Essays on Fourier analysis in honor of Elias M. Stein

an,


= r~
The function
F(O, w)
70

CHAPTER 3

where we have written w = x + I y. The support of ,p, is contained in the square {1 ~ x ~ IYI ~ §}. Therefore, .jX is well defined on the support of ,p,. Using the mean value property of F in the first variable, we obtain

f,

11

2Jl

F(O, r) = - 1

21l'

WEe

F(.jXe iO , x

Now, we change variables by letting x F(O, r) =

!.. 1l'

+ IY),p,(X + IY) d()dxdy.

0

-I ( 1 lZEC

= t 2 and then we let z = te'B. We obtain

F(z,

Id + ly),p,(ld + ly)dJ..(z)dy

yER

where d)..,(z) denotes Lebesgue measure on the complex plane. The map (z, y) 1-+ (z, Izl 2 + IY) for z E C and y E R is a parameterization for ~ and dJ..(z)dy is comparable to surface measure on ~, denoted by da. Therefore, we obtain

(1)

F(O, r) =

l

F(S)K,(S)da(S)

where K,(·) is a smooth function. Note that the support of K,(S) is contained in a ball centered at the origin of radius C y'r (where C is a uniform constant). Thus, K, is our desired local representing measure for holomorphic functions on the Heisenberg group (near the origin). The same argument with inequalities replacing equalities implies that if u is a non-negative plurisubharmoriic function on n which is continuous up to ~, then (2)

u(O, r)

~

l

u(S)K,(S)da(l;).

The key idea in the above analysis is to slice n with analytic discs whose boundary is contained in ~ (in the above case, the analytic disc is the map l; 1-+ (zl;, Id + iy) for l; in the unit disc in C). Then we average our holomorphic F (or subaverage our plurisubharmonic u) over the boundary of the discs. A similar slicing argument can be used for a more general convex or strictly pseudoconvex domain and a representation analogous to (I) and an estimate analogous to (2) can be derived. The case of weakly pseudoconvex domains of finite type presents additional difficulties since the boundary cannot always be locally convexified, but here, too, one can obtain local representing measures by imbedding suitable analytic discs (see [BDN]). In this paper, we generalize these ideas to the case of "wedge domains" with an "edge" given by a CR submanifold M C en of real codimension greater than I. Our goal is to find local representing measures on the edge for points in the wedge near the edge. We restrict our attention to the case where the edge of the wedge has type 2, which is the analogue of the case of a domain with an edge given by a strictly pseudoconvex boundary (as discussed above). We have been

REPRESENTING MEASURES FOR HOLOMORPH1C FUNCTIONS

71

greatly influenced in our work by E. M. Stein's seminal observation that a strictly pseudoconvex boundary can be modeled at each point by a nilpotent Lie group, the Heisenberg group, and that the boundary behavior of holomorphic functions in strictly pseudoconvex domains is intimately connected with the approximating group structure on the boundary. In this paper, we carefully analyze a "model case" for our wedge domains where the edge is a nilpotent Lie group of step 2 (analogous to the Heisenberg group). This is done in section 4 of this paper. In section 2, we state our theorem on representing measures for more general wedge domains, and in section 3, we state an application to the boundary behavior of HP functions defined on wedge domains. The details of the results given in sections 2 and 3 will appear in a later paper. The results in section 2 for general wedges with type 2 edges are obtained by a three­ stage process which is again inspired by the work of Stein and his collaborators (see for example [FS] and [RS]): (i) we pass from the original object of study to a "free" object by adding appropriate variables; (ii) we solve the problem on the freed object by approximating it suitably by the model case; (iii) we return to the original object by integrating out the extra variables. The results of section 3 are related to recent work of Rosay [R].

2 THE STATEMENT OF THE GENERAL THEOREM In order to state our theorem, we first recall some basic facts about CR manifolds. For more details, see [B]. Let M be a Coo submanifold of C n of real codimension d. For p EM, let TpM denote the real tangent space to M at p. Then the maximal complex subspace of TpM is T; = TpM n J(TpM), where J is the complex structure map given by multiplication by I = M is called aCR submanifold, with CR dimension m if for all p EM, d imc T; M = m. M is called generic if for all p E M, TpM + J (TpM) = cn. If M is a generic CR submanifold of real codimension d and CR dimension m, it follows that n = m + d. For p E M, let YpM denote the orthogonal complement to T; Min TpM, and let NpM be the orthogonal complement of TpM in TpC n (i.e., NpM is the normal space of M at p). We have

R.

TpM = T; EB YpM Tpc n = TpM EB NpM where all the direct sum decompositions are orthogonal. Let 1l'p: TpC n 1-+ NpM be the orthogonal projection. Let H~·o(M) denote the subspace of the complexified tangent space to M at p spanned by tangent vectors of type (1,0). If Z E Hl.o(M), then the vector field [Z, Z] is a real tangent vector field.

*

-

--,.,-- -

~-.,.~::---------_._-

_.::~-

-----;:-------:::

--

;--

- -

-

.-.

-

_.

~

----

._""'-~

72

CHAPTER 3

Definition 2.1. The Levi form is the well defined quadratic mapping £p: H~·o(M) ~ NpM given by

£p(Z)

= -1r p(J( -1 [Z, ­Z]p». 21

-

----:,.--.::

73

REPRESENTING MEASURES FOR HOLOMORPHIC FUNCTIONS

of finite type (see [NSW] for more details). Suppose L 1 = X I + 1 YI, ... , L m X m + 1 Ym is a basis for HI.o(M) on an open set w eM.

=

Definition 2.4. For pElt) and 8 > 0, let

where Z E HI.o(M) is any vector field extension of Zp. The convex hull of the image of £p is a cone in NpM and is denoted by r p'

R(P,,)

~ (exp[t. jXj +PjYj +YT}P);

.j,Pj,Y E R

Definition 2.2. Let M C C n be a generic CR submanifold. Then M is of type 2 at p

E

M if and only -if

r p has nonempty interior in N pM.

with

This definition is equivalent to the condition that the real and imaginary parts of the tangential Cauchy-Riemann vector fields, together with their commutators, span the real tangent space to M at p. For any subset K C NpM and any E > 0, let K, = {z E NpM I Z E K, and Izl < E}. If Yl and Y2 are two cones in NpM, we say that YI is smaller than Y2 and write YI < Y2 if YI n S is a compact subset of the interior of Y2 where S is the unit sphere in N pM. We now describe the type of wedge domains that we will study.

Definition 2.3. Let M C C n be a generic CR submanifold oftype 2. An open set Q C C n is a domain with edge M if: (1) M C Q;

(2) For each p E M and for each cone Y < containing p and an E > such that

°

It)

+ Y,

r p there exists an open set It)

C M

C Q.

Property 2) roughly states that near a point p E M, Q locally contains translates of M in directions strictly interior to the cone r p' The set It) + Y, is parameterized in a natural way by It) x Y,. After shrinking E if necessary, we can consider the orthogonal projection IT of It) Y, onto M so that for z E It) + Y"

+

z - n(z)

E

Nrr(z)M.

Iz - n(z)1

where the absolute value denotes the length of a vector in Nrr(z)M. We shall need to use the nonisotropic pseudometric and corresponding non­ isotropic balls on M induced by the ambient complex structure on We only summarize the construction, which can be carried out for any CR submanifold M

cn.

... ,

There also exists a pseudo-metric D: M x M

B(p, 8) =

{~ E

~

< 8,

lyl

< 8

2

\,

Lm . R so that

M; D(p, {) < 8}.

These balls have Euclidean dimension 8 in the m complex tangent space directions of M at p and so D behaves roughly Euclidean in these directions. In the d totally real tangent directions, these balls have Euclidean dimension roughly 82 (since M has type 2) and so D behaves roughly like the square root of the Euclidean distance in the totally real directions. We can now state our main result on the existence of a local integral represen­ tation formula for holomorphic and plurisubharmonic functions in domains with type 2 edges.

Theorem 1. Let U C C n be an open set and let M CUbe a generic CR submanifold oftype 2. Let Q be a domain with edge M. Let p E M, let Y < r p, and let L I , .•. , L m be a basis for the space HI.o(M) near p. Then there exist a neighborhood It) C M of p, a constant E > 0, a constant C < 00, and a Coo function

K:

We write

r(z) =

where T is a fixed vector field transverse to L I,

tajl, l,Bjl

{It)

+ y,}

x M ~ [0,00)

with the following properties: (1) For z E It) + Y, fixed, the function K z ({) = K(z, {) has compact support in B(IT(z), Cy'r(z». (2) For every noncommuting polynomial peL, I) of degree k ::: in the vector fields L I , ..• , L m , II, ... , I m , there is a constant Cp so that

°

IP(L, I)(Kz)({)1 ::: C pr(z)-k/2IB(IT(z), Jr(z»I-I.

74

CHAPTER 3

(3) If F is a function continuous on the closure ofthe set nand holomorphic on n, thenfor z E ill + y, F(z) =

L

K(z, OF(O da(O

where da is the surface area measure on M.

n, then for z

u(z)

s

E ill

L

n

+ y,

K(z, Ou(O da(O

where da is the surface area measure on M.

It is easy to see that the kernel for the Heisenberg group constructed in the introduction satisfies the properties listed in the above theorem. The function (z, y) f-+ ¢r(IZ!2 + iy) appearing in that kernel has support in the rectangle {I z I s C y'r, IyI s C r} for a suitable uniform constant C. This set is comparable to the projection of the nonisotropic ball B(O, C y'r) onto the tangent space of 1; at the origin. In addition, from the definition of ¢r we obtain the estimate l¢r(w)1

s

C r2

C ~ IB(O, y'r)1

where I . I denotes Lebesgue (surface) measure. In section 4, we shall prove theorem 1 for a model case for wedge domains, which is a generalization of the Heisenberg group.

= {z

E ill

+ y,;

D(n(z), q)2 sa dist(z, M)}.

Recall that n is the projection from n to M which is defined at least near M, and D(p, q) is the nonisotropic distance between p and q. Also dist refers to the Euclidean distance. Since D is roughly Euclidean in the complex tangential direc­ tions. the approach regions A allow quadratic approach to M along these directions. Since D 2 is roughly Euclidean in the totally real directions. these regions allow only nontangential approach in these directions. Thus the regions A(y, a, q) are the analogues of admissible approach regions for strictly pseudoconvex domains. It is easy to see that if n is a domain with edge M, and if F E Hl~c(n, M). then F has polynomial growth locally near M. To be precise, given p > 0, and given a point q E M and a compact cone y < r q, there is a real number s. a neighborhood ill C M of q and an E > 0 so that for every F E Hl~c (n, M) there exists a constant C so that W(z)1

s

C dist(z, M)-S

forz E ill+Y,. This polynomial growth in turn implies that every F E Hl~c(n, M) has a distributional limit F* along M in the sense that if rp E Cgo(M) then

JM F(~ + EZ)rp(O da(O

lim [ ,t-+O

3 BOUNDARY BEHAVIOR OF H P FUNCTIONS As an application, Theorem 1 can be used to study the boundary behavior of H P functions defined on a wedge relative to a type 2 edge. We begin with the definition of this class of functions. Definition 3.1. Let n c c be a domain with edge M. Thenfor 0 < p < 00, Hl~c (n, M) is the space ofholomorphicfunctions on n such thatfor every q E M and every y < r q there exist E > 0 and a neighborhood ill C M of q so that n

sup ZEy,

1 W(~ +

zW da(O <

+00.

~EW

If P = 00 we require that sup W(z)1 <

+00.

75

Definition 3.2. Let n C C n be a domain with edge M. Assume that M is a generic CR submanifold of type 2. Let qo E M; let y < r qO ,' and let a > O. By the definition ofa domain with an edge, there is a neighborhood ill of qo in M and an E > 0 so that ill + y, C n. Let q E ill. Then define A(y, a, q)

(4) If u is a non-negative function which is continuous on the closure ofthe set

and plurisubharmonic on

REPRESENTING MEASURES FOR HOLOMORPHIC FUNCTIONS

= F*(rp).

Our main results deal with the existence of pointwise and dominated limits. rather than distributional limits, and we obtain a partial characterization of the boundary value distributions. Theorem 2. Suppose M is a type 2 CR submanifold of en and suppose n is a domain with edge M. Let f E Hl~(n, M), 0 < P S 00. For almost all q E M the following holds: Given y < r q and a > 0, then lim

Zt-+q zEA(y,a,q)

fez)

exists.

This limit defines an element of Lt'oc(M). Conversely, if I S P S 00 and f E Lioc(M) is a CR distribution, then f has a unique holomorphic extension F E Hl~(n, M)for some open set n with edge M.

ZEy,+W

Next we define certain nontangential approach regions in domains with edges.

The key ingredient of the proof of this theorem is the following maximal function estimate for plurisubharmonic functions which is derived from the estimates given

--=--------_._--_._._------_._-_._----_._--

-;..; -

- --- _.

~.=

76

-­

CHAPTER 3

in Theorem 1 in section 2. Let u be non-negative and plurisubharmonic on the wedge Q and continuous up to M; then sup

u(z) ~ ey,aM(u)(q)

where M denotes the maximal function associated to our family of nonisotropic balls and ey,a is a constant which depends only on the cone y and the aperture a. This maximal function estimate (with u = IF IpJ2) is used in the proof of Theorem 2 in much the same way as the analogous maximal function estimates are used in the classical analysis of the boundary values of H P functions on domains in the complex plane.

4 THE MODEL CASE In this section, we derive the reproducing kernel for Theorem 1 of section 2 for m2 the following model case. We let M m ~ e denote the complex vector space of m x m complex matrices, and let Hm denote the real vector subspace of m x m Hermitian matrices. For any r x s complex matrix A, let A * denote the conjugate transpose s x r matrix. For W E Mm, set

= ~ (W + W*);

I(W)

= -

1

21

REPRESENTING MEASURES FOR HOLOMORPHIC FUNCTIONS

Definition 4.1. Define p: em x M m p(z. Z)

~m

= {(Z; = {(z;

Define a quadratic form Qm : em

Qm(Z)

H-

H m by setting

= R(Z)

- Qm(Z).

(W - W*).

=

Z) E em x M m; R(Z)

zz*}

,

Z) E em x Mm; p(Z; Z) = O}

Q m = {(Z; Z) E em x

= {(z; Z)

E

M m; R(Z) > zz*}

em x Mm; p(z; Z) > O}

Here, p(z; Z) > 0 means that the matrix p(z; Z) is positive definite. It is easy to check that ~m is a generic CR submanifold of em x Mm ~ e m+m2 of type 2, and that Q m is a domain with edge ~m. We often identify ~m with em x Hm via the correspondence em x H m 3 (Z, Y) ++ (Z; Zz*

+ IY)

E ~m.

A typical point in Q m is of the form (z; Z) where Z = zz* + X + I Y with X, Y E Hm , and X > O. There is a natural projection IT: Q m H- ~m given by

+ X + IY)) =

(z; zz*

+

IY).

We are interested in the existence of analytic discs in Q m with boundary in Every analytic disc in em x Mm is a continuous map A = (Z; W): D H­ em x Mm which is holomorphic on D. We write ~m.

ZI(S)] Z(O

then

H-

77

and

IT((Z; zz*

Then for W E Mm , R(W) and I(W) are Hermitian; W = R(W) + zI(W); and M m = H m EB I H m. This exhibits H m as a maximal totally real subspace of Mm. We shall view elements of em as m x I complex matrices, and so if

z= [ ]

.. "..::_;,;..~K~~..:ci-~,i:;;;.;~iF:~~~~.;..;;_~~j.i;,:fu;~'''''_::_,ib;;jill"_:_.,,,.''''',i~;;;ji~;

Then set

ZEA(y,a,q)

R(W)

=~,=~~=""~.'- ~·-"--"'=-i~o;;'i~.i;.:r.tia:::i;;5;:~'~;.i,~:L'?'''''";;j-~';;w:~''ii;Nj~i,,~;,;r
=

r

:

'

Zm(S')

z* = [ZI, ... ,Zm]'

where each Zj(.) is a (scalar) holomorphic function. W(.) is an m x m matrix­ valued holomorphic function. Such an analytic disc maps the boundary of the unit disc D into ~m if and only if

Hm given by

ZIZ]

Z]Z2

ZIZm

Z2Z1

Z2Z2

Z2Zm

ZmZI

ZmZ2

ZmZm

= zz* =

R(W(e,II))

=

Z(e'Il)Z(e lll )*

for 0 ~ (j ~ 2n. We shall introduce the notation Ao(S) for the special analytic disc Ao(S) = (Zo(S); WoeS))

-

,-.

78

CHAPTER 3

r

{

{2

=

I

{m

L­

and r

1

0

0

... 0

2{

1

0

...

2{2

2{

1

... 0

2{m-l

2{m-2

2{m-3

Wo({) =

F«O; l)

= (

JEm

0

F({)Ko({) dAC{)

where d"A is Lebesgue measure on I:m = em x Hm • Since F is holomorphic on the subset {(O; W) E em x M m; R(W) > OJ, we can use the mean value property to conclude that

L

F(O; l) = (

JM

Note that

F(O; I

+ w)q:>(w) dw,

m

-

R(Wo({»

=I

{

-2 {

...

-m-l {

{

1

{

...

-m-2

{2

{

1

...

-m-3

{

wher~

oE

H, Y

q:> is any radial function with compact support in a small neighborhood of

Mm and with total integral equal to one. Write w = H + I Y where both E Hm • For a suitable function q:> with compact support we have

{

F(O; I) =

11 Hm

{m-2

{m-3

n

-2

-3

...

2-3

... { 2{

Also,

{{

{2"f

Qm(ZO({» =

I{I = 1, R(Wo({»

2-2

{ {

3-2

{3"f

{ {

- {m"f

{m{

-2

{{

{ {

3-3

{ {

-3

{m{

(l

-m

{{

m

Qm(ZO({»

+ H) + IY)q:>(H,

f) dH dY.

...

{3{

...

{m"fm

and so Ao maps the boundary of the unit disc to I:m • If ~ = (~l,"" ~m) E em, then (R(Wo({»~,~) is a harmonic function of {, while (Qm(ZO({»~,~) = I L:j=l {j ~j 12 is a subharmonic function of {. It is easy to check that

= (l

+ X)2,

H = IJI(X) = 2X

+ X2

where X E Hm, and IJI: Hm ~ Hm. This mapping is a diffeomorphism from some small neighborhood of the origin to a neighborhood of the origin. Hence there is a smooth function 1/J with support near the origin of H m x H m so that

F(O; l)

= Zo({)Zo({)*

+ H)

that is,

=

11 Hm

=

F(O; (l

Hm

Note that the integrand has support in a small neighborhood of the origins of each copy of H m • Next, we make the change of variables

{m-l L

Thus when

79

(R(Wo(O» - Qm(ZO(O)))~,;) = 1;1 2 and hence it follows from the mini­ mum principle for superharmonic functions that R(Wo({» - Qm(Z({» > 0 for I{I < 1. Therefore, the analytic disc Ao maps the open unit disc to Qm. We now construct a local integral representation fonnula for Qm. Let F be holomorphic on Qm, and continuous on Qm' Let I denote the rn x rn identity matrix, so that (0; I) E Qm. Our first goal is to construct a repre­ senting measure for the point (0; l) E Q m with compact support and e oo density; i.e., we wish to find a e oo function Ko with compact support, such that

where

Zo({)

REPRESENTING MEASURES FOR HOLOMORPHIC FUNCTIONS

F(O; (l

+ X)2 + IY)1/J(X, Y) dX dY.

Hm

Let U(rn) denote the group of rn x rn unitary matrices, and let dg denote Haar measure on U(rn). Since Zo(O) = 0 and Wo(O) = I, for every g E U(rn) we have

F(O; (l

+ X)2 + If) = F«(l + X)gZo(O);

(l

+ X)gWo(O)g*(l + X) + If)

.....

~

...

~~,.~-'.,'~-_._---

..

- _ .. _ - - - - - - - - ­

,~

---_._. __ ..~"".,"""""'~

---_.

