro, On relations between operators on RN, TN

and ZN, Studia Math. 101 (1992), 165-182.

[4] E. Berkson, J. Bourgain, and T. A. Gillespie, On the almost everywhere

convergence of ergodic averages for power-bounded operators on LP-sub-

spaces, Integral Equations and Operator Theory 14 (1991), 678-715.

[5] E. Berkson and T. A. Gillespie, Spectral decompositions and vector-valued

transference, in Analysis at Urbana II, Proceedings of Special Year in

Modern Analysis at the Univ. of Ill., 1986-87, London Math. Soc. Lecture

Note Series 138, Cambridge Univ. Press, Cambridge (1989), 22--51.

[6]

, Mean-boundedness and Littlewood-Paley for separation-preserving

operators, Trans. Amer. Math. Soc. 349 (1997), 1169-1189.

[7]

, Multipliers for weighted D' -spaces, transference, and the q-variation

of functions, Bull. des Sciences Math 122 (1998), 427-454.

Berkson, Paluszyriski, and Weiss

[8] E. Berkson, M. Paluszynski, and G. Weiss, Transference couples and their

applications to convolution operators and maximal operators in "Interac-

tion between Functional Analysis, Harmonic Analysis, and Probability,"

Proceedings of Conference at Univ. of Missouri-Columbia (May 29-June

3, 1994), Lecture Notes in Pure and Applied Mathematics, vol. 175, Mar-

cel Dekker, Inc., New York (1996), 69-84.

[9] K. de Leeuw, On Lp multipliers, Annals of Math. 81 (1965), 364-379.

[10] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities

and Related Topics, North-Holland Mathematics Studies 116 = Notas de

Matematica (104), Elsevier Science Publ., New York (1985).

[11] R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities

for the conjugate function and Hilbert transform, Trans. Amer. Math.

Soc. 176 (1973), 227-251.

[12] R. Hunt and W.-S. Young, A weighted norm inequality for Fourier series,

Bull. Amer. Math. Soc. 80 (1974), 274-277.

[13] M. Jodeit, Restrictions and extensions of Fourier multipliers, Studia

Math. 34 (1970), 215-226.

[14] C. Kenig and P. Tomas, Maximal operators defined by Fourier multipliers,

Studia Math. 68 (1980), 79-83.

Chapter 5

Periodic Solutions of Nonlinear

Wave Equations

Jean Bourgain

We are investigating here the construction by the [5] method of space and

time periodic solutions of one-dimensional nonlinear wave equations of the

following form

Dy + y3 + F(x, y) = 0, F(x, y) = 0(y4).

This more resonant case, due to the absence of a linear term p y, was left aside

in [5]. Second, we consider periodic solutions of certain non-Hamiltonian

NLW, with nonlinearities involving derivatives (of first order). The model

case discussed here is the equation

Ely + py + (yt)2 = 0,

p#o,

but the method is by no means restricted to that example. It mainly demon-

strates the applicability of the Lyapounov-Schmidt decomposition method to

construct periodic (or quasiperiodic) solutions independently of Hamiltonian

structure. In the last section, we construct smooth families in a of periodic

solutions of NLW

Dy+py+y3+ef(x,y) =0

with p = p(E) a parameter depending smoothly in e. This result is in the

spirit of [12, 13]. We also comment on issues concerning admissible frequen-

cies for the perturbed invariant tori.

The author is grateful to C. Wayne for many comments and improvements

of earlier versions of the paper.

Chapter 5: Periodic Solutions of Nonlinear Wave Equations

5.1

Periodic Solutions of Nonlinear Wave

Equations of the form y + y3 + 0(y4) = 0.

Consider the 1D NLW equation

ytt - Yxx + y3 + F(x, y) = 0,

(5.1)

wherel

F(x, y) = 0(y4).

(5.2)

Thus we have no linear term my, m # 0, at our disposal as in an equation

of the form

ytt - yzx + my + 0(y

= 0 (5.3)

as considered in [5]. We assume F(x, y) is an even periodic function in x and

in fact take for simplicity F of the form

F(x,y) =

aj(x)y',

(5.4)

4<j<d

where the aj are trigonometric cosine polynomials in x (see comments later

on). Our aim is to construct periodic solutions of (5.1) with periods in a set of

positive measure, using the method from [5] combined with some ingredients

from [11]. In [5], equation (5.3) with m 54 0 was considered and the more

resonant case (5.1) left open. In [11], periodic solutions of the equation

ytt-yxx+y3=0

(5.5)

are produced for periods of bounded type (this avoids the small divisor prob-

lems). As in [11], our starting point is the explicit solution by the 4K-periodic

elliptic function

cn ',

2 2a '

ex n-t s +e-x n-t z

COS 2K (2n - 1)t;

7 n=1

of the differential equation

Cu+C3=0.

'It should be mentioned that, if F(x, y) is x-independent, one may easily construct

periodic solutions of the form y(x, t) = y, (z + At) since the corresponding ODE satisfied

by yl is Hamiltonian. This observation is due to C. Wayne.

