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 5: Periodic Solutions of Nonlinear Wave Equations

satisfying

11011 = O(a2 + e) .

(5.175)

Fix a large number No and let

q=Nol and a"q, IrI <q3.

(5.176)

From (5.169) and (5.176)

Ip-A2+lI

g2.

(5.177)

Hence, for A E A,,oi it follows from (5.172) and (5.171) that

IDf,kI > I - A 2 k

+ n2 + pI - C712 >

I - A2(k2 - 1) + n2 - 1I - Cq2

clkl_a

- Cq2

and thus

IDn,kI > cNo

a for

InI < No.

(5.178)

Therefore the inverse of TNo ==

TIIfI<NO,lkl<No may

be controlled by a

Neumann series and in particular

(5.179)

Since supp y, C B(0, M'') for some constant M, the preceding allows us

to carry out the Newton iteration as long as M,. = Mf < No, leading to an

approximative solution up to e-(2

< e(n)",

say. The control of TT' for

N > No requires further conditions. The main point here is an observation

made by S. Kuksin.2

Assume moreover

ID,,,aI >

I I,/2

for InI < N.

Then (for q small enough)

(5.180)

II DN'SNDN1II <

100

(5.181)

The point is that if some site (n, k), n > No, is "singular" in the sense that

I - A2k2 + n2I < 1, hence IIakII < -L, then for (n1, k1) # (n, k) and nearby

2Private communication.

Bourgain

(n, k), necessarily by (5.160)

- Ak2 + nil

Ik1IIIAk1II > cIk1IIk -

k11-2

cIkI1-. Hence, if IDn,kI

< 1, one gets IDnl,k1I > cInI1- for (nl, kl)

(n, k)

nearby (n, k). Property (5.181) is then easily derived from the preceding,

(5.180) and since IISII < 1j2. It follows that TN1 may again be controlled by

the Neumann series

TN1

= (DN + SN)-1 =

DN'

E(-1)1(SNDNI)'

(5.182)

j>0

(see [3] for all details). We have in particular

1/2

II TN' II

772

(5.183)

In order to fulfill (5.180), we need to restrict the parameters (A, e, a, p). More

specifically, (5.172) yields

C7Dn,k

= 0(1)

(5.184)

and hence, for given (n, k), n > No, (5.180) amounts to removing a set of

,:S 2

measure

-1n7,7,-. Since (5.180) only needs to be verified for In2 - A2k2I < 1,

thus IIakII <

and A E A,,6, there is the following estimate on the total

measure to be removed from the parameter set

el-th < 772N0 1/3 = 9/7/3 .

[ 01/2

Np <l<N

1 clysdic

(5.185)

On the other hand, (5.169) and (5.176) yield for p a variation range of size e.

Observe that when the parameter set excisions first occur, already IIoyII <

e'(1/n)". Thus these excisions are such that the solution to the P-equation

may be extended smoothly to the entire parameter set. There is also the

issue of varying symbol ¢ = c,. given by (5.174). Since at stage r, we let

N = M,. and II0,._1-0,.II < e-N`, this change of symbol clearly does not lead

to problems when fulfilling (5.180). We are reviewing here several issues from

[5]. In conclusion, we may obtain a solution y = ya,E,a,p to the P-equation of

the form (5.163), provided that the conditions

IDn,kI >

X1/2 for Int < M,, (5.186)

(only to be verified for N0 < I nI < MT) are fulfilled and where moreover

ya,E,a,p is smoothly defined on the full parameter set.

Fix A E A,,6. We choose a number v(A) such that

v(A) N'I2

(5.187)

Chapter 5: Periodic Solutions of Nonlinear Wave Equations

and

_)2(k2-1)+n2-1+v(A)I> 10I i1i2 fork01,-1.

(5.188)

The same calculation as (5.185) shows that this is possible.