.;;;;;.;...:;.=~--~.-~....",..._----------

;

CHAPTER 3

80

and so F (0; I) is given by the integral

{

{

(

F((l

F(O; /) =

(/ + X)gWO(O)g*(/ + X) + 1y)'I/J(X, Y)dgdX dY. E

(

Jr.",

.

~~ ~~

-- ------------

81

F(~)Ko(~) dA(~)

= {

+ IY)

(

F(z, Y)Ko(z, Y) dY dz;

Jc'" JH",

Hm • By averaging over the analytic disc

3 ~ 1-+ (XgZo(S); XgWo(Og*X

D

--""_..

(1) For everyfunction F continuous on Q m and holomorphic on Q,

+ X)gZo(O);

+ X by the variable X

-~-~_._--"".-_._~---~~------~-

REPRESENTING MEASURES FOR HOLOMORPHIC FUNCTIONS

JH", JH", JU(m)

We replace I

-

(2) For every non-negative function u continuous on Q m and plurisubharmonic on Q,

we see that F (0; I) is given by the integral u(O; I):5 (

_1 {

{

(2Jr:

{

Jr.",

F(XgZo(e'o );

2iT JH", JH", JU(m) J o Xg Wo(e,o)g* X

Note that for any X, Y

E

Hm , g

+ I Y)'I/J(X E

= { Jc'"

- I, Y)d8 dg dX dY

U (m) and 8

E

+ IY)

belongs to :Em. We now study this mapping W: H m x Hm x U (m) x [0, 2iT] :Em. We have

1-+

Lemma 4.2. For each fixed go E U(m) and 80 E [0, 2iT], the mapping W has maximal rank at the point (/,0, go, 80). :Em

(

In,,,

u(z, Y)Ko(z, Y) dY dz.

[0, 2iT], the point

W(X, Y, g, 8) = (XgZo(e'o); XgWo(e,o)g*X

Proof. After we identify

u(~)Ko(~) dA(~)

Our discussion so far has dealt with representing the value of a holomorphic function, but if the function is plurisubharmonic, all equalities are replaced by inequalities and estimate (2) can be established. The representation formula (1) and the estimate (2) are generalizations of (1) and (2) given in section 1 (with r 1) for the Heisenberg group. We can obtain a representation formula for other points in Q m by making use of the group structure and dilations on :Em. Let g be an element of U (m). Both :Em and Q m are invariant under the rescaling map

=

with em x Hm , the mapping W becomes (z; Z) 1-+ (gz; g Zg*).

W(X, Y, g, 8) = (XgZo(e'o), Y) E em x Hmo

In addition for (zo; Zo)

Thus it suffices to show that the mapping

e2J O

IE

I

the map

(z; Z) 1-+ (z

e'o (X, g, 8) 1-+ Xg

E :Em,

em

mlo e

has rank m at the point (/, go, 80)). This is easily shown by restricting X to diagonal matrices and g to diagonal multiples of go. •

+ Zo; 2zz~ + Z + Zo)

also preserves :Em and Om. (This is the map induced by translation under the group

structure of :Em.) Using lemma 4.3 and the translation map and rescaling map,

we obtain the following representation fonnula for holomorphic functions at an

arbitrary point in Qm' To state the result, note that an arbitrary point in Q m can be

: written as (zo; Zo) = (zo; zozo + Xo + I Yo) E Q m where Xo is positive definite

~,(and therefore, Xo has a unique positive square root X0 1/ 2 E Hm ). Theorem 3. There is a non-negative,

Using this lemma, we can integrate out the extra variables in the integral formula for F (0; I) and obtain the following. Lemma 4.3. There is a non-negative function Ko

E ego(:E m ) such that:

that

e oo function K

: Q m x :Em 1-+ [0, 00) such

if (zo; Zo) = (zo; zoz~

then:

+ Xo + I Yo)

E Qm

82

CHAPTER 3

(1) For every function F continuous on Qm and holomorphic on Qm,

F(zo; 2 0 ) =

f f le lH m

F(z, Y)K(zo; 2 0 ), (z,

Y») dY dz.

m

4

(2) For every non-negative function u continuous on Qm and plurisubharmonic on Qm, u(Zo; 2 0 )

::::

1 lH[ em

u(z, Y)K(zo; 2 0 ), (z,

Y») dY dz.

Some New Estimates on

m

Moreover. K(zo; Zoza

+ X o + lYO),

(z,

Oscillatory Integrals

Y»)

= det(Xo)-m-l Ko(X~I/2(Z - zo), X~1/2(y - Yo - 2zza)X~I/2).

Jean Bourgain

If z = (Zo; 2 0 ) = (zo; Zoza + Xo + l Yo) is a point in Qm, then the quantity r(z) = IIXol1 is proportional to the distance from z to :Em. Moreover, in a fixed compact subcone, det(X o) is proportional to IIX oli m • In view of the fact that K o is a bounded function, the above formula implies the following estimate for K:

C IK(zo, z)1 :::: r(z)m 2 +m The CR (complex) dimension of :Em is m and the real dimension of the totally real tangent space of :E (= dimR H m ) is m 2 • Therefore, the right side of the above estimate is proportional to 1/IB(zo, Jr(z»l. This establishes the k = 0 part of the estimate on K given in theorem 1. The estimate for k ::: 1 can be obtained by differentiating the above expression for K by holomorphic tangent vector fields.

Texas A & M University University ofWisconsin, Madison

REFERENCES [BDN] A. Boggess, R. Dwilewicz, and A. Nagel. "The hull ofholomorphy ofa nonisotropic ball in a real hypersurface of finite type." Trans. Amer. Math. Soc. 323 (1991), 209-232. [FS] G. Folland and E. M. Stein. "Estimates for the "fh complex and analysis on the Heisenberg group." Comm. Pure Appl. Math. 27 (1974), 429-522. [NSW] A. Nagel, E. M. Stein, and S. Wainger. "Balls and metrics defined by vector fields I: basic properties." Acta Math. 155 (1985), 103-147. [R] J. Rosay. "On the radial maximal function and the Hardy-Littlewood maximal function in wedges." Preprint. [RS] L. Rothschild and E. M. Stein. "Hypoelliptic differential operators and nilpotent groups." Acta Math. 137 (1976), 247-340.

1 INTRODUCTION This paper is mainly a summary of recent work of the author on the subject of the title, and most of its content relates to [B 1] and [B2]. The main objects underlying the problems discussed here are certain operators expressed by an oscillatory integral, roughly of the form

Tf(x) =

; (1.1)

!

eirp(x,y) f(y)dy

where rp will be a smooth function of x, y (this phase function rp will be more specified later). Our aim will be to understand the mapping properties of T, namely the pairs of (LPl, LP2)-spaces where T acts as a bounded operator, and to estimate its norm. This question is presently only very partially understood, . even in the simplest cases. In this report, we specify the problem to the following , themes: (i)

Restriction and extension problems;

(Ii) The Bochner-Riesz summation operators;

(iii) The (more general) oscillatory integrals considered by Hormander [Hor].

We do not intend to give a survey of these topics but, instead, to comment on some progress made in [B 1], [B2]. We also want to mention the paper [Stl], which the reader may like to consult for more background and related material. (i) The model cases (in the compact setting) are the operators (1.2)

f

t---+

j I Sd-l

(restriction)

84

CHAPTER 4

Considering the form (1.9), restricting the variables x, y to disjoint balls at distance '" N > 1 and rescaling, one gets operators

and (1.3)

J.t E M(Sd-l) t----+

fl

(extension).

'Lt

Here Sd-I = S the unit sphere {x E jRd I xl = 1} in jRd with its invariant measure a = ad-I, and the measure J.t on S considered in (1.3) is assumed absolutely continuous with respect to a. The operators (1.2) and (1.3) are of course formally dual to each other. The main relevant geometrical feature of the sphere is that it is a compact smooth manifold with nonvanishing curvature. Considering locally such a hypersurface as a graph, (1.3) may be expressed by an operator of the form (1.1), namely (1.4)

Tf(x) =

f

ei(XtYt+"'+Xd-IYd-l+xd'/t(Y)) f(y)dy

where x E jRd, Y is taken in a neighborhood of 0 function on that neighborhood satisfying (1.5)

det ( aa;2

E

R d- I ,

1/1

is a smooth

1/1) i= 0,

and f is a function of the y-variable corresponding to the derivative ;; . (ii) The Bochner-Riesz multipliers are defined by the formula

(1.6)

[

m).(~)

~1

:

_

1~12»)'

if I~I ~ 1

with A > O. We are interested in the corresponding Fourier multiplication operator

f

t----+

(i.

m).r,

more precisely, when it acts as a bounded operator on LP(jRd). Thus the operator is given by a convolution with kernel (1.8)

e i1xI m).(x) '" Ixl(d+I)/2+A'

One has obtained estimates using various methods, either dealing with the Fourier-multiplier definition (1.7) or the convolution operator (1.9)

Tf(x) =

f

eilx-YI

f(y) Ix _ yl(d+I)/2+A dy.

TN f(x) =

(1.10)

f

f(y)eiNlx-Yla(x, y)dy

where a is a smooth function on a neighborhood of (0, 0) in jRd x jRd vanishing on a neighborhood of the diagonal. The approach of [C-S] and [Hor] consists of first passing from this jRd _jRd problem to an jRd-1 _jRd problem, by fixing one coordinate of the y-variable, say Yd = 1, keeping x, (YI, ... , Yd-d in a neighborhood of 0 E jRd (resp. jRd-I). The phase function Ix - YI becomes (1.11)

[ (XI - Yl)

2

+ ... + (Xd-l

- Yd-I)

2

+ (Xd

- 1)

2] 1/2

.

Making an asymptotic expansion of (1.11), eliminating the purely x and y terms and performing changes of variables in x and y separately, one is led to a phase function of the form (1.4), but with the presence of non-linear x­ terms 0 (Ix 12 1Y 12). This generalization of (1.4) gives precisely the operations considered in [Hor]. (iii) In [Hor], one studies the behavior of operators TNf(x) =

(1.12)

f

eiN",(x'Y)a(x, y)f(y)dy

where a E CO'(jR2d-I) and ({J E c oo (jR2d-l) is a real valued function satisfying the conditions for (x, y) E supp a: rank a2({Jjaxay = d - I

(1.13)

ifl~I>1

(1.7)

85

SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS

(1.14)

~ ( a({J , t) = ay ax

0, 0

i=

t

2 E ]Rd

===}

det ( a 2 (a({J ay ax

,t)) i= 0,

condition (1.14) meaning that the map y 1-+ (*", t) has only non-degenerate critical points. After coordinate changes, ({J may be given the form (1.15)

({J(x, y) = xlYI

+ ... + Xd-IYd-1 + xd(Ay, y} + O(lxIIYI(lxI 2 + IYI2»

where A is a symmetric matrix with det A i= O. The number N > 1 is a parameter and one seeks uniform estimates of the form (1.16)

IITNfllq < C",N-d/qllfllr

for certain pairs (q, r). Those pairs are specified in [Hor] for d = 2 and the higher-dimensional setting is proposed as a question. Thus the setting (iii) generalizes both (i) and (ii).

86

CHAPTER 4

It is conjectured that m .. defines a bounded Fourier multiplier on LP(~d) iff ). > 0 and (1.17)

2d

d

+ 1 + 2),

2d < P < d - 1 - 2), .

We assume). :s d 21 and p =I- 2 (trivial case). The condition). > 0 is necessary because of C. Fefferman's solution ofthe ball multiplier problem [Fl]. It follows from (1. 8) that the range (1.17) is optimal. It is conjectured that (1.2) maps boundedly from Lq' (~d) to p' (Sd-d or, more generally, (1.4) fulfills an inequality

exploited to disprove the ball-multiplier conjecture. A systematic investigation of the geometric structures in order to approach the Fourier Analysis problems mentioned above was proposed in [F2]. The paper [Bl] contains the first results along these lines of investigation (for d > 2). Surprisingly, for d ::: 3, inequality (1.16) may fail under the conditions (1.19) and hence this approach to the Bochner-Riesz multipliers for d > 2 is a bit too general. In fact, it turns out that if d > 2 is odd, (1.16) may only hold if q ::: 2~~/), even for r = 00. In [B2], we study the case d = 3, showing that for a "generic" phase function ({J one has (1.20)

IITfIILq(IRd) :s Cllfllu(sd_Il

(1.18)

3

for (1.19)

87

SOME NEW EST/MATES ON OSCILLATORY INTEGRALS

< inf

{q

IllTNfll q < CN- 3 /qllflloo for N > I} <

4

II 2d

q > d - 1

and

d+l (d - 1)q

+~:S1. r

The condition (1.19) is the best possible one may expect (recall that o-d-l IXI ,Ll/2 for Ixl ~ 00). One considers the same range (1.19) for the inequality (1.16) (proposed for d > 2 as a question in [HorD. In dimension d = 2, problems (i), (ii), (iii) are essentially completely un­ derstood and the optimal results stated above are known to be valid. Thus the multipliers m A ,). > 0 act boundedly (as a Fourier multiplier) on LP(~2) for p V p' < 4 and (1.16), (1.18) hold for (r, q) interpolated between the pairs (2, 6); (4,4) (not allowing (4,4». See [Stl] for a more complete bibliography. Appar­ ently, the only results (preceding [BID in dimension d :s 3 were those derived from a purely "L 2 -method." Thus (1.18) was shown for r = 2, q > d~1 by P. Tomas [T] and E. Stein (see [StID if r = 2, q ::: d~l' Also (1.16) is shown in [Stl] to be valid in this case. Following [F2], one may then obtain the bound­ edness of (1.7) under the condition (1.17), provided p* = p V p' ::: 2~~/). The argument is based mainly on an L 2 -factorization of the LP-operator, invok­ ing Parseval's theorem and the L 2-restriction result for spheres of [T] mentioned earlier. The author obtained in [B 1] results beyond the range of the L 2- tec hniques. It is shown, for instance, that if IL E M(Sd-d, ~~ E U'O(S), then the Fourier trans­ form fi is p-integrable for some p < 2~~11). Also the Bochner-Riesz conjecture is verified for certain p with p* < 2~~/). The approach makes essential use of certain new estimates on higher dimensional Besicovitch-type maximal operators, which will be described later. The role of such geometric objects became clear in [Fl], where the existence of the 2-dimensional Kakeya-Besicovitch set was

11

2(d

2d

+ 1)

d - 1

d - 1

Here the relevant geometric structures are generalizations of the Besicovitch struc­ tures, in the sense that straight lines are replaced by curves r y' described by the equation (1.21)

B({J

By

(x, y)

= a(y).

For such systems of curves, certain "compression phenomena" may occur, leading to the failure of (1.16) for r = 00 and certain q > }:!.l (with various degrees of strength). Thus the non-linear x-terms in ({J given by (1.15) playa significant role if d > 2. Very recently the author observed that if d is even one may obtain (1.16) when r = 00 for certain q < 2~~/) whenever ({J satisfies the conditions (1.13), (1.14) (this is not the case for d odd, as said previously). The main reason for this is a different behavior of the families of curves r y defined by (1.21), in the sense that they cannot be contained in a set A C ~d of dim A = di 1 (which is possible if d is odd). At the end of the paper, we give a sketch of the reason for this fact for even d. More precise statements and details will appear elsewhere. We also give an application of the non- L 2 restriction results in the context of a.e. convergence of solutions of the 2-dimensional Schrodinger equation

I

!J..u = iB,u

(l.22)

u(x, 0)

for t

~

=

f(x)

0, improving on Vega's result (for d

= 2) (see [B3], [VD.

88

CHAPTER 4

2 ESTIMATES BASED ON L 2 -METHODS

Expand and estimate the left member of (2.8) as

I~ was shown by Tomas [T] that for f E LP(~d), P < 2~:31), the restriction f lSd-I yields an L 2-function on the sphere. In [Stl], complex interpolation is used togettheendpointresuhp S 2~:31). Also (1.16) is proved for r = 2,q ~ 2~~/)

(see [Stl], Th. 10). The aim of this section is to make some further comments on those estimates, that will playa role later on (we will not be concerned here with the endpoint behavior). The discussion below will provide further information on distribution of level sets. Puning x = ~,Ix' I « N, we replace the operator TN defined by (1.12) by

T~f(x')

(2.1)

=

f

eiN
denote the level sets. Assume R > 1: (2.4)

IA).I

s ).. -2[R +

N and IT~f(x)1 > )..}

Ilf 112 s

p-(d-l)/2 sup IA).

n

B(z,

P)IJ.

Izl
p) the ball centered at z of radius p.

Proof. Denote A = A). and consider a collection {Ba } of subsets of A such that

diam Ba < R

(2.5)

>

R

(2.7)

for

a

[UaBal '"

Let Xa stand for the indicator function of Ba. Denote may assume Re L(Tf, Xa} > (2.8)

1IT*lIiz . L

IAal

(Rl

+L

c(a,

f3)IA a l • IAfJl.

a# Here IIT*IILz denotes the norm of T* acting on L 2-functions of norm ~ 1 sup­ (R) ported by a ball of radius R; c(a, 13) = c(p), P = dist(A a , AfJ) where c(p) denotes a uniform bound on the kernel K(x, x') of TT* for Ix - x'i > p. K is given by the oscillatory integral

f

K(x, x') =

II~ T*Xall:

diAl

2 < c)..2IAI .

=1=

IK(x,

eiN[
The square of IIT*IILz (R)

may be evaluated by the L 2-norm of the operator with

kernel of the form

IAI· by T. From (2.7), one

f

L(y, y') =

eiN[
where z E ~d, Izi < Nand bR denotes a smooth function such that bR(x) = 1 if Ixl < R, bR(x) = 0 if Ixl > 2R and fulfills the obvious derivative estimates. By Schur's lemma, this L 2-norrn is bounded by

f

(2.14)

s~p

Since IL(y, y')1 < cR d [1

+ Rly

13

T~

x')1 < Ix _ x'l(d-l)/2

c(p) < cp-(d-l)/2.

(2.12)

(2.13)

IAI denotes the measure of A and B(z,

{disl[B•• BpJ

XfJ} S

(ct. [Stl], Prop. 6). Thus

I. Then the following inequality holds for

p dyadic

(2.6)

a¥fJ

C

R
Here

IIT*Xall~ + L(TT*Xa,

(2.11)

{Ix I <

L

La

From the hypothesis on q; one has the estimate

Proposition 2.2. For 0 < ).. < 1 let A). =

(2.9)

(2.10)

where a is supported by a neighborhood of 0 in ~d-l .

(2.3)

89

SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS

IL(y, y')la(y') dy'. - y'U- d, (2.14) is at most

c.R.