Bourgain

(See [7] pp. 343, 345).

Rescaling (5.1), letting

y -4 by,

(5.8)

we get the equation

lift - yzz +

62y3 + 63F, (X, y) = 0

(5.9)

(d will be restricted to a neighborhood of 0).

Defining

yo(x,t) = 4K c(4K(x + At)) = c(x + At),

(5.10)

where

A =

l + b2, (5.11)

we get thus a solution of the equation

°yo + o2yo = 0, (5.12)

which is 1-periodic in x and T = A-' periodic in t.

The main idea is to

perform a perturbative procedure starting from yo in order to obtain a so-

lution of (5.9). The perturbation scheme is based on the Newton iteration

procedure as in [5], yielding a quadratic error at each step, hence doubly

exponentially fast convergent. It is therefore less sensitive to small divisor

phenomena than if one would perform an expansion in a 8-series (as in [11]).

During this process, a sequence of restrictions will appear that essentially

will restrict 8 (and hence A) to a Cantor-like set of positive measure (as in

[5]).

The solution y of (5.9) will be given by a Fourier series of the form

y(x,t) _ E &, k) cos 27r(nx + kAt). (5.13)

n,kEZ,n>O

Observe that by (5.6) and (5.10)

yo(x, t) = 8,/2- a E

e7r(n-1/2) + e-a(n-1/2)

cos 21r(2n - 1) (x + At). (5.14)

Thus the assumption on F to be an even and periodic function in x is

consistent with a representation of y in the form (5.13), in the sense that

F(x, y(x, t)) remains a Fourier series of the form (5.13). As will be apparent

later on, treating the problem in generality would lead to certain resonance

problems we do not intend to analyze here.

Chapter 5: Periodic Solutions of Nonlinear Wave Equations

We replace yo by a "truncated" version of (5.14), redefining

yo(x, t) = 8v'2-7r

n=1

cos 27r(2n - 1) (x + At)

(5.15)

eir(n-1/2) + e-,r(n-1/2)

taking A .' log

so that in particular

Dyo+62yo =O(63)

and

Dyo + 62yo + 63F1(x,Yo) = O(63).

(5.16)

Following the procedure described in [1,2,4], we construct a solution y of

(5.9) satisfying

Iy(n, k) I < e-(InI+IkU`

(5.17)

where 0 < c < 1 is some constant that will depend on F (in particular on d).

The estimate (5.17) results from the fact that the correction try to the

approximate solution yr_1 obtained at step r - 1 of the iteration will satisfy,

say,

supp try C BZ2(0, (10d)r log 1/6), (5.18)

while

109 Iltryll <

100)rlog 5.

(5.19)

Thus with this procedure, supp yr is contained in some finite region in Z2 at

each stage. It is also possible to obtain a real-analytic solution y (observe

that c given by (5.6) and yo are real-analytic) and assuming more generally

that F(x, y) is real-analytic, even periodic in x, but this requires keeping

track as in [5] of a sequence of real-analytic norms

HOP = E PlnI+IkI

I y(n, k)I, p > 1,

n,kEZ2

(5.20)

along the iteration.

Let yr_1 be the approximative solution to (5.9) obtained at stage r - 1.

The main point in obtaining yr = yr_1 + try by Newton's method is to

control the inverse of the linearized operator

T(try) = O(try) + 352 yr_1(try) + b3(8yFi(x, yr-1)) (try)

(5.21)

Bourgain

with Z2-lattice representation (expressing T passing to Fourier transform)

T == D + 362 Sy2_1 + 63

where D is the diagonal operator

Dn,k = 41r

2 (-k2 A2

+ n2)

(5.22)

(5.23)

and S, denotes the (Toeplitz) operator with matrix elements

SO (x, x') = (x - x') (5.24)

expressing 4-multiplication in Fourier space.

In order to obtain A ,y with the desired properties, we ensure bounds and

off-diagonal decay estimates on the inverse of restrictions

TM = TI

lnl<M, Ikl<M

(5.25)

of the form

IITMifI < 6-2M° (for some constant C)

(5.26)

and

ITMi(x,x')I < b-2e-zlx-a'I`

for say

Ix - 4 > Ma

(5.27)

(cf. [1]).

One then defines try satisfying TM(try)+PM[b2yr_1+63F1(x, yr_1)] = 0,

where PM refers to the projection on Fourier modes InI < M, IkI < M.

At the first step, we consider for T the operator

D + 362 Syo

(5.28)

to obtain thus

IIAIyII = O(51-)

(5.29)

and

El yl + 52yi + 63 F1 (x, yi) = 0(64-).

(5.30)

The main point is the discussion of T-i. The structure of "singular sites"

when InI+1kI -+ oc has here a simple pattern of lacunarily separated elements

as in [5]. However, there is also the behavior of T,yo (Mo a sufficiently large

constant) that has to be taken into account, depending on specific properties

of yo and thus (5.6).