Next we specify (5.162) requiring p and a to solve the equations

-a2 + 1 + p + 1 [;3(1,1) + e f (x y)(1,1)] = 0

(Q-equation),

-A2+1+p+3y2(0,0)+EOyf(x) y)(0,0)=v(A),

(5.189)

where y is substituted from solving the P-equation. If the latter equation in

(5.189) holds, then at stage r, Mr > No, we get from (5.188)

IDn,kI >

I - a2(k2 - 1) + n2-1 + v(A)I + O(IIy - Y'-11)

/ -Mr

> 10 jnj1ze

>InJ/iz

and (5.186) is fulfilled. Since the equations in (5.189) appear in the form

lsa2+0(a3+e)

= 0

4a2 + 0(a3 + e) = v(A)

with

(5.190)

p= A2_ 1 +0

(5.191)

one obtains from the implicit function theorem p = p(e), a = a(e)

v(.1) - 77 as smooth functions of E.

References

[1] J. Bourgain, Construction of quasiperiodic solutions of Hamiltonian per-

turbations of linear equations and applications to nonlinear PDE, Interna-

tional Math. Research Notices 11 (1994), 475-497.

[2] ,

Construction of periodic solutions of nonlinear wave equations

in higher dimension, GAFA 4 (1995), 629-639.

[3)

, Nonlinear Schrodinger equations, IAS Park City Mathematics

Series, vol. 5 (1999), 5-157.

Bourgain

(4)

, Quasi-periodic solutions of Hamiltonian perturbations of 2D lin-

ear Schrodinger equations, Annals of Math. 148 (1998), 363-439.

[5] W. Craig and C. Wayne, Newton's method and periodic solutions of non-

linear wave equations, Comm. Pure and Applied Math. 46 (1993), 1409-

1501.

[6] L. Eliasson, Perturbations of stable invariant tori, Ann. Sc. Super. Pisa,

Cl Sci. IV, Ser. 15 (1988), 115-147.

[7] A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher tran-

scendental functions (Bateman ms.) (1953).

[8] J. Frohlich, T. Spencer, and P. Wittwer, Localization for a class of one

dimensional quasi-periodic Schrodinger operators, Comm. Math. Phys.

132, no. 1 (1990), 5-25.

[9] S. Kuksin, Perturbation theory for quasi-periodic solutions of infinite di-

mensional Hamiltonian systems and its applications to the Korteweg - de

Vries equation, Math. USSR Sbornik 64 (1989), 397-413.

[10]

, Nearly Integrable Finite-Dimensional Hamiltonian Systems,

Springer LNM 1556 (1993).

[11] B. Lidskij and E. Schulman, Periodic solutions of the equation ii - uxx +

u3 = 0, Funct. Anal. Appl. 22 (1988), 332-333.

[12] J. Moser, On the theory of quasi-periodic motions, Siam Review 8 (1966),

145-172.

[13] ,

Convergent series expansions for quasi-periodic motions, Math.

Annalen 169 (1967), 136-176.

[14] C. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equa-

tions via KAM theory, CMP 127 (1990), 479--528.

Chapter 6

Some Extremal Problems in

Martingale Theory and

Harmonic Analysis

Donald L. Burkholder

6.1

Introduction

The main new results are Theorem 6.3 and its corollary for stochastic inte-

grals, Theorem 6.6. However, we begin with a little background with roots

in some of the work of Kolmogorov and M. Riesz of more than seventy years

ago. Let n be a positive integer, D a domain of R", and H a real or complex

Hilbert space. Suppose that u and v are harmonic on D with values in H.

Let JVul = (E

Jauiaxk12)1/2.

Then v is differentially subordinate to u if,

for all x E D,

IVv(x)I < +Du(x)l

Fix a point C E D and let Do be a bounded subdomain satisfying

CEDoCDoU8DocD.

Denote by p the harmonic measure on 8Do with respect to . If 1 < p < oo,

let

r/ 11/p

IIull,=supIJ

lul'dpI

LL ODo

where the supremum is taken over all such Do.

Research partially supported by NSF grant DMS-9626398.

100

Chapter 6: Some Extremal Problems in Martingale Theory

Theorem 6.1. [5]. If Jv(l;)J < lu(C)l and v is differentially subordinate to

u, then

i(IvI ? 1) < µ(Iul + Ivi > 1) < 2IIuuti

(6.1)

and even for the inequality between the left and right sides of (6.1) the con-

stant 2 is the best possible. Furthermore, if 1 < p < oo and p' = max{p, q}

where 1/p+ 1/q = 1, then

IIvII, < (p' -1)IIullp.