So (2.8), (2.9)

imply (2.15)

)..21A1 2 ;S RIAl

+

L R
p-(d-I)/2

L

IAallAfJl

dist(Aa,A~)~p

from which (2.4) immediately follows. If we let in particular in (2.4) R '" ).. -4/(d-l), it follows that (2.16)

IA).I ;S ).. -(2(d+l))/(d-l)

90

CHAPTER 4

implying a bound on T acting between L 2 and LP'OO(lRd), p = 2~~/). The preceding shows, moreover, that if IA" I "'-' A- P , then also IA" n B I "'-' A- P for some ball B of radius A-4/(d-I). This statement is easily seen to be sharp. Denote, for instance, JL to be the measure on Sd-I where ~. = IA~1/2 XA, A the e-cap centered at the north pole (0, ... ,0, 1). Thus IAI "'-' ed-I and fi(x) "'-' e(d-l)/2 = A in the tube of essential shape f- 1 x ... X e- I , xe- 2 , of measure e-(d+l) = A-Po d-I

The tube is contained in an e- 2 = A-4/(d-l) ball. In [B4], it is shown that for d = 3, (p = 2~~/).= 4) and ~~ "extremal" case

= TATm XA, A a measurable subset of S2, the

lIilIlL4(]R3)

(2.17)

= 0(1)

91

SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS

such sets was exploited in [FI] to disprove the ball multiplier conjecture, i.e., the operator

f

(3.1)

f---+

(iIB(o.l)f

is unbounded on LP(lRd ), d > 1 (except p = 2 of course). Two-dimensional Besicovitch sets are necessarily of full Hausdorff dimension (see [FaI] for more details on this and related facts). This result is unknown in dimension d 2:: 3 (at least at the time of this writing). A positive answer to this would be implied by the Fourier Analysis conjectures described in the introduction, more specifically, the restriction conjecture: (3.2)

d U(lR ) ---+ LI(Sd-I):

f

f---+

jls

More precisely, there is the •

is bounded for all p < }:I' In [B 1], new information on these Besicovitch structures was obtained, leading to certain (sharp) maximal inequalities that will demonstrate their importance later

Proposition 2.18. For 0 < 8 < 1 denote by C8 a collection of 8-caps < of S2 forming a covering of bounded multiplicity. Let r denote the exponent ~ and L~ the space U «) where < is endowed with normalized measure. Then for A and JL as above

on. Next we summarize the main geometric measure theory results from [B 1]. Their proof is of a more combinatorial nature. We will consider averages over tubes in IRd of unit length and width 8, with direction ~ E Sd-I (see Fig. 1). Given a bounded measurable function on IRd , we define two maximal functions N, N* which we refer to as the Kakeya (resp. Nikodym) maximal function. These names are those appearing in [B 1]; in other places they may have been used differently.

may, essentially speaking, only occur for caps. following estimate.

(2.19)

lIilll4

~ c{ L

rJ;adi~

4 8

L

4 II

TEC,

dJL

du

11

L~

}1/4

Thus, if (2.17) holds, one gets from (2.19) the following distributional property for the set A: (2.20)

L

sup

0<8<.1

TEC,

IA n <\ IAI 4 /51<11/5

= 0(1).

8 dyadIc

Hence for some cap < on S2 (2.21)

IAI "'-' IA n
(e, 0) - tube

from which follows the previous claim. Proposition 2.18 is shown by direct calculation of the L 4 -norm. It seems of ) interest to have an analogous statement in arbitrary dimension (p = 2(d+I) d-I .

3 BESICOVITCH-TYPE MAXIMAL OPERATORS Recall that the classical Kakeya-Besicovitch set in 1R2 is a measurable set of zero measure containing a line in every direction. As said before, the existence of

~

o Figure 1.

._~~_~":::"::?,":"~~~~~;::::';:':~~::-.:;"l'.'~'~:::~~""'::~'~~!-':<:::~':"'''::~''':',~:~

92 (i)

CHAPTER 4

ft(~)

= sup _11

rl

T

N*(x) = sup T

~ Irl

1

f(x)dx

etc.

T

1 T

1 )dIP-1+E

8"

II ft* II LP(lRd)« (

I )dIP-l+E

8"

IIf11u(lRd)

p :::: Pd

where the exponent Pd is given by the recursive formula

= 2; Pd{2pd-l

= (d + 2)Pd-l

- I}

- d.

+ L)

XElRd

defined on the Grassmannian G(d, k). From Proposition 3.7 one may derive the following corollary, for instance. Proposition 3.12. (i) There is an a priori inequality P IIF*IIL G(4,2)

(3.13)

Proposition 3.7. For given dimension d > 2, (3.5), (3.6) hold provided

P2

f(x)dx

F*(L) = sup F(x

(3.11)

II fII LP(lRd)

for I :::: p :::: d. These inequalities are valid for d = 2 and shown by Fourier Analysis methods. In [B 1], the following result is obtained.

(3.8)

i

and the corresponding maximal operator

and, similarly, (3.6)

F(L) =

(3.10)

Considering the case of radial functions and the existence of Besicovitch sets, it seems natural to conjecture the following inequalities IIfnu(sd-J)« (

Those numbers also yield lower bounds on Hausdorff dimension of Besicovitch sets in corresponding dimension d. More generally, consider for a k-plane in JRd the Radon transform

f(y)dy

where now the supremum is taken over all (~, 8)-tubes r centered at x (thus fixed center x and varying direction ~).

(3.5)

= 49

PS

where the supremum is taken over all (~, 8)-tubes (fixing ~ and considering different translations). Irl denotes the measure'" 8d- 1 of r. We call N the Kakeya maximal function of f for 8 eccentricity. (ii) fr (the Nikodym maximalfunction) is defined on JRd,letting (3.4)

93

155

ft is defined on the sphere Sd-l, letting (3.3)

(3.9)

SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS

::::

cpllfll p

for P > 2 and assuming f supported by the unit ball of JR4. (ii) Let A be a measurable subset of JR4 offinite measure. Then for almost all L E G(4, 2) each translate of L intersects A in a set offinite 2-measure and SUPXElRd

IA

n

(x

+ L)I

<

00.

This result is in the spirit of the work of Oberlin and Stein [O-S]. Following [Fal], call a (d, k)-Besicovitch set a measure zero subset of JRd containing a translate of every k-plane. Thus Besicovitch's result affirms the existence of (2, I)-Besicovitch sets. It follows from Proposition 3.12 that there are no (4, 2)-Besicovitch sets. More generally, one shows in [BI] Proposition 3.14. Assume (d, k) fulfill the condition

Thus, in particular, P2 = 2

P3 = P4

=

(3.15)

d :::: 2k -

1

+ k.

7

Then the (d, k)-property holds, i.e., there are no (d, k)-Besicovitch sets.

30

The proof relies on Proposition 3.7 and Fourier analysis techniques similar to the ones used in [O-S].

"3 11

CHAPTER 4

94

4 ESTIMATES ON OSCILLATORY INTEGRALS

Define for fixed x

We will show in particular how the results of the last two sections may be applied to get new information on the higher-dimensional restriction problem. Consider more generally q; of the form (1.15) and define for given N > 1

Consider the oscillatory integral (2.1) -Ff(x)

(4.2)

= TNf(x) =

I

II T fIIU(B(O,R»

(4.11)

ei!/t(x.y) f(y)dy

for certain

P<

2(d

+ I)

= 1/J(x, Ya) + (Vy1/J(x, Ya), Y -

where the O(IY - Ya 2) = 0 ( Ixl < R. Define the operators 1

Taf(x)

(4.5)

=

l

)

Ya}

+ O(ly -

2 Ya1 )

I: ei!/t(x. ya)Taf(x).

The next point consists in introducing a new variable Z E B(O, -JR). Because of the presence of Y - Ya, Iy - Yal < R.- 1/ 2 in the phase function of Taf, one may, roughly speaking, write for Ixl < R I / 2 Tf(x

+ z)

rv

I: ei!/t(x+Z.Ya) Taf(x) a

f

12

»'

2d 2 < - - < P < PI d-l-

2(d

+ 1)

== -d­ -l

to be specified later. At this point, we just use L 2-theory (see remarks at the end of this chapter) and interpolate between the known L 2 - LPI result and the L 2 - L 2 bound (4.14)

III eiry(Z,Y)g(y)dyll L2(B(O.R 1

(4.15)

1f 2 ))

~

4

R ' / 1IgI12

1-

e

e

--+-, 2

PI

it follows (4.12) ~ R(I-O)/4 . IIgl12

(4.16)

and thus from (4.11), (4.8) and the definition of the function g P II Tfll LP(B(O,R» (4.17)

and estimate (4.3) as

<

'V

R- d/ 2 . R«d-I)/2)p . R«(I-O)/4)p

·1

B(O,R)

R- d/2+(p/4)(d-O) { B(O,R)

J

(I: ITafI2)P/2 a

Next we will replace suitably

(4.8)

IITfII~p(B(o.R»

I I eiry(z,y) g(y)dy I U(B(O,R

(4.12)

P

a

(4.7)

a

observed in section 2 (see proof of Proposition 2.2). Hence, putting ei ('V y!/t (x ,Ya),y-Ya) f(y)dy

and write Tf(x) =

ei!/t(x.Ya)Taf(x).

and we have to bound

in boxes

Qa

(4.6)

=

I: ei!/t(x+Z.Ya) Taf(x) = I: eiry(Z,Ya)g(Ya) '" R(d-I)/2 I eiry(z,Y)g(y)dy

(4.13)

*) will be dropped because of the assumption

1

glQa

The exponent P will be taken in the range

d-I

JR

1/J(x, y)

- 1/J(x, y)

= g (y) as

a

This computation is summarized next. Consider a partition of the y-domain (= neighborhood of 0 in IRd Qa of size centered at points Ya' Write for y E Qa (4.4)

and the function g

= 1/J(x + Z, y)

Because in (4.9) Z is restrained to the ball B(O, R I / 2 ), one may again roughly set 17(z, y) rv 17(z, Y + u) for lui of size R I / 2 • Hence,

(f supported by a neighborhood of 0). Fix 1 < R « N. Our purpose is to get a new distributional estimate on T fIB(o.R) by estimating

(4.3)

17(Z, y)

(4.9)

(4.10)

= Nq; (~ , y) .

1/J(x, y)

(4.1)

95

SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS

rv

II:

R- d / 2 { {( B(O,R) B(O,R 'f 2 ) a

J

J

P ei!/t(x+Z.Ya) Taf(X)I dZ}dX.

(4.18)

ITa f1 2; Taf(x) =

~ ei('Vy!/t(x.Ya),Y) fa (y)dy

=

gx

Ilgx liP2 dx

<

'V

%

CHAPTER 4

97

SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS

Proposition 4.27, For T given by (4.2), R < N, one has the boundfor IfI < 1

where Q = B(O, R- 1/ 2); fa(Y) = f(Ya

(4.19)

+ y)

for

(428) •

Y E Q.

1/2 R(3d+I)/(4 p)-(3(d-I)/81IM R _1/2 11 L(p/2J' ->LI

II Tfll LP(B(O,R)) '" <

where 2 S p S 2~~/) and the maximal operator M8 is given by (4.24).

Clearly, one has fa~ 1

2

(4.20)

ITafl 2 (x) =

I

(V y 1/l(X, Ya))

.

Since supp fa C Q, one may write I fa 12 as an average of functions of the form b( ~ ) where b is a standard bump function and the average is taken over points ~,with averaging weight R-(d-l)/21Ifall~. At this point we make the following assumption on f

(4.21)

f

E

oo

L ,

If I

s

1

It remains, of course, to find the estimates on M8, depending on the given phase function f{J. In the case of the restriction problem, f{J is linear in x and hence 1/1 = f{J in (4.1). The operator M8 defined in (4.24) becomes (4.29)

M8g(y)

= SUp15-(d-l) ~

{

JIIVy",(x.Y)-~I<8}

Ig(x)ldx.

Letting f{J be as in (1.4), Le.,

2

improving on the L -hypothesis. Since then Ilfall~ oS R-(d-l)/2, ITa fI 2(x) may be recaptured (as a convex combination) from functions of the form R-(d-I) . b (R- I/ 2(V y 1/l(x, Ya) - ~a)) .

(4.22)

Thus for some choice of points

{~a},

(4.23)

R- d/2+(p/4)(d-B)-(p/2)(d-I). {

JB(O,R)

~a)

b ( R- I / 2 (Vy 1/1 (x, Ya) -

)J

p/2dx.

a

f{J(x, y)

= XIYI + ... + Xd-IYd-1 + xd1/l(y),

the gradient equations become (4.31)

(4.17) is estimated by

[L:

(4.30)

Xi =

-ai 1/l(Y)Xd + ~i

(i = I, ... , d - 1).

Because of (1.5), the map Y f-+ V 1/1 (y) yields a diffeomorphism on a neighborhood of 0 E lR.d - 1 and it is therefore easily seen that M8 corresponds to the Kakeya maximal function from Section 3. Thus, applying Proposition 3.7, letting the exponent p from Proposition 4.27 be

of the integrand in (4.23) by duality. Consider the We estimate the l_power p following related geometric maximal operator

p

= 2p~

(Pd defined by (3.9)),

it follows that M8g(y) = sup (o-(d-I) {

(4.24)

~

assuming 8 bound (4.25)

»

JII82Vy1{F(8-2x,Y)-~I<8}

N- 1/ 2. Making a change of variable x

Ig(X)ldX}

= Rx', (4.23) has the upper

R d/ 2+(p/4)(d-B)-(p/2)(d-l) ~ R-(d-I)/2(M -1/2g)(Ya)

[

R

pn ]

1 ) (d/(pI2)')-1 (4.32)

II M 8II Up/2>'->L' S II M 811(p/2)'

oS ( "8

We should observe at this point that Pd given by (3.9) satisfies Pd > di l , hence P < 2~~/). Substitution of (4.32) in (4.28) gives that (4.33)

II T fllu(B(o,R))

«

R(d+I)/(4 p)-(d-l)/8+
We apply (4.33) in conjunction with Proposition 2.2. Observe that from (4.33) for for some g

E

L(p/2)' (B(O, 1)) of norm 1.

The sum in (4.25) may be replaced by the y-integral. Hence collecting previous estimates, it follows that if IfI S 1 (4•26)

IITfl1 LP(B(O,R»

1/ 2 < R d/(2 p)+I/4(d-B)-(d-l)/21IM R _1/2 11 Up/2J' ->LI'

'"

Finally, substituting the value of 0 given by (4.15), (4.26) yields

IfI < 1 (4.34)

IA>.

n B(O, p)1

« ).. -p p(d+I)/4-«d-I)/8)p+<,

which yields after substitution in (2.4) (4.35)

IA>.I

oS

)..-2

[R

+ ).. -p R-(d-3)/4-«d-l)/8)P] .

98

CHAPTER 4

Choosing R optimally, we get the following estimate on the measure of the level set AA' assuming IfI < I: (4.36)

IAAj < A-2(2(d+l)+p(d+3»/(2(d+I)+P(d-I»-€,

A <

where p = 2p~. Observe that if we let p = PI = 2~~/), we get IA;l.1 < A-Pi-€. Hence, since now p < 2~~JI), the exponent in (4.36) will improve. Thus, if we take 2 J.t E M(Sd_l) with E LOO(a) rather than ~~ E L (a), better integrability properties for it are found. For instance (cf. [B4]),

*

99

SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS

*

sets is of bounded multiplicity). This estimate is more refined than invoking in the L 2 -extension theorem. It is possible to improve a bit the exponent Proposition 4.37 by using Proposition 2.18 in developing a substitute for (4.17).

5 APPLICATION TO THE BOCHNER-RIESZ SUMMATION OPERATORS

The second part of the statement follows from the Maurey-Nikishin factorization theorem for general operators from L 00 to LP (p > 2) and considerations of rotational invariance (thus using more specifically the symmetry properties of the sphere).

In [B I], we used a variant of A. Cordoba's 2-dimensional argument (see [Co]), breaking up the Fourier multiplier m A in rectangular pieces and applying the new information in the restriction problem, together with an estimate on the Nikodym maximal function described in section 3. We follow here a different approach, namely the Carleson-Sjolin-Hormander reduction, evaluating (1.9) from estimates on oscillatory integrals and, more precisely, again using Proposition 4.27. In fact, this will give us a better result than proved in [BI], since it will establish the for d = 3. conjecture provided again p* = p V pi 2: Let b be a standard bump function localizing to a neighborhood of 0 in IRd. Using a dyadic partitioning, estimate (1.9) by

Remark. In proving Proposition 4.37 by the above method, the use of the crude

(5.1)

*

Proposition 4.37. Let J.t be a measure carried by S2 with ~~ E L 00(S2)' Then it E U(lR.3) for p > « 4). The same conclusion holds assuming ~~ E LP(S2).

inequality (4.17) is a weak point in the approach. In the case of the restriction­ extension problem, the situation is the following: one considers a system of boxes {Ba} approximating a spherical shell and functions fa' supp f: c Ba(see Fig. 2). The square function replacement (4.17) is the first step in the argument and consists in comparing

IlL fat

with

!!(L

2Y/2t·

*

L 2k::::O

k

«d+I)/2+J.)

III

f(y)eilx-ylb

k

y»)

dyll

. LP(dx)

=

Fix N 2k and rescale by putting x then becomes Nd/p+dIlSNgli p

(5.2)

(2- (x -

= N x', y = N y'. The LP -norm expression, where

g(y')

= f(Ny')

Ifa I

In the 2-dimensional case, we let p = 4 and essentially obtain equivalence, because of the geometric properties of the sum sets Ba + Bp (this family of

and the operator S N is given by SNg(X) =

(5.3)

I

g(y)eiNlx-ylb(x - y)dy.

Thus, an estimate of the form IISNglip

(5.4)

Ba

Figure 2.

« N- d/P+€lIgli p

would yield from the preceding a bound on (5.1), provided p fulfills the condition < A + d I . Here p was assumed> 2 and this is thus (1.17). Again by general factorization theory and the rotational invariance, it suffices to get the L 00 - LP inequality

if,

(5.5)

i

IISNflip

:s N-d/P. IIf11oo.

100

CHAPTER 4

As proposed above, we perfonn the reduction to an lRd- 1 -lR d problem, freezing the Yd-variable and obtain the phase function (letting Yd = 1) rp(x, y)

(5.6)

=

d-I [

L(Xj - Yj)2

+

(1 - Xd)2

J=I

Here Y = (YI, ... , Yd-d and x are taking values in a neighborhood of O. Let 8 > N- I / 2 , M = 82 N and d-I

T/(x,

(5.7)

y) = [ ~(Xj -

+

(M - Xd)2

VyT/(X, y)

= M~

Ty.~ =

{IXj - MYj

+ (1 _

II!

(5.12)

~~12)1/2 (Xd -

M~

> 2 2(d 2(d

P

then

the regions involved in the definition (4.24) ofthe maximal function by

T=

1 )d/p-I+e

8"

IIgll p

for p < Pd, Pd defined by (3.9). Substituting in (4.28) where T is given by (4.2), rp given by (5.6), yields the same estimate (4.34) on the measure A). n B(O, p) for the level set A). of T I, II I :'S 1. Hence (4.35), (4.36) are equally valid. Therefore, if (5.11)

Rewriting the gradient equations

(5.9)

IIM~gIlLP(lRd-I)« (

] 1/2

MYj)2

J=I

(5.8)

missing y-dimension), one may obtain the same bounds (5.10)

]I~

101

SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS

are given

+ 1) + qed + 3) . + I) + qed - 1)'

eiNrp(x,y) l(y)a(Y)dyt

_ 2 I q - Pd

;S N-d/Pllflloo.

Integrating over variable Yd (taken here to be 1) leads to the required inequality (5.5). So finally we get

M)I < 8 (l :'S j < d)}.

(Observe that necessarily I~ I = 0(1) if we want T to intersect a neighborhood of o [see Fig. 3].) Thus the regions are neighborhoods of straight lines. Observe that if we fix a point i in a neighborhood of 0 and let Y (YI, ... , Yd-I) vary, the tubes Ty.~y satisfying i E Ty,~y will describe all direc­ tions. By a straightforward adaptation of the proof of Proposition 5.6 in [B 1] on the Nikodym maximal function (to which M~ is obviously related, but with a

Proposition 5.13. m).(~) = (l - I~ 12)~ defines a bounded multiplier on LP (lR d ) if p* satisfies (1.l7) and also (5.11). In particular,for d = 3, one gets p* ::: as an extra assumption on p satisfying (1.17).

¥s

6 BEHAVIOR IN THE GENERAL CASE It turns out that in dimension d ::: 3, Honnander's question on the estimates (1.16) (6.1)

I

f

e
"C N- d /, II f II,

has a negative answer, even if we let r = 00. In fact, there are examples of phase functions rp where the LOa - U estimate only holds for q ::: 2~~/) (thus in the L 2 -range), if we let d be odd. Define rp(x, y) as follows (6.2)

rp(x, y) = xlYI

My Figure 3.

lR

+ ... + xd-IYd-1 + 2Xd(YIY2 + Y3Y4 + ... + Yd-2Yd-d +Xd2(YI2

d·l

Take (6.3)

I(y)

= eiN(y;+yJ+"'+Y;_,)

2 ) + Y32 + '" + Yd-2 .

102

CHAPTER 4

so that the integral in (6.1) may be written as

f

(6.4)

ei N {[(XdYI +Y2)2+(X, YI +X2Y2)]+" '+[(Xd Yd-2+Yd-1 )2+(Xd_2Yd_2+Xd_1 Yd-I)]J a(x, y)dy.

For 0 > 0, consider the set

n

(d-I)/2

(6.5)

Q8

=

[!X2j' Xd -

X2j-11

< 0]

n B(O,

I)

Put ~ = ~Y = (0, -2Y2). For this translation, one thus gets curves rye JR.3 contained in the 2-dimensional surface Xl = X2X3. This compression phenomenon is responsible for bad behavior of the maximal operators M8 defined by (4.24), in 8- 1/ q . the sense that for q :::: 2, IIM 8 11 q --->q The previous behavior is not generic, however. In fact, one shows in [B2] that, if d = 3, in the generic case the best q satisfying (6.1) with r = 00 lies in the open interval ] }!.l' 2~~/) [ (and depends on cp). More generally, in the example (6.2), the gradient curves r yare translated in the -dimensional surface "v

di l

j=l

of measure

IQ8 1

(6.6)

"v

Letting 8

«

(6.14)

8(d-I)/2

~,the expression between brackets in (6.4) can be replaced by

(6.7) [(XdYl

103

SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS

X2j-l

=

j = 1,2, ... ,

X2j • Xd

d-I

2

di'

A compression in a set of -Hausdorff dimension seems impossible if d is even. We give a rough sketch of the argument for d = 4.

+ Y2)2 +X2(XdYl + Y2)] +... + [(XdYd-2 + Yd_d 2 + Xd-l (XdYd-2 + Yd-dl

for x E Q8. Putting Zl = XdYl + Y2, essentially amounts to

=

Z3

. n (1

(6.8)

+ Y4, .•. , Zd-2

XdY3

=

XdYd-2

+ Yd-I. (6.4)

Proposition 6.15. Let l • define

cp

r y ,8

(6.16) eiN(Z]+XHIZj)a(Zj)dz j ).

be given by (1.15) (d = 4). For Y in a neighborhood of

o E IRd -

where the ~Y

E jRd-1

=

[IVycp(x, y) -

(6.8)

(6.9)

"v

N- l / 2 , hence

N-(d-I)/4.

"v

8]

are chosen arbitrarily. Consider the set

j=I,3, ... ,d-2

The individual factors in (6.8) are of size

~yl <

(6.17)

Q8

=

Ur

y ,8.

y

This set has8-entropy (in the metrical sense) >

8-«d+I)/2)-r for

some L =

L'I'

> O.

Thus (6.1) admits the lower bound

IQ 81 1/ q

(6.10)

• N-(d-I)/4

N-(d-l)/(2q)-(d-I)/4

"v

restricting x to Q8, using (6.6), (6.9). Inequality (6.1) may thus only hold if

d-I

(6.11)

-2q

thus q > -

4

d

> ­ - q

2(d+I). d-l

In particular, for d = 3, (6.12)

d-I

+ --

cp(X, y)

and the gradient equations (6.13) {

cp

=

The general idea of the proof is as follows. We will first observe that the entropy conclusion is true with L = O. Assuming then Q 8 of o-entropy '" O-(d+l)/:, appearing as an "extremal" case, new geometrical information on the structure of the r Y is gained, allowing us to see eventually that the o-entropy number of Q 8 needs to be ;(, 0-(d/2)-1. The argument produces the conclusion of 6.15, with a L = L'I' which is numerical, provided bounds are imposed on sufficiently many derivatives of cpo To make matters a bit more concrete, take cp of the form

is given by XlYl

VyCP

+ X2Y2 + 2X3YlY2 + xiY;

= ~ are

Xl

=

-2Y2 X3 -

X2

=

-2YlX3

2Ylxi

+ ~2.

+ ~l

(6.18)

cp(x, y)

=

d-I

I

LXiYi -

2:Xd(Y;

+ ... + Y~_I) + 0(lx1 21Y1 2 + IxIIYI 3 )

i=l

so that V y cp =

~

(6.19)

=

Xi

becomes YiXd

describing the curve

+ 0(lx!2IYI + IxIIYI 2) + ~i

r y.

(i=I, ... ,d-l)

- - - - - - - ------------- ------------------

104 (i)

CHAPTER 4

If one specifies a point i (6.20)

x; =

E

YiX~

----- --------------------_.

- -----

-------

-- -

------------

- -

-

-- - - - - - - - - - -

-- - ---------------------------­

105

r y and lets x' = x - i, (6.19) clearly becomes ad

+ 0 (lx'llyl(lil + Ix'i + Iyl)) .

From (6.20), Ix;J ;S IYllx~1 (1 ::: i ::: d - 1). Also, if we fix x~ away from 0, application of the implicit function theorem yields a diffeomorphic correspondence between (x;, ... , x~_I) and y. For instance, Jy = x~ld + o(lx'l) and hence is invertible from the preceding (see Fig. 4). Consider in the y-variable a net £ of 8-separated points. Thus 1£1 = card(£) '"" 8-(d=-l). Let £1 c £ such that the r y,8, y E £(, have a common point. It follows easily from (i) that (6.21)

- -

SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS

2

(ii)

--- - -

e8 (

xd

I

-.

I Xd

---1

• _

H2

HI

Figure S.

U r y,8) .(; rII£d yEE:,

denoting by e8UJ) the 8-entropy numbers (= minimal number of 8-balls required for a covering of Q). Hence, if we assume Q8 defined by (6.17) satisfies (6.22)

e8 (Q8 ) ......, <

8-(d+I)/Z ,

;s e8(Q 8 n HI) . e8(Q8 n Hz). Because of the free choice of the pair (Xt~' xJ), this forces (6.24)

8-(d-l) '""

e8({y})

e8(QJ) ......, > 8-(d+I)/Z .

(6.25)

it follows that (6.23)

It foIl ows again from (i) that y may be derived from its image ina Li pschitz way and therefore

1£11

;s r(d-l)/Z

Next, we will assume

for such family £1. (iii) Fix separated values xJ, xJ of Xd such that the corresponding hyperplanes H(J: Xd = x~ (a = I, 2) get intersected by all r y in points P; (see Fig. 5). Consider the map y ~ (pi, pi) E (Q8 n HI) x (Q 8 n Hz).

,

ry (d - I)-dimensional

8-(d+I)/Z

in order to derive more information on the structure of the curves r y , and finally get a contradiction. (iv) It follows from (iii), assuming (6.26), that a typical [Xd = Xd ]-hyperplane H will satisfy (6.27)

Xd

e8(Q8) '""

(6.26)

e8(Q8

n

H) '""

8-(d-I)/Z.

Fixing H, there is thus a 8-separated system P of ~ 8-(d-I)/Z points P E H such that r y,8 n P =f. 0 for each y. Also, from (ii), each point PEP will belong to r y,8 for;S 8-(d-I)/Z values of y E £. This leads to a partition of (a large subset) of £ in sets £p (6.28)

£ =

U £p;

I£pl '""

8-(d-I)/Z

PEP

and for PEP a "bush" 13 p of curves equation Figure 4.

(6.29)

{ry

lYE £p}, where

Vyrp(x, y) = Vyrp(P. y)

ry

has the

.----

~---~~~~---::;;;::-~~....:;:==,...--==;;;,=---=-=---==--::--=--------=;=-~~-=-~.

-~~~-===~,-~==:-~---=-~~----~==~~

106

CHAPTER 4

------=-------==-r

107

SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS

Since I[~'I < 8-(d+I)/2

(X )d-I, (6.26), (6.32), (6.33) imply that

~ 8- 1

L

1[~'11/2 ~ 8- 1

(

i

-(d-1)/2

1['1'"

)

8- 1 E-(d-I)/2K

ZEI

and hence (6.31) follows. (vii) It follows from (v), thus (6.26) and (6.30), together with general measure theoretic considerations on set intersections, that (6.34)

p

H

p' Figure 6.

(v)

for y E [po The set Q.s is contained in a 8-neighborhood of UPEpB p (see Fig. 6). If we fix an individual point P, it follows from (6.28) that (6.30)

e.s(B

8- 18-(d-I)/2 =

8-(d+I)/2

Indeed, let pi (6.28)

IP n B(Q, E)I < '"

= Pn

B(Q, E),

Us

=

each S of dimension 2 generated by moving the curve r y described by (6.29) along some other curve, as indicated above.

(E ) (d-l)/2

Ipil =

1[11 '"

Bp

SESp

and hence the 8-neighborhood B~ represents a "large" portion of Q.s. (vi) Let 8 « E and fix an E-ball B(Q, E) in H. Then (6.31)

8-(d+I)/2

for'" 8-(d-l) pairs (P, Pi) in the product P x P. From (vi), one can, moreover, assume that the points P, pi are far apart. If (6.34) holds, one gets additional information on the curves r y for y E [p, since a large subset of B p admits a foliation in 2-dimensional surfaces S obtained by fixing some y' E [po and varying r y containing P and intersecting r y' (see Fig. 7). Based on this consideration, a straightforward construction permits us to replace B p by a union of'" 8-(d-3)/2 surfaces S ESp, (6.35)

p) ~

e.s (~ n B~.) '"

"8

K and ['

UpEP'[p. Thus by

K8-(d-I)/2.

Consider an E-net T in [' and let [' = UZEI[~' be a partitioning of [' in sets of diameter E. By construction, if y E [~', the curve r y will be in an IOE­ neighborhood of r z , since the basepoints in H are E-close and Iy - zl < E. Therefore (6.32)

e.s(1J, r

y)

'"

~e.s(~, r

P'

y ).

Also, the same considerations as in (iii) show that (6.33)

e.s(U r YEf~'

y )

~ 8-11[~'11/2.

P Figure 7.

_';5II1;:II

108

,'1'mI!<'"'9,;_"",fitl<'I;:Ioj/,,-':~",,;{'~""j""""~'--Y:"'r.1,r,'''4~{'''R'''';;-;;''1r,.·:-'T'''';;:~;;,'

CHAPTER 4

'1i"'~,,-:ri';·~"}"~"';·"'~';.j;:;J'Jl':·:"iS"'''*'=;,",",",77;:''''"'

"~:'-":"'.ir,",'=g·~'GI;;;;:,-;:;'·i;;:P~,:",-:YO""'-'l'>',-'iP~::"i_'"-1·"N"":';1?'''=-'''''''f>'"'!iI';'if.''l'B)7''.'AI'=~,'''''':'Il:'"·",,t.''?~~'~=;1'J>

Wi:!';;;j'!fl',:';;,9,,:,"-';i!:
109

SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS

(viii) Fix a smooth 2-dimensional surface S away from H, a point i E S, and let Tj be the tangent plane of S at i. Consider the family of curves r y through i given by (6.20). By (6.20), the tangent vector"iy = ; = (tl, ... , td-I, 1) of r y at i is described by the equations (6.36)

ti = Yi

+ O(ltIIYI(lil + lyl)

i = 1, ... , d - 1.

Hence (6.37)

(ix)

ti = Yi

.

+ O(lyl(lil + lyl)·

Restricting t to lie in Tj , a smooth I-parameter family of directions is ob­ tained, corresponding to a curve YI in the y-variable and a curve Y2 = try n H; Y E yd of points in H. Consequently, one may find a neigh­ borhood S' of i in S such that if the basepoint P of r y in H lies away from Y2, r y may only have a "transversal" intersection with S, assuming r y intersects S'. From (6.34), (6.35), one may find a point PEP such that

L

(6.38)

(sn8~1)

e&

'"

P'

0-(d+I)/2

FigureS.

SESp

for'" 8-(d-I)/2 points P' satisfying (6.39)

P. From (6.33), there is some S

E

e&

(S n ~I) '"

E

Sp

In the general case, this last construction will yield a representation (6.35) where now the S are 3-dimensional. One repeats the transversality analysis (viii) and gets 2-dimensional exceptional sets. The estimate (6.35) becomes 8(d-5)/28-(d-I)/2, etc., so for d ~ 6, 4-dimensional structures are produced. Continuing, one eventually gets

2 0­

for'" 8-(d-I)/2 points P' E P. Identifying S with a point neighborhood S' as constructed in (viii), we get a curve Y in H such that the intersection of Sand r y (if any) is transversal, provided its basepoint pi lies away from y. But from (vi), fixing 8 » 8, the number of points pi E Pat 8-distance from Y is at most (6.40)

'"

8- 1

8 )(d-I)/2

(

8"

e&(Q&)

contradicting (6.26) for d = 4.

~

e&

e&(Q&) ;;:

0-[(d-I)/2j-2

=

o-d/2-1

(d is even)

contradicting (6.26). =

8(d-3)/20-(d-I)/2

due to (6.31). Hence, from the preceding, letting 8 be small enough, a point P' E P satisfying (6.34) and lying at 8 distance from Y is obtained (d = 4 > 3). Hence, a large subset of S lies on the r y (y E t' P' )-curves, intersecting, moreover, S in a transversal way (see Fig. 8). Since the corresponding structure contains a diffeomorph of a neighbor­ hood of 0 in]R3, this implies that (6.41)

(6.42)

(~I)

;;:

3 0- ,

7 AN APPLICATION TO TWO·DIMENSIONAL SCHRODINGER OPERATORS Based on the results from sections 2 and 4 of this paper, we sketch an alternative proof of the following result from [B3]. Proposition 7.1. Let u = u (x, t) be the solution ofthe Schrodinger equation with initial data f E S(]R3) (7.2)

~u

= i -ata u; u(x, 0) = f(x).

....··\Y..,:;.-6·'

,:;:"<:~:
110

CHAPTER 4

Then, assuming 1 in the Sobolev space H S (JR 2 ),for some s > p where p there is almost everywhere convergence

t'

= P2

<

lim u(x, t) = I(x)

1--->0

The new element is' the fact that the p-exponent is strictly less than ~. It is known that for dimension d = I, the assumption 1 E H 1/ 4 (JR) ensures (7.3), which is the optimal result (see [Cal, [D-K]). The statement of Proposition 7.1 with p = ~ was shown by Vega [V] in arbitrary dimension. Recall that the solution u of (7.2) is given by the Fourier integral u(x, t)

= / j(~)ej«x,~>+11~12)d~.

sup 0<1<1

I (j(~)ei«X'~>+II~e)d~111 JI~I
~ NP(/ Ij(~)12 d~)1/2

for some p < ~. Choose some q > 2 and estimate the left member of (7.6) by appropriate change of variables (rescaling) (7.7)

N 2- 2/qll sup

I(

0
JI~I<1

j(N~)ej«x'~>+llmd~111

. L«dx)

Using standard considerations, one may then estimate it further by (7.8)

N 2(l-I/q) II

(j(N~)ej«x'~>+t1~12)d~11

JI~I<1

. Lq(dx dt)

Consider the surface (~, 1~12) in JR3, restricting ~ to the unit disc. Since there is obviously curvature and smoothness, the restriction and extension theory applies equally well here as it does for the sphere. Call (p, q) an admissible exponent pair provided (7.9)

IIl1l1 q

~ ell ~~

'

....•.•.. - - ._..__ __,_.,__._._.,..__._,,_,_.,_.,

'----"-~~-~~_._---~"-",_.~.,_._

.. ...

----~-'-~.'.'----'---,-~."--_._. ,- - 0·'··----·----·-··--_•.•._.. .._ •... ,._, __._ __.,_. . ~

.. ..

~

_--~

.

~

..

111

N 2(l-I/q) Ij(N~)1 P d~ )l/P = N 2(1-1/p-l/q) ( / Ij(~)1 P d~ )l/P . (/

the form

t

with p., a measure carried by the 2-sphere S2 or the restricted paraboloid P con­ sidered above. The classical L 2 -restriction theorem states then that (2,4) is admissible, and in section 4 admissible pairs were obtained with q < .4.

XQ

A

1(0 = IQ1 1/ 2

(7.11)

(Q

C B(O, N»

then an improvement will be obtained, unless for some square Q one has

IQI "-' IQI '" IQ n QI·

(7.12)

Remark.

The statement of Proposition 7.1 will result from an inequality of the form

II

_.""~--~_._-,_ __ .._...'.._.' _

For p = 2, q = 4, that is inequality (7.6) with p = ~. It was shown in section 2 (see discussion following Proposition 2.18) that this (2, 4)-estimate may be improved, unless the density "corresponds" to the indicator function of a cap (as a rough statement). More precisely, if we assume j to be of

u*(x) = sup lu(x, t)l. 111>0

(7.5)

..

_~-~"'~~-_."-_

Applying the (p, q) pair in (7.8), one gets a bound

a.e. in x

and a control on the maximal operators (7.4)

..

SOME NEW ESTIMATES ON OSCILLATORY INTEGRALS

(7.10)

(7.3)

(7.6)

._,---_._--.~~~-~ ._._ _. . _, _

.

..

j

may always be broken up in level sets. The meaning of'" actually will allow factors of the form N° for some specific 8 and so yields an "improvement." That this is the result of the reasoning below I leave to the reader to check. So assume (7.12) holds. If IQI "-' N 2 , apply estimate (7.10) for an admissible pair (p, q) with q < 4 gotten from [BI]. Since here (7.13)

(/

Ij(~)IP d~ yiP "-' N 2/p N- 1

we get an estimate of the form (7.14)

N

l

-

2

/q

with I - q~ < 12 • If IQI = N~, with N l = N 1- s , thus Q C ~o follows. Write in (7.7) (7.15)

~ = ~~

+ 1/;

1~12 = 1~~12

+

B(O, Nd, we proceed as

+ 2{~~, 1/) + 11/1 2

1::' ~o I1/ I < !:!J.. h were 50 = N' N' It is clear from (7.15) that the parameter-values needed to recapture the supre­ mum for t E [0, N 2 ] may be taken in a ~ -net and hence the passage to the

t-integral and (7.8) gives a saving factor ( ~ )l/q. Then continue with the (2, 4)­ extension theorem to conclude also that case. If one pursues this argument a bit more explicitly. it easily leads to (7.6) with an exponent p < ~. Institut des Hautes Etudes Scientifiques. Bures-sur-Yvette. France

112

CHAPTER 4

REFERENCES

[Bl] J. Bourgain. "Besicovitch type maximal operators and applications to Fourier analysis." Geom. Funct. Anal. 1:2 (1991), 147-187. ."LP-estimates for oscillatory integrals in several variables." Geom. Funct. [B2] Anal. 1:4 (1991), 321-374. [B3] . "A remark on SchrOdinger operators." To appear in Israel J. Math. [B4] . On the restriction and multiplier problem in JR.3. Lecture Notes in Mathematics, no. 1469. Springer Verlag, 1991. [Cal L. Carleson. "Some analytical problems related to statistical mechanics." In Eu­ clidean Harmonio-Analysis. Lecture Notes in Mathematics, no. 779. Springer Verlag, 1979. n [Co] A. Cordoba. "A note on Bochner-Riesz operators." Duke Math. J. 46, N3 (1979), 565-572. [CoS] L. Carleson and P. Sjolin. "Oscillatory integrals and a multiplier problem for the Anthony Carbery, James Vance,

disc." Studia Math. 44 (1972), 287-299. Stephen Wainger,* and David Watson

[D-K] B. Dahlberg and C. Kenig. A note on almost everywhere behavior of solutions to the Schrodinger equations. Lecture Notes in Mathematics, no. 908. Springer Verlag, 1982. [Fal] K. J. Falconer. "The geometry of Fractal sets." Cambridge Tracts Math. 85. [Fl] C. Feffennan. "The multiplier problem for the ball." Ann. ofMath. 94 (1971), 330­ 336. We would like to give a brief exposition of our recent work on Hilbert transforms [F2] . "A note on spherical summation multipliers." IsraelJ.Math. (1973),44-52. and maximal functions along curves in Rn [CYWW). In particular, we wish to [F3] . "Inequalities for strongly singular convolution operators." Acta Math. 124 describe a theory of dilations that seems useful in this work. This is an outgrowth (1970),9-36. of joint work with Mike Christ that dealt with the case n = 2 [CCYWW). We [Hor] L. Honnander. "Oscillatory integrals and multipliers on F LP." Arkiv Math. II (1973), would like to acknowledge the important contributions Mike made to this project 1-11. [St] E. Stein. "Oscillatory integrals in Fourier Analysis." In Beijing Lectures in Harmonic in its earlier stage. Analysis. E. M. Stein, ed. Annals of Mathematics Studies 112. Princeton University We let r (t) = (t, Y2 (t) ... , Yn (t» be a smooth curve in RR with r (0) = O. Press, 1987. For a function I in Cgo(R n ), we define [T] P. Tomas. "A restriction theorem for the Fourier transfonn." Bull. Amer. Math. Soc. 81 (1975),477-478. dt 1 (1) H r I(x) = I(x [V] L. Vega. "The SchrOdinger equation: pointwise convergence to the initial data." Proc. t / _I Amer. Math. Soc. 102 (1988), 875-878.