(6.2)

To see that (6.2) holds, define U and V on )Ell x lRl by

U(h, k) = p(l - i/p*)P-'(Ikl - (p* - 1)Ihl)(Ihl + Ikl)p-1 (6.3)

and

p h p

k k `

- 1) l l

V(h,

I - (p

) = J

( . )

Then U majorizes V, U(h, k) < 0 if Jkl < Jhi, and U(u, v) is superharmonic

on D. Therefore,

f V (u, v) dy < f U(u, v) dp < U(u(e), v(t;))

0, (6.5)

which gives (6.2). See [5] for the details and an application to Riesz systems.

The proof of (6.1) has the same pattern. If Ihl + Iki > 1, let

U(h, k) = V (h, k) = 1 - 2Jhl.

(6.6)

Otherwise, let

U(h, k) = Jkl2 - jh12 and V(h, k) = -21hl.

(6.7)

It is easy to check that, as above, U majorizes V, U(h, k) < 0 if Ikl < Jhl,

and U(u, v) is superharmonic on D. So (6.5) holds here also and gives (6.1).

That the constant 2 is the best possible even for the inequality between the

left and right sides of (6.1) is shown in Remark 13.1 of [7].

In Theorem

6.1, the usual conjugacy condition on v in the classical setting of the open

unit disk is replaced by the weaker condition of differential subordination, a

condition that makes sense on domains of R". Inequalities similar to (6.1)

and (6.2) can hold even if u and v are not harmonic. As above, fix e E Do

and denote the harmonic measure on 8Do with respect to l; by p.

Theorem 6.2. (7]. Suppose here that u : D -4 [ 0, co) and v : D -5 IRE are

continuous functions with continuous first and second partial derivatives such

that

Iv(0I < Ju(l;)J,

(6.8)

Burkholder

IVvI $ IVul,

IovI < loin.

(i) If u is subharmonic, then

u(1v;1 > 1) < ft(u+ Ivl > 1) < 311uIIi.

(6.11)

Moreover, if 1 < p < oo and p`" = max{2p,q} where 1/p+ 1/q = 1, then

Ilvlly <_ (p** - 1)IluII .

(6.12)

(ii) If u is superharmonic, then (6.11) holds with the constant 2. This is the

best possible constant even for the inequality between the left and right sides.

After appropriate functions U are found the proof follows the same pat-

tern as the proof of Theorem 6.1 (see [7]). These functions U were discovered

originally in a martingale setting [4, 7]. In this setting, p' - 1 is the best

possible constant for the martingale analogue of (6.2), and the same is true

for the other constants mentioned in Theorems 6.1 and 6.2. However, it is an

open question whether or not p' - 1 is the best possible constant for (6.2).

It is also open whether 3 is the best constant for (6.11) and whether p" - 1

is the best for (6.12).

The Beurling-Ahlfors transform (the singular integral operator on IP(C),

1 < p < oo, that maps f to its convolution with -1/7rz2) is important in the

study of quasiconformal mappings and elsewhere; see, for example, the paper

of Iwaniec and Martin [13] and references given there. The LP(C)-norm of this

operator is not yet known precisely, except for p = 2, although this knowledge

would be helpful. The function U defined in (6.3) has been used recently (see

Banuelos and Wang [2], and Bai uelos and Lindeman [1]) to obtain sharper

information than previously known about the norm of the Beurling-Ahlfors

transform. Both papers outline several approaches that might lead to the

precise value of this norm. One of these approaches, attributed to the referee

of [1], is the following. The function U gives rise to a rank-one convex function

on the set of 2 x 2 matrices with real coefficients [12, 1]. It is an open question

[10] whether or not a rank-one convex function on the 2 x 2 matrices is also

quasiconvex. If the answer is positive, then the norm of the Beurling-Ahlfors

operator is p' - 1.

6.2 A Martingale Setting

One of the goals of the next few sections is to explain, justify, and augment

the following equalities and inequalities, which give some insight into the

behavior of martingales and submartingales.

fy(posmar) = /3p(mar) = /3p(possub) if 1 < p < 3/2,

(6.13)

102

Chapter 6: Some Extremal Problems in Martingale Theory

f3P(posmar) = /3P(mar) < 33p(possub) if 3/2 < p < 2,

(6.14)

/3P(posmar) < /3P(mar) < /3 (possub) if p > 2.