5

Dilations Associated

to Flat Curves in IR

rCt» - ,

and

(2)

M r I(x) = sup -I O
l

h

I/(x - r(t»ldt.

0

We are interested in the problem of obtaining estimates of the form

(3)

IIHr IIIU(Rn) S A(p, r)II/IIU(Rn),

and

(4)

IIMr fIIU(Rn) S A(p, r)II/IIU(Rn).

·Supported, in part, by a grant from the National Science Foundation.

"'"~-::"~~~;';:';:~:"~~~~:;;k.-~~ii~~;~~~~~~~7~~;;;~~~~~;;;;~~~~~~;;;'~~~:<;:~·:;,;"",,~-.,,",,~~~~~~~;.~~.\.~;:i.;;'~~~~

114

CHAPTER 5

If the estimate (4) holds for some p <

lim -h1

h--+O

l

for each x in Rn, and

00,

h

0

f(x - f(tndt

=

f(x)

lim T),.x = 0,

(11)

a.e.

for every f locally in LP. Positive results for (3) and (4) have been known for a long time under an appro­ priate "curvature hypothesis" on f. The following theorem was due to the efforts of Nagel, Riviere, Stein, and Wainger in the 1970s. See [SW]. Theorem of the 1970s. Suppose f(t) satisfies the following curvature condition:

For small t, f(t) lies in the span of the vectors f'(O), f"(O) ... ,

(5)

fU)(O), ....

),.--+0

for each x in R n • Note that the examples in (6) are "homogeneous" with respect to a dilation group. That is f(>"t) = T),.f(t)

(12)

where T),. is the contracting dilation group defined by T),.(XI,X2 ... ,Xn )

= (>"XI,>..k2x2 ... ,>..k"Xn).

The examples (7) do not satisfy (12), of course, but still for>.. positive and small and t small

Then

f(At)

IIHr fllu(RG)

~

115

DILATIONS ASSOCIATED TO FLAT CURVES IN JR"

A(p, f) II fIIU(RG) ,

I < P <

rv

T),.f(t).

00,

(perhaps more importantly the set of curves and

IIMr fllu(RG)

~

A(p, Ollfllu(RG),

1<

P ~

Examples of curves satisfying the curvature hypothesis (5), are

(6)

f(t)

= (t, t k2 ... , t kG )

and

(7)

f(t)

= (t, t k2 + t k2H2 ... , t kG + tk"HG)

where the k j and i j are positive integers. Examples of curves not satisfying the curvature hypothesis are

(8)

f(t) = (t, e-I/ltla2 ... , e-I/ItI"")

(10)

>.. ~

forms a compact family of curves on [1, 2].) Groups of dilations played three roles in the Theorem of the 1970s. First the dilations allowed one to "normalize" certain estimates. Furthermore, the homo­ geneity was used as an aid in bounding from below the absolute value of derivatives of ~ . r(t), for vectors ~ in Rn, so that the lemmas of Van Der Corput might be used. Finally, the dilations provided a variant of the Calder6n-Zygmund theory. Let us first give an illustration of what we mean by "normalizing" estimates. It is not hard to see that the L 2-boundedness of H r is equivalent to the uniform boundedness of l dt m(~) = exp(i~ . f(tn - ,

f

t

-I

where a2 ... , an are positive integers. An important tool used in proving the Theorem of the 1970s was that of a contracting group of linear transformations. A collection of non-singular linear transformations T),. on Rn defined for>.. > 0 is called a contracting group of linear transformations if

(9)

f),.(t) = T),.-Ir(At),

00.

T),.Tp, = T),.iL'

for>.. > 0 and JL > 0, Tlx = x,

over vectors ~ in R n • Let

1

mj(~) = .

2-J~ltl~2'2-j

exp(i~

dt . f(tn - . t

One might expect the estimate for mj(~) to be very subtle depending on the curvature of f near the origin. In the examples f (t) = (t, t k2 ... , t k"), for instance, one might expect the behavior of m j (~) to depend in a complicated way on k2 ... ,kn • However, if f(t) is homogeneous with respect to a group T),., we

__-_...._.._.._--

...... ..

;======:..===c:== ..-.,---- . -.. ---------..- . -- -.-..-. ,. -.-.. -....-

116

CHAPTER 5

can argue as follows: mj(~)

= =

1

2-j ::;ltl::;2.2- j

i i i

.

.

t

dt exp(lg. T2-jr(t))­ t

1::;ltl::;2

=

t

dt exp(ig . r(2-Jt))­

1::;ltl::;2

=

dt exp(ig . r(t))­

*

dt

exp(iT2-j~ . r(t)) - . I::; It 1::;2 t

In the last integral, t stays in a compact interval not containing the origin, and we can expect the curvature of ret) to behave in a uniform way as it does in the examples. Thus the various types of behavior that might occur for m j (~) (depending in the examples on k 2 ... , k n) is expressed in terms of T2*-j~' Next, the homogeneity of ret) implies that the curve 7/(t) = r(et) satisfies the differential equation (13)

7/'(t)

ImJC~)1

:::

which together with the trivial estimate CIIT2*-j(~)11

implies that meg) is bounded. It turns out that to obtain LP estimates for p # 2, one requires an appropriate version of the Calderon-Zygmund theory. It is well known that there is a Calderon­ Zygmund theory corresponding to a contracting dilation group. See [R]. In the 1970s LP estimates were obtained by using the complex interpolation of analytic families of operators due to Stein [S]. For example in studying the operator H r , one introduced the analytic family of operators H; defined by

iffj(~) = where mz(g)

= pZ(g)

mz(g)j(g),

f

ItIZeiN(t)

.~_w.'_

..

,...:..,::.._:~,;:~_.·_.~n~_

..;:;:_ _~ __"""~_ _. ~

~.~_._~_~

_ _._~,.. ~

.~...._._.~~.,

117 2 for an appropriate distance function peg). One shows HZ bounded on L for Re z ::: E, for some positive E. Further, one shows that for Re z < 0, HZ is convolution with a kernel satisfying the appropriate Calderon-Zygmund condition, and so is bounded in all LP. Stein's Theorem then implies that H r is bounded in all LP, I < p < 00. More recent arguments inspired by work of Christ [C], and Duoandikoextea and Rubio de Francia [DR], employ a Paley-Littlewood argument. In order to employ the argument, one must show that for a smooth function ¢J with compact support vanishing near the origin, the multipliers DILATiONS ASSOCIATED TO FLAT CURVES IN JR"

(15)

mt (g)

= Erj (t)¢J(T2*-j g)

(where rj(t) denotes the usual Rademacherfunctions) are bounded LP multipliers. One does this by checking that the inverse Fourier Transform of mt (g) satisfies a Calderon-Zygmund condition. For a curve such as r (t) = (t, t k2 ... , t k"), the appropriate dilation group T,l.(x) = (}..x!, }..k2X2 ... , }..k"Xn)

ret) = (t, e-I/ltla, ... , e-I/ltla"). Perhaps a first guess might be to choose T!(x) = (hi, Y2(}..)X2 ... , Yn(}..)X n)

CIIT2*_j(~)II-I/n,

Imj(g)1 :::

....:...:,~~

stares one in the face. However, it is not so clear how to choose dilations for a curve such as

= A7/(t)

for an appropriate n x n matrix A (exp(A log t) = Tt ). The differential equation (13) can be used to obtain estimates from below on derivatives of 7/(t) so that well­ known estimates of Van Der COIput can be used to obtain estimates on m j (g). See [SW]. As a matter of fact one obtains the estimate (14)

___·,·_.._·,·-_·_··'-:-.:.

~t

if ret) = (t, Y2(t) ... , Yn(t)) (p is for "provisional"). We shall see that this provisional guess has some problems, but nevertheless we can gain some insight into matters by considering T!. First of all, we must give up the group property (9). Moreover, it is not even true that r,l.(t) = T,l.-I reAt) forms a compact family on [1, 2]. (For example, el/,l.e-I/,l.t is not uniformly bounded for 1 ::: t ::: 2.) On the other hand, sums of derivatives of r,l. (t) . g do have uniform estimates from below of the type which are useful for the application of Van Der Corput's Lemmas. We can see that instead of the group property (9), we might hope to have (16)

IITs-

1

T,1I :::

C(t/s)€,

s::=: t

for some positive E. This condition was essentially introduced by Riviere [R]. Something like (16) would seem natural if we want to bound a multiplier like mt (g) defined in (15), for by taking adjoints in (16) we can see that the inequality would imply that the supports of the functions ¢J(T2*_jg) would be essentially

.•~ ..~_,-"-':--

,;o~";=;';"-=~,;;:;;"':;;;"-;::;;;:;;:-"::;;:;;;;;;;;;;:=;;;;'7..:;;=~~~",-:;:::::~~-:::;C~~==T";;;:;""-::;:::::;:;;-~~-_-==:;;;;.;;;=,c;;~c..::;::.=."

118

CHAPTER 5

disjoint. Further, while we do not expect (12) to hold, at least we might expect (17)

f(t) = Tte

for some fixed vector e. Also normalizing as in the homogeneous case, we see mj

(~) = [ .

exp(i~

lZ-J :'Oltl:'OZ'Z-i

_=

1 1

exp(i~

I :'OI/I:'OZ

=

t

=

.

-I' dt TZ-iTz_i f(2- 1t ) - .

t

I . dt exp(i. TZ*-i~' TZ-=if(Tlt))-.

t

1:'OItl:'OZ

Thus if we let Uj(T/)

(19)

=

1

Concerning the maximal function we have the following result.

Theorem B [NVWW2]. Suppose y(t) is convex and h(Ct) ~ 2h(t) for some C > 1. Then Mr is bounded in LZ(R z ). Now for the curve ret) = (t, t log Itl), h(t) (t, y(t» in R Z, we might try

t

(25)

where fj(t) = Tz-=}re Tjt ),

we seek estimates of the form (21)

IUj(T/)1 :::s CIIT/II-

°

T)"(Xl, xz)

=(

A

AIn IAI

h =

~) (::) .

Thus, if we want our theory to include the homogeneous curve (t, y(t» (t, tIn Itl), we can not define T).. by the formula

(23)

h(t)

= ty'(t)

- y(t).

t. Thus for curves ret)

(Y~A) h~A»)'

T)..=(~

:k)

which is different from the dilations in (25). However, in this case the two families are equivalent in the following sense: Two families of transformations S).. and T).. are equivalent if there are balls BJ, Bz, and B3 such that T)..B I C S)..Bz C T)..B3.

This means that the kernels treated by the Calder6n-Zygmund theory for T).. and S).. are the same. (Of course

T)"(Xl, xz) = (Axl, y(A)XZ).

Note that the first column of the matrix in (22) is reA), and we ask how to interpret the non-zero entry, A, in the second column. A quantity that had already played a role in the theory was the function

=

Moreover, the proof of Theorem B required the use of certain "balls," and those balls can be obtained by applying T).. to one fixed ball. Of course in the case that ret) = (t, t k ), we have been taking

E

for some € > uniformly in j. Let us tum now to the reason for discarding T/. First of all, the curve ret) = (t, t log It J) is homogeneous and the dilation group G).. is given by the linear transformation (22)

2h(t),

for t > 0.

exp(iT/ . fj(t» dt ,

1:'OItl:'OZ

(20)

~

t

expI(~'

1

= (t, yet)) at f(t).)

= (t,

h(Ct)

(24)

or mj(~)

(-h(t) is the y-intercept of the tangent line to ret) For example we had the following Theorems:

yet)). Assume y(t) is convexfor t > 0, and that yet) is odd. Then a necessary and sufficient condition that H r be bounded in LZ(R z ) is thatfor some C > 1

dt . f(2-1t»­

1:'OItl:'OZ

(18)

119

DILATIONS ASSOCIATED TO FLAT CURVES IN JR"

Theorem A [NVWWI]. Let ret)

. f(t)) dt

.

...

--=_-:;~ ~,~..:::;::,,;:;::;~;;;<:;;;;;;;;;:;;:;:;"~;;;;,,~--;::;:;;;;;;;;:-~~~:::;-,;:.~:;:~.~~,;:;;;;;:

S)..

=

(A 10: IAI

~)

and

T).. =

(~ A10~ IAI)

120

CHAPTER 5

T!,

are not equivalent.) There is another reason for discarding which we shall come to when we discuss the n-dimensional situation. In any case, the following theorem was obtained by using the dilations

T= (Y~A) h~A»)'

for

2 :s

r

:s

n. Our basic convexity assumption will be that for 2 :s r

Dr (t)

(29)

h'(t) 2: fh(t)

(26)

t

(

.

Then (27)

IIMr fIIu(R2)

:s

A(P, r)llfIIU(R2),

:s

00.

1< P <

00.

1< p

If in addition r(t) is extended to negative t as an odd curve, (28)

IIHr

fIIU(R2)

:s

A(p, r)lIfllu(R2),

The condition (26) is stronger than (24). In fact (24) does not suffice for the positive estimates of Theorem C if p < 2 for (27) and for p =1= 2 in (28). See [CCVWW] and [C2]. The next questions are what convexity should mean in R n and how TA in (25) should be generalized from R Z to R n , n 2: 3. It turns out that there are different directions in which to proceed. One pos­ sibility is to proceed inductively. If an estimate is true for a curve r(t) = (t, yz(t) ... , Yn(t» in R n, then the same positive estimate holds for the curve r(t) = (t, Y2(t) ... , Yn-I (t» in R n- 1• Thus one possible avenue is to add as­ sumptions as n increases. This is the approach we shall take. The unfortunate aspect of this approach is that the theory will not be GL(n, R) invariant. In other words if r(t) satisfies the hypothesis and g is in GL(n, R) the curve gr(t) will not necessarily satisfy the hypothesis (even though the estimates for H r and Mr are equivalent to those for Hgr and Mgr ). It will be true that our theorems will be invariant under the group of lower triangular matrices with positive diagonal entries and 1 in the upper left-hand comer. An alternative approach which is GL(n, R) invariant is developed in [CVWW]. In that theory the hypotheses are more complicated. The basic difference is that in our theory the condition (16) is proved, while in the GL(n, R) invariant theory (16) is assumed. Since our theorem is inductive, as explained above, given a curve r(t) (t, Y2 (t) ... , Yn (t» in Rn, it will be natural to consider also the curves

rr(t) = (t, Y2(t) ... , Yr(t»

in R'

:s

n,

r:(t) ) : > 0

= det

A

Theorem C [CCVWW]. Suppose r(t) = (t, y(t» with y(t) convex and that there is an f > 0 such that

121

DILATIONS ASSOCIATED TO FLAT CURVES IN JR"

ry)(t)

for t > O. We are mainly trying to understand what happens if D n (t) vanishes to infinite order at t = 0, so to simplify matters we might assume Dn(t) =1= 0 for t =1= O. Now the inductive nature of our hypothesis leads to the assumption that Dr(t) =1= Ofort =1= 0,2:s r:S n. Finally,byreplacingsomeyjbY-Yj,wearrive at the condition Dr (t) > 0, t =1= 0, 2 :s r :s n. (A similar assumption occurs in the theory of differential equations. See [H].) There are also generalizations of h(t) for curves in R n. Thus if r(t) = (t, Y2(t) ... , Yn(t» and rz(t) = (t, Y2(t», we define hz(t) to be the h(t) for rz(t). To define h 3(t) consider the curve r3(t) = (t, Yz(t), Y3(t» in R 3. We consider the tangent line, L (t), to r 3 at the point r 3 (t) in R 3 • We let - (t) be the point of intersection of L(t) with the plane XI = O. Then under the hypothesis (29) f'(t) is a convex curve in R Z• We let h 3 be the "h" for r(t). We define h r for r :s n in an analogous manner. It turns out that the h j (t) can be expressed as a ratio of determinants that is easy to calculate asymptotically in examples. As in the case n = 2, the hj(t) were known to have had a role in the theory. This is expressed by the following theorem:

r

Theorem D. Suppose r(t) = (t, yz(t) ... , Yn(t» is convex in the sense of (29) and that r (t) is odd. Then a necessary and sufficient condition that H r be bounded in L z is that for some C > 1,

hj(Ct) 2: 2h j (t), for

2 :s

j

:s

t > O.

n.

Thus it seems that the generalization of (25) should be

yz(t)

hz(t)

0

Y3(t)

?

h 3(t)

o o o

Yn(t)

?

?

hn(t)

0

(30)

TA =

122

CHAPTER 5

r..

is obtained from Looking again at (25) we note that the second column of the first column by applying to the first column a first-order differential operator. Namely, if

RW(t)

Theorem 1, Let J be a bounded interval on R. Assume

find first-order differential operators R I ... , Rn-I such that

+ 1st column of TA •

It turns out that the convexity assumption (29) allows us to find unique Rj so that (30) and (31) are satisfied with the question marks filled in. If we denote by TA.i,j the i, jth element of the matrix for TA, we find

TA ,i,1

(32)

= Yi(t),

(40)

M n f (t) is single-signed for t in J,

(41)

{31 (t) = 1 and al (t) S 1/l: for some positive l: and all tin J,

TA,i,j+1

n

L IMJ(t)1 2:

(42)

l:

for t in J,

j=1

and, finally, (43)

l

x

Y

{3j(t)dt 2: E(Y - x) max aj(t) x9~y

. for x < y, x, y in J, and 1 S j S n - 1. Then for A 2: 1,

and (33)

1 S j S n - 1

{3j(t) > 0,

for tin J,

= tZ( W;t))',

R j (jth column of TA ) = j

and

aj(t) > 0

(39)

R (First column of T)J = 2nd column of TA • This suggests that we should try to

(31)

123

DILATIONS ASSOCIATED TO FLAT CURVES IN JR"

= R j+1 TA,i,j

Iif( eiA/(tldtl < C(E n "

(44)

J) _1_.

AI / n

where We apply the proposition with a j (t) = h j (t) and {3 j (t) the h j corresponding to the normalized curve

f(A) )' h7(A) RJ(A) = ( hj(A) hi(A)'

(34)

I

HTs- T,1I

s C(~

r,

It turns out that to verify the crucial hypothesis (42) it suffices to show

s 2: t,

wh~u v~w is the

m

r(t) = T,e,

The next matter is somehow to relate the dilations to decay estimates of the type (21). We found it convenient to prove a variant of the standard lemmas of Van Der Corput. We consider differential operators of the form (37)

M·f(t»)'

C(a)

for 1 S A Sa, where TA is the dilation matrix formed from r j' (This shows then the way the dilations are used to obtain the decay estimate (21).) Further, (45) can be shown to hold under the assumption hi (t) 2: E hjt(t l . The last point concerns the Paley-Littlewood decomposition. We choose a COO(R n ) function¢(x) such that¢(~) = 1 for I~I S 1 and¢(~) = ofor I~I 2: 2, and we assume that we have a family of non-singular linear transformations Ak satisfying the condition

IIAk~IAkli Sa < 1.