(6.15)

Suppose that (1), F, P) is a probability space filtered by (fn)n>0, a non-

decreasing sequence of sub-a-algebras of P. Let fn and gn be strongly in-

tegrable, F -measurable functions on 9 with values in H, n > 0. Then

f = (fn)n>o is a martingale under the additional assumption that the condi-

tional expectation E(fnIFn-1) = fn_1 for all n > 1.

Write

fn = E dk and gn = E ek

k=0

and suppose that g is differentially subordinate to f :

Ienl < IdnI for all n > 0. (6.16)

Let IIf I IP = suPn>0 IIf,,I IP and denote by #,,(mar) the least extended real

number 0 such that

IIgOIP < /3IIfIIP

(6.17)

for all martingales f and g as above. The probability space and the filtration

(Fn)n>o vary with the pair (f, g). However, it is no loss to assume that

the probability space is a fixed nonatomic space such as the Lebesgue unit

interval.

Define /3P(posmar) similarly: f is a nonnegative real-valued martingale

and g is an H-valued martingale that is differentially subordinate to f. Here,

of course, do in (6.16) is real-valued. The sequence g is conditionally differ-

entially subordinate to f if

IE(enl-FnAI s IE(dnlFn-,)I for all n > 1. (6.18)

Notice that if f is a martingale then the right side of (6.18) vanishes. Define

/3P(possub) to be the least extended real number f3 such that (6.17) holds

for all pairs (f, g) where g is &Ei(-valued and both differentially subordinate

and conditionally differentially subordinate to f where f is a nonnegative

real-valued submartingale. Condition (6.16) has the same purpose as the

conditions (6.8) and (6.9); condition (6.18) that of (6.10). For example, if

I If IIP is finite as we can assume, then

EV (fn, gn) <- EU(fn, gn) < ... < EU(fo, go) <- 0 (6.19)

replaces (6.5). This is the way (see [4] and [7]) to prove that 0(mar) < p* - 1

and fi(possub) < p** - 1. Equality holds as can be seen by examples or by

Burkholder

103

an alternative approach (see [6], [7]).

Note that p` = p" if 1 < p < 3/2

and p` < p"' if p > 3/2. Also, the example on page 671 of [3] shows that if

1 < p < 2, then fp(postnar) _ /3p(mar). Therefore, (6.13) and (6.14) hold.

The proof of (6.15) will be complete once Theorem 6.3 is established in the

next section.

The quantities /3p(posmar), ,dp(mar), and, ,6p(possub) do not depend on

the dimension of the Hilbert space H. They also remain the same if g is

restricted to be a ±1-transform of f (ek = ekdk where ek E {1, -1}, k > 0)

and, as we shall see, in other cases as well.

6.3

On the Size of a Subordinate of a

Nonnegative Martingale

Throughout this section, assume that p > 2 and r = (p -1)/2. Also, assume

in the proofs with no loss that )Ell is the real Lebesgue sequence space

Theorem 6.3. If p > 2, then

/3P(posmar) = prp-t. (6.20)

So if p > 2, then (p - 1)/2 < flp(posmar) < p/2 < p - 1. Accordingly,

(6.15) holds.

Proof. The first part of the proof is to show that the inequality

II9lIp 5 prp-tilfllp

(6.21)

holds for all martingales f and g relative to the filtration (-Fn)n>a such that

g is H -valued and is differentially subordinate to the nonnegative real-valued

martingale f

. We can and do assume in the proof that I If I Ip is finite. Then,

by differential subordination, gn is in L":

n n

I9.1 5 E Iekl 5 F, Idkl 5 21: IfkI

k=0 k=0 k-0

Here define V : [ 0, oo) x H -> R by

V (X' y) = Iyl" - prp-txp.

Then the expectation EV (fn, gn) is equal to

nllp -

p-tlifnlip

Il9

and (6.21) follows from the inequality

(6.22)

EV(fn,gn) < 0. (6.23)