(46)

mj(~) = ¢(Aj~) - ¢(Aj+l~)'

(47)

Mj+I!(t) = aj(t) ( ~j(t)

s

(This condition was first introduced by Riviere [R].) We define

MI!(t) = f'(t)

and (38)

II (T{)-I II

(45)

for some l: > O. Clearly also (36)

hi (t) where h j is

rj(t) = Tz-=Jr
It turns out then that the hypothesis that h j (Ct) > 2h j (t) implies (35)

=

and

,

where aj and {3j are smooth functions. Then we obtain the following theorem.

(48)

SJ by

fJ(~) = mj(~) . i(~).

We then obtain the following Littlewood-Paley estimate.

124

CHAPTER 5

Theorem 2. Iffor each integer k, IIAk~IAkll ~

(49)

and (50)

( r

I ~ ISi/(x) I'

IX

L:5 Cpll!II",

1< p <

I h/t)::::

00,

I ~Si/l :5 Cpll (DSj!JI't' II"

11M, fIIu 1< p <

IIUr/IlLP ~ C(p)lIf11u,

1< p <

00

Ur/(x) =

h / t ) .J -- 2 ... ,n,

E -- ,

t

~ t ~

liB, fllu ~

1

00,

Cpllfllu,

1< p <

00.

Also, in examples where r(t) = (t, e- l / tOI

j

where again rj(t) are the standard Rademacher functions. To obtain the estimate (51) we need an appropriate variant of the Calder6n-Zygmund theory (which is not contained in the theory of spaces of homogeneous type of Coifman and Weiss [CW]). We denote by B j, j E Z the "balls" of the theory. We assume for each j E Z there is a "ball" B j' We assume that each B j is open, convex, balanced and bounded. We then have the following result. Theorem 3 [CCVWW]. Assume (i) UB j = R n ; (ii) nB j = {OJ; (iii) B j + 1 C B j .. and (iv) there exists a K < oosothatlBjl ~ KIB j +i1. Let T be a convolution operator bounded on L 2(R n ), with convolution kernel K(x). Assumefor each y E B j (v) J;r¢2B j IK(x - y) - K(x)ldx ~ A. Thenfor fin CO'(W), 1< P <

1< p ~

00,

L rj(t)Sj/(x),

IITfIIU(R") ~ Cllfllu(R"jo

~ Cpllfllu,

and if in addition r(t) is odd,

where (52)

= (t, Y2(t) ... , Yn(t» be convex in the sense of (29).

then

The key step in the proof of Theorem 2 is to show that for 0 (51)

Main Theorem. Let r(t) Iffor some E > 0,

< 1, then

125

DILATIONS ASSOCIATED TO FLAT CURVES IN JR"

••• ,

e l //

O ")

for t > 0,0 < al < ... < an, it is easy to see that the hypotheses of the Main Theorem are satisfied. We would like to add a remark pointing out a connection between our work and the theory of uniform asymptotic stability of systems of ordinary differential equations. Suppose one considers the n x n system of differential equations y'(t) = B(t)y(t)

(53)

where y(t) is a vector in R n and B(t) is an n x n matrix. Let F(t) be a fundamental matrix for the equation (53). That is, the columns of F(t) are solutions of the system and det F(t) =f. O. Then the system is uniformly asymptotically stable if IIF(t)F- 1 (s)1I ~ Ce-e(t-s),

t:::: s,

for some positive E. See [CO]. Thus after a change of the time scale the adjoint of F (t) satisfies the estimate (16). In particular our results imply results on uniform asymptotic stability for systems in which the elements of the coefficient matrix are unbounded. For example, the system z' (t)

= B(t)z(t)

00.

The bound C depends only on p, n, A, the L 2 operator norm of T, and the family of balls B j, but not on f. The above considerations lead to the following Theorem.

with

B(t) =

(_~ a(t)

l+la

(t»)

a(t)

~_~~~_

__ -==

=:::_-=-~== =-c::-~-------=_ -===-==-=-===-=-=..:;;;;;;;-~=_~::;;;-~=-===~,

126

-=-==--=-==--­

-----,,;=-=--~~"~"~~~~,=~~~-~~~~~--=~---~._~=~-----~-~~

-=--~ .~-~~=~~--=-~~.:;;~~~-~~~~~~=~-~

CHAPTER 5

is unifonnly asymptotically stable under the hypothesis 0 ::: a (t) ::: C, for some positive C. University ofSussex, Brighton, United Kingtom Wright State University University ofWisconsin, Madison Rutgers University, Camden

6

Nonexistence of Invariant

REFERENCES [C] M. Christ. Personal communication. [C2] M. Christ. "Examples of singular maximal functions unbounded in LP." Preprint. [CCVWW] A. Carbery, M. Christ, J. Vance, S. Wainger, and D. Watson. "Operators asso­ ciated to flat plane curves: LP estimates via dilation methods." Duke Math. J. 59(1989),675-700. [CO] W. A. Coppel. Dichotomies in Stability Theory. Lecture Notes in Mathematics, no. 629. Springer Verlag, 1978. [CVWW] A. Carbery, J. Vance, S. Wainger, and D. Watson. "The Hilbert transform and maximal function along flat curves, dilations, and differential equations." To appear. [CW] R. Coifman and G. Weiss. Analyse harmonique non commutative sur certains espaces homogenes. Lecture Notes in Mathematics, no. 242. Springer Verlag, 1971. [DR] J. Duoandikoextea and 1. L. Rubio de Francia. "Maximal and singular integral operators via Fourier transform estimates." Invent. Math. 84 (1986), 541-561. [H] P. Hartman. Ordinary Differential Equations. John Wiley, 1964. [NVWW1] A. Nagel, J. Vance, S. Wainger, and D. Weinberg. "Hilbert transforms for convex curves."Duke Math. J. 50 (1983), 735-744. [NVWW2] A. Nagel, J. Vance, S. Wainger, and D. Weinberg. "Maximal functions for convex curves." Duke Math. J. 52 (1985), 715-722. [NVWW3] A. Nagel, J. Vance, S. Wainger, and D. Weinberg. "Hilbert transform for convex curves in R" ." Amer. J. of Math. 108 (1986) 485-504. [R] N. Riviere. "Singularintegrals and multiplier operators." Arkiv Math. 9 (1971), 243-278. [S] E. Stein. "Interpolation oflinear operators." Trans. Amer. Math. Soc. 83 (1956), 482-492. [SW] E. Stein and S. Wainger. "Problems in harmonic analysis related to curvature." Bull. Amer. Math. Soc. 84 (1978), 1239-1295.

Analytic Hypoelliptic Differential Operators on Nilpotent Groups of Step Greater than Two Michael Christ*

INTRODUCTION

The development of a large part of the theory of subelliptic partial differential equations has been guided by fundamental papers of Stein and collaborators. Among these are the explicit computation by Greiner and Stein [GS] of the Szego kernels for certain three-dimensional CR manifolds; the work of Fol­ land and Stein [FS] which developed aspects of hannonic analysis on the Heisenberg group and established its connection with the equation on strictly pseudoconvex domains; the proof by Rothschild and Stein [RS] of sharp reg­ ularity estimates for sums of squares of vector fields and their forging of the link between these operators and harmonic analysis on graded nilpotent Lie groups; and the estimation by Nagel, Stein, and Wainger [NSW] of the fun­ damental solutions of such operators in a very precise way. These works have pointed the way to a far-reaching domain of research centered on the confluence of partial differential equations, complex analysis in several vari­ ables, hannonic analysis on Euclidean spaces and on nilpotent groups, and real

a

variables. 'This research was supported by the Nalional Science Foundation and the Inslilut des Hautes Etudes Scienlifiques.

128

CHAPTER 6

The present note serves to announce further progress in this direction. Detailed proofs will appear elsewhere [C5], [C6]. We are concerned with five interrelated themes: • Analytic hypoellipticity of subelliptic partial differential operators, especially those which are left-invariant and homogeneous on some nilpotent Lie group. • Hypoellipticity of certain PDE with complex coefficients, in the Coo sense. • Certain one-parameter families of irreducible representations of nilpotent Lie groups and algebra~. • Existence of eigenvalues for certain non-selfadjoint linear operators. • A scattering problem for ordinary differential equations. Our investigations have been profoundly influenced by all four of the papers cited above. Experts should not be misled by our title, in which at least three distortions are perpetrated in the interest of brevity.

ANALYTIC HYPOELLIPTIC DIFFERENTIAL OPERATORS

In order to motivate later considerations, let us recall how it may be proved that a non-elliptic operator with constant coefficients, in ]Rd, admits solutions which are not analytic. Let L be such an operator. One approach is to note that if L were analytic hypoelliptic, then for any open sets Ql <E Q2 <E Q, there must exist C < 00 such that for every u E Coo (Q2) satisfying Lu == 0, for every multi-index

a, (1.1)

II

~ull a ox

LOO(Q])

~ C\+lallal!lIullL''''(~h)'

Assume that L has positive order n, and let P be its symbol. P is a polynomial, which may be considered to be defined on Cd, and then P(~) = 0 for certain ~ E Cd. If L is not elliptic, then a sequence {~j} C Cd may be chosen such that P(~j)

= 0,

I~j I ---+ 00,

(1,2)

1£f(~j)1 ---+ 0 I~jl

1 REVIEW To begin, we recall some facts concerning constant-coefficient, homogeneous dif­ ferential operators on ]Rd. A differential operator L is said to be (COO) hypoelliptic in an open set Q if whenever Q' c Q is open, u is a distribution in [C
Moreover, if L has constant coefficients but is not necessarily homogeneous, then (2) and (3) remain equivalent. For a certain very restricted class of nilpotent Lie groups of step 2, including the Heisenberg groups [M], these conditions (1) and (2) remain equivalent for homo­ geneous, left-invariant differential operators, and are equivalent to a modification of (3), the Rockland criterion; the latter and (1) remain equivalent for all graded nilpotent Lie groups. We will be concerned in this chapter with the non-validity of the equivalence between (2) and the other conditions, for groups of step three or higher.

129

as j ---+

00.

In order to violate the Cauchy estimates (1.1), set

/j(x) = exp(i(x, so that L/j

== O. With a

~j)),

= a(j) a multi-index to be determined, I

aO;a Ii (0) 1 =

I~jl

while for any bounded neighborhood Q of 0, Cl+1a1Ial!llliIlLoo(Q) ~ C1+1a11al! exp(CiI£f(~j)1) with C\ = C\ (Q) fixed. If each ~j E ]Rd, then (1.1) is cl~arly contradicted by fixing any a satisfying a j i=- 0, and exploiting the condition that I~j I ---+ 00. In the general case, if the imaginary parts are nonzero, then exp(C\I£f(~j)l) will grow as j ---+ 00, so that both the left- and right-hand sides of (1.1) grow with j and the situation is more delicate. It can be shown that the third condition in (1.1) still permits the choice of a sequence aU) such that (1.1) is violated for all finite C, as j ---+ 00; we shall not reproduce the details. However, for the sequel it is instructive to note that for such a construction to succeed in contradicting the Cauchy estimates, it is permitted that II /j II LOO(Q) grow as j ---+ 00, but is equally essential that it not grow too rapidly, in a suitable sense.

130

CHAPTER 6

From the viewpoint of hannonic analysis, what we have done is to decompose the regular representation of lR d on L 2(lR d ) into its irreducible unitary components, which are parametrized by ~ E lRd ; to complexify so as to obtain non-unitary rep­ resentations depending holomorphically on the parameter ~ E Cd; to associate to each representation n and differential operator L an operator n(L) on the Hilbert space 1fT( = L 2 (lRO) = C; and to exploit those n for which n(L) fails to be injective. Ellipticity of L is determined solely by real ~, that is, by the unitary representations, but the nonunitary ones were vital in the analysis of analytic hy­ poellipticity. Note that.for homogeneous operators, the symbol is a homogeneous polynomial, so that if it is elliptic, then it is nonvanishing in a conic neighborhood of IRd in Cd, and hence no sequence satisfying (1.2) exists. A similar situation occurs for homogeneous, left-invariant partial differential operators on graded nilpotent groups. The set of all irreducible unitary represen­ tations n may be described quite explicitly [K]. To each such L and each n is associated a differential operator n (L) acting on a dense subspace of 1fT( = L 2(IRk) for some ken). As conjectured by Rockland [R], and proved by Rothschild and Stein [RS] in the negative direction and by Helffer and Nourrigat [HNI], [HN2] in the positive direction, hypoellipticity of L in the Coo category is equivalent to the injectivity of n(L) (on its domain) in 1fT( for every irreducible unitary representation n. This is the modification of (3) mentioned above. Thus it is natural to expect that the study of analytic hypoellipticity for such operators should be related to the properties of certain non-unitary representations. The subject was first approached from this point of view by Helffer [He]. It proved problematic to determine whether n(L) failed to be injective for some n, or not, and that is the problem to be addressed here. We are able to treat only a very restricted class of groups, for which all the relevant representations satisfy 1fT( = L 2(lR 1), the simplest case which does not arise for abelian groups. Then n(L) is an ordinary differential operator, about which much can be said by elementary means. One of our goals is to develop in detail the analysis of holomorphic one-complex-parameter families of such representations. Some notation: "A ....., B" means that the ratio of these two positive quantities is bounded above and below by positive constants. (z) = 1 + Izi. The natural logarithm is denoted In, the real part by!)i, and the imaginary part by:S. C'" denotes the class of real analytic functions.

2 THE MAIN RESULT g will denote always a (finite-dimensional) nilpotent Lie algebra, and G a con­ nected Lie group whose Lie algebra is g. Elements of g will be identified with

ANALYTIC HYPOELLlPTlC DIFFERENTIAL OPERATORS

(real) left-invariant vector fields in a neighborhood of the identity element 0 via the exponential map.

131 E

G

Definition 2.1. (a) A Lie algebra g is said to be stratified ifit admits a vectorspace decomposition asg = EBj~lgj,with[gj, gk] C gj+k!orallj, k E Z+,suchthatglgenerates g as a Lie algebra. (b) A stratified Lie algebra g is said to have k generators if dim(gd = k. (c) A nilpotent, stratified Lie algebra g is said to be of step r if gr+l = {OJ but gr =1= {OJ. (d) If g is stratified, then Un denotes the linear space ofall left-invariant differen­ tial operators on G which may be represented as homogeneous polynomials ofdegree n in the elements of gl. Definition 2.2. For any formal polynomial P in two non-commuting variables, with complex coefficients, P denotes the naturally associated polynomial in two complex variables. If g is stratified with two generators, and if a basis {X, f} for gl is fixed, then to each such polynomial, homogeneous of some degree n, is associated the element P(-iX, -if) of Un' Conversely, any element £, of Un may be so expressed; though P will not be uniquely determined, it is easily verified (by consideration of the abelian group whose Lie algebra is g/EBj~2gj) that P is uniquely determined by £'. Definition 2.3. Let g be a stratified Lie algebra with two generators X, f E gl. (a) A homogeneous polynomial P ofdegree n is said to be generic ifthe polynomial z ~ P(z, I) in one complex variable has degree exactly n, and moreover has n distinct complex roots. (b) £, E Un is said to be generic if it equals P(-iX, -if) for some generic polynomial P. It may be verified that the property of being generic is independent of the choice of basis for gl, so long as the basis itself is suitably generic. Since it is determined by P, it is an intrinsic property of £, = P(-iX, -if), independent of the choice of the representing polynomial P. Our principal result is:

Theorem 2.4. Let G be a Lie group whose Lie algebra g satisfies both: • g is stratified and has two generators.

132

CHAPTER 6

• g is of step greater than or equal to three.

Then any generic homogeneous, left-invariant differential operator .c on G fails

to be analytic hypoelliptic.

Remarks. 1. The proof yields a stronger result in the scale of Gevrey classes. For some open set Q and real number s ~ 1, G S (Q) denotes the class of all Coo functions defined on Q such that, for each compact subset K C Q, there exists C < 00 such that for every multi-index a and every x E K, laa f(x)1 ~ cl+lal(l

+ lal)sla l .

For s = 1 this is simply the class of real-analytic functions, while for s > 1 there are strict inclusions C W C G S c Coo. Then our method establishes existence of a solution to.cF == 0, in a neighborhood of 0, such that F fails to belong to G S for any s < 3. 2. The case .c = X 2 + y2 was treated in the union of the two papers of Helffer [He] and of Pham The Lai and Robert [PRJ some time ago, so that Theorem 2.4 is no surprise. However, their argument depends on certain particular features of X 2 + y2 which are absent for general .c. 3. The principal hypothesis is that g have step strictly greater than two. There exists a class of nilpotent groups of step two, for which every left-invariant, homogeneous differential operator that is Coo hypoelliptic is automatically analytic hypoelliptic [M]. In particular, this occurs for the Heisenberg groups lHln • 4. The hypothesis that g have two generators may be weakened to dim (g2) = 1, with an appropriate redefinition of genericity. It would be interesting to decide whether the result remains valid without any such hypothesis; this cannot be done by our method as it stands. S. The hypothesis that.c be generic is made to simplify certain technical aspects of the proof, and might conceivably be eliminated by a more elaborate argument along the same general lines. 6. In order to simplify the exposition slightly we assume henceforth that the roots of z f-+ P(z, 1) have distinct real parts.

3 OUTLINE OF THE PROOF The first step is to decompose the regular representation of G on L 2 (G) into irreducible components, and to focus attention on three particular families of com­ ponents. For our purpose this amounts simply to separation of variables and may be achieved in the following concrete fashion. Let n j denote the dimension of

ANALYTIC HYPOELLIPTIC DIFFERENTIAL OPERATORS

133

gj, let N = L nj, and let Xjk be coordinates in ~nj for 1 ~ k ~ nj. In the coordinates given by the exponential map, G is identified with ~N, and X, Y take the form

a

LPjk(X)-, .k aXjk J,

where the p jk are polynomials, homogeneous in a natural weighted sense. More­ over, Pjk depends only on those variables Xii for which i < j. Thus the space of all functions on G which depend only on those variables x jk for which j ~ 3, is invariant under X and y, hence under.c. It suffices to produce a function in this space which is annihilated by .c in some open set, and which is not analytic. In the same way, we may restrict attention to functions independent of all variables X3k with k ~ 2. Since gl and g2 have dimensions 2 and 1, respectively, our functions now depend on four variables which will be called x, y, s, t. Y takes the form c 1ax + C2 ay + l(x, y)as + p(x, y, s)a, where Ci are constants, lis linear, and p is a homogeneous polynomial of degree two in :x, y plus a constant times s. X takes a similar form. Mter composition with an appropriate polynomial diffeomorphism, X = ax

and

Y = ay - xas - x 2 a,.

In effect, we are now working on a particular four-dimensional nilpotent Lie group of step three. We consider three families of irreducible representations of this group, or more concretely, the action of X, Y on three classes of functions F on ~4: (I)

F(x, y, s, t) =

ei(~,(x,y))

with

~ E ~2,

(II) F(x, y, S, t) = f(x)eii.s with 0 -::j:. A E ~, (III) F(x, y, S, t) = f(x)eiZYeirt with 0 -::j:. r E ~ and Z E iC.

The actions of X, Y on these classes offunctions generate representations of the Lie algebra which they generate as (possibly unbounded) operators on iC l = L 2 (~o), on L2(~1), and on L2(~1), respectively. X acts as ~[, -i d/dx and -i d/dx, and Y acts as b -AS and (z - rx 2 ), respectively. The representations (I) factor through g/EBj ::2gj, those of type (II) through g/EBj ::3gj, and those of type (III) through g/EBj::4gj. To these representations of g correspond representations of G. In case (III), when z E iC\~, these representations are not unitary, and indeed elements of G are represented as unbounded linear operators defined on a certain subspace of L2(~).

It is a general principle [G], [HN1], [R] that one should study in succession the representations of type (I), then type (II), then type (III) in order to obtain increas­ ingly more refined information about.c. For those oftype (I),.c acts as P(~l, ~2);

134

if f>(~)

CHAPTER 6

= 0 for some 0 i- ~ then.c annihilates exp(ir(~, (x, y)}) for all r E ll~+. By the argument outlined in § 1, it follows that .c is not analytic hypoel­ E lR. 2 ,

liptic, nor even Coo hypoelliptic. Therefore we may assume henceforth that P is an elliptic polynomial. The following is well known, and expresses the analogue of the last paragraph for the representations of types (II) and (III). Lemma 3,1. (i) If there exists· f E Loo(lR.), not identically vanishing, such that P(-i 1:x: ,x)f == 0, or such that P(-i :f:x:, -x)f == 0, then there exists a function F, defined on an open subset of G, such that P(-iX, -iY)F == 0 but F is not Coo. Therefore.c = P(-iX, -iY) is not analytic hypoelliptic, nor even Coo hypoelliptic. (ii) If there exist z E lR. and 0 i- f E L 00 (lR.) for which P (-i 1:x: ' (z - x 2 f == 0, then .c is again neither analytic hypoelliptic nor Coo hypoelliptic. (iii) If there exist z E C and 0 i- f E L 00 (lR.) for which P( -i 1:x: ' (z - x 2 »f == 0, then .c is not analytic hypoelliptic.

»

The proof is analogous to that for type (I). As in the argument outlined in § I , some growth restriction on the solution f is essential; in order to contradict the Cauchy estimates one considers the solutions of L

f

(.1/3 X )eir1fJ;;y e irt

where. is a positive parameter which will tend to 00, and where f, z are as in (iii) and are fixed. Thus the behavior of f (x) for large x comes into play. The condition that f be bounded may be relaxed in case (iii) to If(x)1 = O(exp(lxI 3 - e for some e > 0, but it turns out that (if none of the roots of P are purely imaginary) every solution will either decay rapidly to zero as Ix I -+ 00, or will grow like exp(8lx 13 ), that is, too rapidly to lead to the desired negative result. Thus it is irrelevant whether one seeks solutions in the Schwartz class, in L 2, in L 00, or in some class permitting moderate growth at infinity. As far as the existence of such bounded solutions for representations of type (II) goes, only the sign of A is relevant, not its magnitude; this reflects the existence of a group of automorphic dilations of G. A bounded solution f for A = I leads by an appropriate dilation of the x variable to a bounded solution for any given A > O. Similarly, it is no loss to set. = + I in case (III).

»

Definition 3.2. .c is said to be nondegenerate if P is elliptic and there exist no boundedfunctions in the nullspaces of P(-i 1:x: ' ±x).

ANALYTIC HYPOELLIPTIC DIFFERENTIAL OPERATORS

135

There are certain cases in which a very superficial analysis shows .c to be degenerate. Let {Yj: I ::::: j ::::: n}

be the roots of Z

f-+

P(z,

I), recalling that

Pis assumed to be elliptic.

Proposition 3.3. Assume.c = P( -i X, -i Y) to be homogeneous, left-invariant and generic on a nilpotent group whose Lie algebra is stratified with two gener­ ators, and assume P to be elliptic. Then.c is degenerate if any of the following occur: (1) The degree of P is odd. (2) Some root Yj is imaginary. (3) The degree n of P is even and the number ofindices j for which ffi(Yj) > 0 is not exactly n /2. The idea of the proof will be, indicated later. There is the following consequence: Corollary 3.4. Let.c = P (-i X, -i Y) be a generic homogeneous, left-invariant differential operator on the Heisenberg group 1Hl 1 ofdimension three. If the degree of P is odd, then .c fails to be Coo hypoelliptic. We assume henceforth not only that P is elliptic, but also that none of these conditions are satisfied. These are by no means the only causes of degeneracy: the operators X 2 + y 2 + ia[X, Y] on 1Hl 1 fail to be Coo hypoelliptic for a discrete set of real values of the parameter a, while Pis independent of a. A fundamental issue in our analysis is the difficulty of detecting such more subtle degeneracies. For our purpose the essential analysis takes place on the family of representations of type (III), which reflect the fact that the group is not of step 2. Let us consider a more general situation by letting m E (2,3,4, ... } be arbitrary and setting

L;;

=

d P(-i - , (z - x m dx

I

».

Definition 3.5. z E C is said to be a nonlinear eigenvalue of the operator family {L;;: z E q If there exists f E L 00 (lR.), not identically vanishing, such that L;;f = O. In fact, the nonlinear eigenvalues are genuinely eigenvalues for a related problem; see [KeJ, [PRJ. The set of nonlinear eigenvalues has some structure. Proposition 3.6. Assume that P is homogeneous and generic, and that Pis elliptic and that none of the roots Yj are imaginary. Let m E (2,3,4, ... }. Then the set

:.=_.­

-:--

-~---_.-._----------~~

.-

~,

136

--".~-,~::;,

CHAPTER 6

of nonlinear eigenvalues of (Lzl is equal to the set of all zeroes of an entire holomorphic function W, which satisfies

= O(exp(C1zl m /(m-l)) Moreover when m = 2, W(z) == IW(z)1

for some C < 00. constants Cl, C2.

Cl

exp(c2z 2) for some

,.~-

.

""!"~

ANALYTIC HYPOELLlPTIC DIFFERENTIAL OPERATORS

137

an entire holomorphic function of order exactly m/(m - 1), and since this order is not an integer, W has a nonempty, in fact infinite, set of zeroes. As discussed earlier, this implies that .c is not analytic hypoelliptic; if any of the zeroes happens to be real, then it is not Coo hypoelliptic. A small extension of the same reasoning establishes Theorem 3.9. In 1R3 , with coordinates (x, y, t), set

In §4 we shall explain how W is constructed. A class of examples in which it can be computed explicitly is d P(-i - , (z - x m- 1 » dx Then

(3.7)

W(z)

={

d

= [- - + (z dx

i:

- x m-

1

2

e (zs-m-

s m)

I

d )] 0 [ -

dx

+ (z

- x m-

1

)].

±l

for m odd.

On the other hand, if m is even and d d P(-i - , (z - x m - 1» = [ dx dx

+

(z - x m -

1

d )] 0 [ - -

dx

+ (z

_ x m- 1)],

then W(z)

== O.

It was observed in [CG] that the entire function j':x, exp(2(zs - m- I sm» ds has order m/(m - 1). Thus at least three behaviors are possible: W can be identically zero, a nonzero constant, or an entire holomorphic function of order exactly m/(m - 1). In general, W will be of a transcendental nature, as illustrated already by (3.7), and will not be effectively computable. On the technical level our crucial result is Proposition 3.8. Fix an integer m ~ 3 and consider L z = P( -i :fx ' (z - x m- I » where P is generic, homogeneous, and nondegenerate. Then there exists 8 > 0 such that !W(z)! ~ 8 exp(8Izl m/(m-I)

as

IR+

=

ax

and

Y

= ay

-

xm-1at •

Let P be a generic, homogeneous polynomial. Then for any m E {3, 4, 5, ... }, .c = P(-iX, -iY) is not analytic hypoelliptic. Some additional work is required because the type (II) representations do not arise directly as in the previous discussion, but only as a limit of the type (III) representations (with (z - x 2 ) replaced by (z - x m - 1 in the fashion described by Helffer and Nourrigat [HN3]. Therefore, in the degenerate case, their results imply only the weaker conclusion that .c fails to be maximally hypoelliptic; see [HN3] for the definition. In fact, just as in the group-invariant setting, .c turns out to be hypoelliptic in neither the Coo nor the analytic sense. By two different arguments [PRJ, [HH], [C2], [C3] the case L = X 2 + y 2 has already been treated. Both used particular properties of this operator not present in general.

»,

for m even

ds

x

3 Z ~ 00.

Theorem 2.4 follows at once. If.c is degenerate, then analysis on the level of either type (I) or type (II) representations establishes the existence, on an open subset of G, of a function F 1- Coo such that.cF == 0; hence .c is hypoelliptic in neither the Coo nor the analytic categories. If.c is nondegenerate, then W is

4 THE WRONSKIAN W AND THE SCATTERING PROBLEM Throughout this section it is assumed that P is nondegenerate and generic, and for simplicity of exposition that none of the roots Yj are imaginary. The principal idea of the proof is to define W as the determinant of what can be thought of as a scattering matrix. The ordinary differential equation L z has n linearly independent solutions, and if r denotes the number of indices j for which ffi(Yj) > 0 then it happens (see below) that there exist exactly r linearly independent solutions which remain bounded as x --+ +00, and exactly n - r independent solutions which remain bounded as x --+ -00. This happens for every z E Co Now, the former solutions have an asymptotic behavior which may be understood rather precisely as x --+ +00, and similarly for the latter as x ~ -00. Therefore the only possibility for a globally bounded solution is that these n solutions should become linearly dependent, for some value of z. The situation is mostly parallel for the level (II) representations, but certain differences have their significance. The number of independent bounded solutions for P(-i:fx, -2x), as x ~ +00, is again r, but is also r as x --+ -00. For

138

CHAPTER 6

Ix '

P( -i 2x), the number is n - r in both directions. Therefore if r > n/2, one has 2r > n solutions, which perforce must be linearly dependent. Hence there exists a bounded solution for P(-i -2x). The same goes for P(-i 2x) if r < n /2. Taking the case where some Yj has vanishing real part into account, this argument establishes Proposition 3.3. A precise formulation for the level (III) representations is as follows. Order the roots Yj so that ffi(Yd < ffi(Yj) whenever k < j. Denote

Ix,

4>j.(x)

= Yj(Zx -

+ Pj In(z

- x 2)

where the Pj are certain scalars, and that branch of the logarithm function is chosen whose imaginary part is equal to i 7T: on the negative real axis.

Proposition 4.1. Suppose that P is nondegenerate and generic. Then for each j there exists Pj E C such thatfor each z E C, there exists afunction 1/IZ E COO(IR) Lz

o (exp(C1d/2 )).

Comment. The growth estimates for W are largely artifacts of its normalization, rather than an intrinsic feature of the representations under consideration. In the case m = 2, all representations with z E IR are mutually unitarily equivalent; an intertwining operator is induced by the change of variables x 1--+ x + z. Thus one might expect W to be constant, but in reality it is aGaussian. To approach the lower bound of Proposition 3.8, in the case m = 3, assume z E 1R+ is large, let A denote a sufficiently large positive constant, and denote = ±ZI/2.

xt

x ~

1/lZ == 0

1/IZ(x) =

e"'j(X)(1

+ O(x- I ))

as

x

~ +00.

E

xii + AZ- I / 4 and all 1 :s j :s n, d1

-1/I+(x) dx1 Z}

(4.4)

and

=

1R+ ,for all 0

l-I(Z - x 2 )l-l e <%>j(x)(l

:s e <

n ,for all

+ O(A -I)).

}

I 4 For Ix - x+1 o -< AZ- / ,

1/1"};. is uniquely determined, while in general each 1/IZ is uniquely determined modulo linear combinations of {1/I;t: k > j}. There exists a constant C I < 00 such thatforall z E C,forallx ~ CdZ}I/2 and all 0 :s k < n, 1 dx k

IW(z)1 =

Proposition 4.3. For all sufficiently large z

satisfying

dk

The last clause of the proposition implies that W is an entire holomorphic function. Straightforward estimates yield the upper bound

Ix,

tx3)

139

ANALYTIC HYPOELLlPTIC DIFFERENTIAL OPERATORS

1

(4.5)

1

n-I I -1/I+'(x) d I "-'" n-I Z-l/4 I -d "Z-l/4 ,',+'(X+ ~ dx1 ~ ~ dxlP~ 0 l=O l=O

+ AZ- I / 4 ) I '

uniformly in z. (1/I+'(x) -

y k - I (z

Z}}

_x 2 )k-I e <%>j(X») I

< C(lxl

-

+ (z) 1/2)-II(z -

x 2 )k-I e <%>j(x) I

•

The 1/1Z may be constructed so that each is an entire holomorphic function of z, modulo (jor each z) a linear combination of {1/I;t : k > j}.

There exist also solutions 1/1; possessing analogous properties as x ~ -00. There is one formal change in the statement: 1/Iz} is unique modulo linear combi­ nations of the more rapidly decaying solutions {1/Izk : k < j}, and the final clause of the proposition should be modified accordingly. Assume that the number of Yj with positive real parts is n/2.

:s

o'

Z

1

Definition 4.2. W(z) = determinant

In summary, one has excellent control of the 1/IZ for x - xii » Z-I/4, one loses much of this control for Ix - xii I Z-I/4, and one has no control for x :s xii - AZ- I / 4 . Thus the situation is analogous to a scattering problem, in which the solutions 1/1Z undergo scattering by an obstacle which is localized where Ix - xiii :s AZ- I / 4 • The same loss of control occurs for the solutions 1/1;, with xii replaced by x Hence the situation is more accurately viewed as one of scattering by two obstacles, widely separated for large z. Normalize the 1/IZ by multiplying by scalars, depending on z, so that the new solutions t/J thus obtained satisfy

dl - l 1/1;(0): j ( dx

:s

1 d ) n/2; - l 1/IZ(0): j > n/2; dx

"n -I -l/4 I _d ,1,+ (x) I "-' 1 ~Z dx 1 PZ} l=O O:":l
v Ix -

I 4 x+1 o <- AZ- / ,

uniformly in z E 1R+ provided z is sufficiently large.

__

__

IiiiliJ

140

__

CHAPTER 6

Define g~(y) = tz~(rl/2

+ r- I / 4 y).

Lemma4.6. As JR.+ '" Z ~ 00, the g~formanequicontinuousfamilyoffunctions on every compact subset of R The same holds for all oftheir derivatives. For any sequence r of values of z tending to +00, if {g~ : z E r} converge uniformly on compact subsets of lit then the limit is a solution g~,j of P(-i -2x) which does not vanish identically. Moreover, g~,j (x) decays rapidly to 0 as x ~ +00, for all j > n12.

:x'

There are similar functions gti, and possibly limits g~,j' to which an analogous statement applies. Let us imagine for a moment that circumstances were simpler, that there was only the single obstacle near xii, that the 1/tz} satisfied estimates similar to those for the 1/t~, for Ix - xiii ::s AZ- 1/ 4 , and that the gti were redefined accordingly, This is actually the situation, except for in terms of the behavior of 1/tti near very minor alterations, when (z - x 2 ) is replaced by (z - x m - I ), with m even. There are then two possibilities:

~

:x'

IW(z)1 ~ z-co exp( ~

L

Iffi(Yj)lz3 / 2 )

j

as JR.+ '" Z ~ 00, for some constant Co E JR.+; or as z ~ 00 through a sequence for which (4.7) fails for a sufficiently large but fixed Co, fixing a subsequence for which the g~ and gti all converge to limits g~,j and g~,j' respectively, the solutions (4.8)

{g~.j: j

::s n12} U {g~.j:

j > n12}

are linearly dependent. In the case (4.8) there exist scalars, not all equal to zero, such that (4.9)

L

Cjg~,j ==

j ~n/2

L

j>n/2

Cjg~,j"

Defining f

=

L

Cjg~,j'

j~n/2

fx '

f is a solution of P( -i -2x). It decays rapidly as x ~ -00 because each of the g~,j does (because ffi(Yj) < 0); it also decays rapidly as x ~ +00, because of the alternate representation afforded by (4.9). It follows from the estimates (4.4) at xii + AZ- 1/ 4 that the g~.j must be linearly independent, hence (granting the same for the g~.j in this hypothetical situation) that f does not vanish identically. Thus we would have Proposition 3.8.

2

This scheme does not apply directly to P(-i (z - x » because we know 14 I 4 essentially nothing about the 1/ttl in the large interval [x + AC / , xii - AZ- / ]. To overcome this we introduce a third set of solutions 1/t~, whose behavior is well understood in exactly this interval. We introduce two additional determinants of the same nature as W, one governing the scattering of the 1/tti into 1/t~j at X the other governing scattering of 1/t~ into 1/t~ at xii. An inequality relating the three shows that if W is substantially smaller than the estimate in (4.7), then one of the other two determinants must also be anomalously small. The argument of the as appropriate. One preceding paragraph may then be implemented at either of =f2x). By means ofachangeofvariables, obtains abounded solution of P(-i this produces a bounded solution of P(-i =fx).

o

o'

:x'

xt,

:x'

5 RELATED QUESTIONS In this section we consider in more detail some issues related to the operator

xt.

(4.7)

141

ANALYTIC HYPOELLJPTIC DIFFERENTIAL OPERATORS

£

= B; +

(By - x m - 1Bt )2

in JR.3 , and the ordinary differential operators

d2 L z = - - + (z _x m- I )2. dx 2 We assume always that m ~ 2 is even. It is useful to introduce L/3"

= Lz

where

z = re i /3.

Fixm. Definition 5.1. A ray JR.+ . e i /3 in C is said to be ofpolynomial growth ifthere exist a E JR. and c > 0 such that for all sufficiently large r E JR.+,

(5.2)

II L /3"fIIU(IR) ~ erG IIf11u(lR)

for all f in the domain of L /3,"

It is permitted that a be negative. As proved by Pham The Lai and Robert [PRJ, whenever two rays of polynomial growth can be found such that 1131 - 1321 < 1C m;;; I , then there exists at least one nonlinear eigenvalue in the sector bounded by the two rays. When m is odd, any ray with 1C 12 < 13 < 31C 12 is a ray of polynomial growth, as is JR.+ (with a = (m - 2)/2(m - 1)); this yields sufficiently narrow sectors to guarantee the existence of nonlinear eigenvalues, for odd m ~ 3. This is how the failure of £ to be analytic hypoelliptic was first proved for odd m. When m is even, JR.± are still rays of polynomial growth, with a = (m - 2) 12(m - I), but it is not apparent that there exist any other such rays.

142

CHAPTER 6

means that they are extremely close to being linearly dependent for large r. The together in functions fr required in Proposition 5.4 are obtained by pasting the interval Ix - Xo I S ).,r-(m-2)/2(m-l) via a partition of unity. The functions 1/1;= extend to entire functions of x E C, their Wronskian is constant. and (5.6) follows by analyzing the equation obtained from Lp.r by shifting from x E IR to x E IR . eia for a certain ex =F O. In a sense, the solutions 1/1;= constitute the only possible obstruction to the

One may ask (we do so only for even m): • • • •

t;=

Where are the nonlinear eigenvalues located? Do there exist any rays of polynomial growth. besides JR±? How does W behave off of the real axis? What can be said about Coo hypoellipticity of operators

Do

= a; + (eiOa y -

143

ANALYTIC HYPOELLIPTIC DIFFERENTIAL OPERATORS

x m - 1at )2,

which are sums of squares of non-real vector fields satisfying the condition of Hormander? This last question is related to the others; to study Coo hypoellipticity of Do one should analyze the one-real-parameter family oflevel (III) representations in which Do corresponds to {L o.r : r E 1R) acting on L 2 (JR). Some partial answers may be obtained.

Proposition 5.3. For m ~ 4 and even, there are no nonlinear eigenvalues z satisfying I arg(z) I < I m~l or IJr - arg(z) I < I m~l .

inequality 5.2:

Proposition 5.7. For each even m ~ 2 there exists a ~ 0 such that for any f3, for any sufficiently large r E 1R+, there exists a subspace V of codimension one in the domain of L p.r' such that IILp.r fIIU(IR) ~ era II fIIu

Vf

E V.

For m ~ 4, (J' is strictly positive. Proposition 5.4 plus standard arguments yield

Proposition 5.8. For any e such that eiO ¢ lR, De fails to be Coo hypoelliptic at This suggests that there may be a corresponding estimate which would affirm that every ray in these two sectors would be a ray of polynomial growth. but this hope is dashed by the next result.

Proposition 5.4. For any even integer m ~ 2, the only rays ofpolynomial growth are JR±. More precisely,for any f3 E JR satisfying 0 < 1f31 < Jr, there exists 8 > 0 such that for every r E JR + , there exists fr E L 2 (JR) satisfying II fr II U = I but

II Lp.r fr IIU(IR) S C exp( _8r m /(m-l».

t;=

xt

t

Proposition 5.5. The Wronskian of the normalized solutions satisfies (5.6)

This should be contrasted with the situation for operators which are sums of squares of real vector fields satisfying the Hormander condition. In that case. any small perturbation of the vector fields in the C k topology, for suitably large k, produces an operator which is still hypoelliptic, even though the perturbation will not in general be of lower order in any sense.

6 FINAL COMMENTS

fr is constructed by the method of the preceding sections. Solutions 1/Ir± of Lp.r, which decay rapidly as x -+ ±oo, respectively, are constructed as before. by multiplying by scalars so that are suitably normalized on the Defining interval {Ix - xol S ).,r-(m-2)/2(m-l»), where Xo = (r cos(f3»I/(m-1) plays the in the previous analysis, and defining W to be the Wronskian same role as determinant of r± so that IWI S C uniformly in r. one finds

t;=

the origin.

\WI s C exp( _8r m /(m-l».

t;=

We emphasize that the normalizations are carried out so that the are es­ sentially bounded below by I near xo, and similarly for their derivatives. so (5.6)

d

For homogeneous. constant-coefficient differential operators in IR , there is an equivalence between ellipticity, Coo hypoellipticity. and analytic hypoellipticity. As presented. our results indicate that this equivalence, with ellipticity replaced, of course. by the Rockland condition that Jr(L) be injective for each irreducible unitary representation Jr , breaks down for the class of groups in question. However, the analogy with the abelian case goes further, from an alternative point of view. Consider again the case of JR d , and let L be homogeneous and elliptic. with constant coefficients. Then, although L will be analytic hypoelliptic. still the Cauchy estimates will be violated ifthe constant C1+lal is replaced by Ce • £Ial for all sufficiently small £, no matter how large Ce is chosen to be. This amounts to the fact that while the symbol of L is free of zeroes in a conic neighborhood of IR d C Cd, yet it still does have zeroes.

144

CHAPTER 6

Consider now the four-dimensional nilpotent group of step 3 which stood at the center ofour analysis, and let L be a (generic) homogeneous, left-invariant operator which satisfies the Rockland condition. Then we have proved the existence of certain non-unitary representations lr on which lr(L) is not injective in L 2 , an analogue of the fact that every nonconstant polynomial in one complex variable has at least one zero. This means not only that L is not analytic hypoelliptic, but that again, the constant Cl+lal appearing in the Gevrey estimates of order s = 3 for solutions of L cannot be replaced by C, . elal for arbitrarily small e. It can be shown tha.t for any such operator on this group which satisfies the Rockland criterion, all solutions of Lu == 0 in an open set must belong to the Gevrey class G3 . Thus the situation is quite analogous to that for IRd , provided only that the class G 1 of analytic functions is replaced by G 3 • Substantial progress on related problems has been achieved in the interval be­ twee1! preparation of this manuscript and its publication [C7],[C8]. More detailed information on the nonlinear eigenvalues has been obtained by Ching-Chau Yu in a UCLA PhD dissertation (in preparation). University ofCalifornia, Los Angeles

REFERENCES

a

[C I] M. Christ. "On the equation in weighted L 2 nonns in C I." J. Geom. Anal. 1 (1991),193-230. [C2] . "Some non-ana1ytic-hypoelliptic sums of squares of vector fields." Bul/. Amer. Math. Soc. 116 (1992), 137-140. [C3] . "Certain sums of squares of vector fields fail to be analytic hypoelliptic." Comm. Part. Dijf. Eq. 16 (1991), 1695-1707. [C4] . "Analytic hypoellipticity breaks down for weakly pseudoconvex Rein­ hardt domains." Int. Math. Res. Not. 1 (1991), 31--40. [C5] . "A family of degenerate differential operators." J. Geom. Anal. 3 (1993), 579-597. [C6] . "Analytic hypoellipticity, representations of nilpotent groups, and a nonlinear eigenvalue problem." Duke Math J. 72 (1993),595--639. [C7] . "A necessary condition for analytic hypoellipticity." Math. Research Letters 1 (1994),241-248. [C8] . "The Szego Projection Need Not Preserve Global Analyticity." Preprint. [CG] M. Christ and D. Geller. "Counterexamples to analytic hypoellipticity for domains of finite type." Ann. ofMath. 235 (1992), 551-566. [CL] E. Coddington and N. Levinson. Theory of Ordinary Differential Equations. McGraw-Hill, 1955. [FS] G. B. Folland and E. M. Stein. "Estimates for the b complex and analysis on the Heisenberg group." Comm. Pure Appl. Math. 27 (1974), 429-522.

a

ANALYTIC HYPOELLlPTIC DIFFERENTIAL OPERATORS

145

[G] D. Geller. Analytic Pseudodifferential Operators for the Heisenberg Group and Local Solvability. Mathematical Notes 37. Princeton University Press, 1990. [GS] P. C. Greiner and E. M. Stein. "On the solvability of some differential operators of type Db." Several Complex Variables, Proceedings of International Conferences, Cortona, Italy, 1976-77, Scuola Normale Superiore, Pisa, 197, 106-165. [He] B. Helffer. "Conditions necessaires d'hypoanalyticite pour des opCrateurs invari­ ants a gauche homogenes sur un groupe nilpotent gradue." J. Diff. Eq. 44 (1982), 460--481. [HH] N. Hanges and A. A. Himonas. "Singular solutions for sums of squares of vector fields." Comm. Part. DijJ. Eq. 16 (1991), 1503-1511. [HN1] B. Helffer and J. Nourrigat. "Hypoellipticite pour des groupes nilpotents de rang 3." Comm. Part. Diff. Eq. 3 (1978), 643-743. [HN2] . "Caracterisation des opCrateurs hypoelliptiques homogenes invariants a gauche sur un groupe nilpotent gradue." Comm. Part. DijJ. Eq. 4 (1979), 899-958. [HN3] . Hypoellipticite Maximal pour des Operateurs Polynomes de Champs de Vecteurs. Prog. Math. vol. 58. Birkhauser, 1985. [K] A. A. Kirillov. "Unitary representations of nilpotent Lie groups." Russian Math. Surv. 17 (1962), 53-104. [Ke] M. V. Keldysh. "On the completeness of the eigenfunctions of classes of non­ selfadjoint linear operators." Russian Math. Surv. 26 (1971),15-44. [M] G. Metivier. "Hypoellipticite analytique sur des groupes nilpotents de rang 2." Duke Math. J. 47 (1980), 195-221. [NSW] A. Nagel, E. M. Stein, and S. Wainger. "Balls and metrics defined by vector fields I: Basic properties." Acta Math. 155 (1985), 103-147. [PRJ Pharo The Lai and D. Robert. "Sur un probleme aux valeurs propres non lineaire." Israel J. Math. 36 (1980), 169-186. [R] C. Rockland. "Hypoellipticity on the Heisenberg group-representation-theoretic criteria." Trans. Amer. Math. Soc. 240 (1978),1-52. [RS] L. P. Rothschild and E. M. Stein. "Hypoelliptic differential operators and nilpotent groups." Acta Math. 137 (1976), 247-320.

------.~----... ~----=----=~==~--====--=.:;;~;~==~<~~~=-:=;:;:=~===~=.~"§.t:~~~g;j::"'E~,,~~':£~~~~~~~_==_~.::..'"~~~~~~~~-=--='55i''''''''.,

.. ~~~::==~~~~=~:=,;;;;

147

OPERATEURS B/LlNEAlRES ET RENORMALlSATlON

LI(lR n ), sans appartenir a 1-{l(lR.n ). Si la somme de ces n tennes appartient a 'HI (lR. n), ce fait remarquable est dfi ala presence de cancellations, provenant des

hypotheses (1.2). Ces cancellations seront analysees en appliquant un algorithme de renonnalisation achacun des produits Ej(x)Bj(x). Un algorithme de renor­ malisation pennet d'ecrire Ie produit ponctuel uv entre deux fonctions u (x) et v (x) sous la fonne

7

= P(u, v) + R(u, v)

uv

(1.3)

Operateurs Bilineaires

ou l'operateur bilineaire P, definissant l'algorithme, satisfait les conditions suivantes:

et Renormalisation

P(u, v) = P(v, u)

(1.4)

R. R. Coifman, S. Dobyinsky, et Y Meyer

(1.5)

P(u, v)

= 0

chaque fois que la fonction u(x) est une constante

et ou I'operateur bilineaire R possMe les proprietes suivantes: (1.6)

si u j, j

1 INTRODUCTION Une fois renonnalisees, certaines expressions bilineaires se comportent mieux que prevu et la fonne precisee du lemme du div-curl nous pennettra d'illustrer cette remarque. Cette fonne precisee, conjecturee par P. L. Lions et demontree dans [2], utilise l'espace 'HI (JRn) de E. Stein et G. Weiss. Rappelons que l'espace de Hardy 1{1(JRn) est I'ensemble des fonctions I, appartenant aLI (JRn) , dont les n transfonnees de Riesz R I (f), ... , R n (f) appartiennent egalement aL I (IR n ). La nonne de I dans 'HI (lR.n ) est la somme 11/111 + IIRI(f)lh + ... + IIRn (f)lll et, muni de cette nonne, 'HI (lR.n ) est un espace de Banach, compose de fonctions d'integrale nulle. Comme I'ont montre C. Feffennan et E. Stein [4], Ie dual de 'H I (JRn) est l'espace B MOde John et Nirenberg. La fonne precisee du lemme du div-curl est l'enonce suivant:

Theoreme 1. Soient E(x)

=

=

(E I (x), ... , En(x» et B(x) (B I (x), ... , B n (x», X E IR n, deux champs de vecteurs verifiant les proprietes suivantes:

(1.1) (1.2)

Ej(x) E L 2 (JRn ),

Bj(x) E L 2 (IRn)

div E(x) = 0,

Alors E(x)· B(x) = EI(X)B1(x)

+ ... +

u E L2

pour

1S j S n

curl B(x) = O. En (x)Bn (x) E 1{1(lR n).

Les derivees qui interviennent dans (1.2) sont prises au sens des distributions. Chacun des tennes Ej(x)Bj(x) du produit scalaire E(x) . B(x) appartient a

(1.7)

et ~

vEL

1, et v j, j

en nonne L 2 et si Uj alors R(uj, Vj)

Nous avons note Ii

J Ii (x)rp(x) dx

--->..

--->..

--->..

~

R(u, v) E 'HI

impliquent

I, sont deux suites bomees

u (j -+ +00), Vj

U(u, v)

I (j

2

--->..

V (j -+ +00),

(j -+ +00).

-+ +00) la convergence au sens des distributions:

-+ J I(x)rp(x) dx pour toute fonction de test rp. Dans la con­ clusion de (1.7), on peut supposer que rp(x) appartienne aI'espace V M 0 qui est la fenneture, pour Ia nonne BMO, de I'espace vectoriel des fonctions continues et nulles a I'infini. Signalons enfin que 1{1 est Ie dual de V MO. Voici la signification des deux tennes de la decomposition (1.3). Le tenne P(u, v) est la partie principale du produit uv. Si au lieu d'appartenir a L 2 (JRn), les deux fonctions u et v appartenaient a LP(lR.n) et si p < 2, alors Ie produit uv n'aurait plus de sens, en tant que distribution, et it conviendrait de lui soustraire certaines quantites infinies pour retrouver un terme suffisamment oscillant pour etre une distribution temperee. Ces quantites infinies sont presentes dans Ie tenne P(u, v). Le tenne R(u, v) appartiendra a l'espace de Hardy 'H p / 2 lorsque u et v appartiennent aLP et que pest inferieur a2. Le terme R(u, v) est donc Ie produit uv, renormalise par soustraction de P(u, v).

Cette renonnalisation doit etre la plus simple possible: nous ne voulons sous­ traire que Ie strict minimum et, la plupart du temps, ne rien soustraire du tout. Si par exemple la fonction u (x) est une constante, Ie produit entre cette constante et une fonction arbitraire vex) n'a pas besoin d'etre renonnalise et P(u, v) sera donc

148

CHAPTER 7

nul, comme l'indique la condition (1.5). Cette meme condition (1.5) nous perrne­ ttra de verifier (section 8) que Ie produit entre une fonction u(x) qui est reguliere et a support compact et une fonction arbitraire vEL 2 conduit a une correction P(u, v) faible, au sens que l'operateur qui a v associe P(u, v) est compact. L' operation de renorrnalisation doit perrnettre de multiplier deux distributions temp6rees arbitraires. Elle s'apparente au paraproduit, tel que J. M. Bony l'a defini. La condition (1.7) signifie qu'en se limitant a des suites bornees dans L 2 , la renorrnalisation est compatible avec la convergence au sens des distributions. Signalons enfin qu'il existe plusieurs solutions au probleme de la renorrnalisa­ tion du produit. Nous avons choisi celIe, presentee dans la section 3, qui conduit aux calculs les plus simples. Revenant au theoreme 1, on a automatiquement et sans tenir compte de (1.2), (1.8)

Ej(x)Bj(x) - P(E j , Bj)(x)

E

HI (lR") ,

1::: j ::: n.

Demontrer Ie tMoreme 1 revient donc a prouver que la somme S(x) = P(EI, BI) + ... + P(En , Bn ) appartient aussi a HI lorsque les conditions (1.1) et (1.2) sont satisfaites. En fait, on a beaucoup mieux et cette somme S (x) appartient ai' espace de Besov homogene iJ~' I qui sera defini soigneusement dans la section suivante et qui est un sous-espace "rudimentaire" de HI (lRn ). L' enonce du theoreme 1 prete done a confusion puisque Ie rOle joue par l' espace de Hardy HI (lRn ) y est relativement super/lCiel. alors que l' espace de Besov iJ?·1 est au CfPur du probleme. Cette meme analyse (1.3) nous foumira une demonstration particulierement elegante du lemme "classique" du diY-curl, tel que F. Murat et L. Tartar 1'ont enonce. II s' agira de montrer que, sous les hypotheses (1.2), Ie produit scalaire E· B possede egalement la propriete de continuite faible decrite dans (1.7). Dans la sec­ tion suivante sont rappeles quelques resultats generaux concernant les op6rateurs bilineaires et l'espace de Besov homogene iJ?·I. Nous pourrons alors construire toute une collection d'op6rateurs P ayant les proprietes (1.4) a(1.7). A I'interieur de cette collection, nous choisirons un exemple explicite, se reliant aux "parapro­ duits" de J. M. Bony et conduisant a une demonstration particulierement simple du theoreme 1 et du lemme de Murat et Tartar. Nous conclurons en indiquant (sans demonstration) les liens qui existent entre la renorrnalisation et les series d'ondelettes.

2 LA DEFINITION GENERALE DES OPERATEURS BILINEAIRES P Pour la commodite du lecteur, nous rappelons quelques enonces figurant dans [1] ou [5].

I

149

OPERATEURS BlLINEA/RES ET RENORMALISATION

Les symboles bilineaires servant aconstruire les operateurs bilineaires sont des fonctions r(~, TJ) definies sur lRn x lRn prive de (0,0) et verifiant les conditions usuelles

la;affr(~, TJ)I ::: Ca,p(I~1

(2.1)

+

ITJ\)-laI-iPI

ou a = (ai, ... ,an) est un multi-indice de longueur lal = al

a

a~

=

(a )a a~1

l

...

(a)~ a~n

+ ... + an

et ou

.

En particulier, les conditions (2.1) sont satisfaites si r(~, TJ) est indefiniment derivable dans lRn x lRn prive de (0, 0) et si r (A~, ATJ) = r (~, TJ) pour tout A > O. On associe aun symbole bilineaire r (~, TJ) l' operateur bilineaire T(j, g)(x) = (2nT 2n

(2.2)

II ei(Hry)·xr(~, TJ)j(~)g(TJ) d~

dTJ

j, g sont les transforrnees de Fourier de f et g. Les conditions (2.1) impliquent que T, qui est evidemment defini si f et g appartiennent ala classe S (lRn ) de Schwartz, se prolonge en un operateur (encore note T), defini sur L 2(IRn) X L 2(lRn ), a valeurs dans L I (IRn). Un cas particulier evident est Ie symbole r = 1 conduisant a T (j, g) = f g. La continuite de T: L 2 X L 2 ~ L I sous les hypotheses (2.1) est donc une generalisation de l'inegalite de Holder. Cela amene aconjecturer que T se prolonge en un operateur lineaire continu de LP x L q a valeurs dans L' si 1 ::: r < 00, 1 < p ::: 00, 1 < q < 00 et + ~ = ~. Ceci est vrai (valeurs limites comprises) et la demonstration se trouve dans [5]. Une fois pour toutes, les conditions (2.1) seront supposees satisfaites dans les enonces qui suivent (lemmes 1, 2, et 3). Dans ces conditions, la fonction r sera appelee Ie symbole bilineaire de l' op6rateur bilineaire T.

OU

i

Lemme 1. Une condition necessaire et suffisante pour que l' operateur bilineaire T: L 2

X

L 2 ~ L I, defmi par (2.2) ait la propriete suivante T(j, g) E H1(IR n)

(2.3)

lorsque f

E

L 2 et g

E

L2

est que son symbole bilineaire r (~, TJ) verifie r(~, -~)

(2.4)

= 0

pour tout

~ =1=

O.

Les conditions (2.3) et (2.4) sont aussi equivalentes a (2.5)

(

JJR.

T(j, g)(x) dx = 0

pour toute f

E

L 2 et toute g

E

L2

150

CHAPTER 7

ou encore acette meme condition (2.5) forsque f et g appartiennent a fa classe de Schwartz.

151

OPERATEURS B1LINEA1RES ET RENORMALISATION n

On designe par 1/I(x) une fonction de la classe de Schwartz S(lR ) dont la transforrnee de Fourier est nulle si I~ I ::: 2/3 ou si I~ I ~ 8/3 et verifie, si ~ =I 0, 00

Ce lemme foumit une demonstration (fausse) du theoreme 1. Pour verifier que E . B appartient a'HI, il suffirait donc de montrer que la condition (2.5) est satisfaite. Or on a B = V U (x) puisque curl B (x) = O. II vient E (x) . B (x) = div[U(x)E(x)] puisque div E(x) = O. Finalement J E(x) . B(x) dx = O. Cette demonstration est insuffisante parce que Ie produit scalaire E . B n'est pas un des op6rateurs bilineaires auxquels s'applique Ie lemme I. La preuve du lemme I consiste a evaluer l'integrale I = T(f, g) . u(x) dx lorsque u (x) appartient v M 0 et que II u II B M 0 ::: 1. Mais on a I = L(f, u)gdx oil I'operateur bilineaire Lest defini par Ie symbole A(~, lJ) = r(~, -~ - lJ). On a alors IIL(f, u)1I2 ::: ClJfll2l1ullBMo en appliquant Ie theoreme 34 du chapitre VI de [I]. Le lemme suivant n' intervient pas dans la demonstration du theoreme 1. II nous apprendra que tous les choix de l'operateur T que nous ferons sont equivalents.

J

a

J

o<

(2.9)

c :::

L 1~(2-j~)I. -00

j On designera alors par 1/1j (x) la fonction 2nj 1/1 (2 x) et par /:!" j l' operateur de

convolution avec 1/1 j' Vne fonction f ELI (Rn ) appartient a B?' I si et seulement si 00

L lI/:!"j(f)111 <

(2.10)

et cette condition ne depend pas du choix des operateurs /:!" j' Vne seconde definition equivalente s'obtient en decomposant f(x) dans une base orthonorrnee d'ondelettes 1/IA' A E A, de regularite r ~ l. Alors f E B?,I si et seu1ement si (2.11)

f(x)

=L

U A1/IA(X)

et

AEA

Lemme 2. En conservant les notations du lemme 1, une condition suffisante pour que T (f, g) appartienne arespace de Besov homogene B?' I (Rn), pour tout couple (f, g) de deuxfonctions de L 2 (Rn), est que l' on ait r(~, -~)

(2.6)

~

pour tout

= 0

=I

0

= 0 = r(O, lJ)

pour

~

i- 0,

00.

AEA

Selon la terrninologie de G. Weiss et de ses collaborateurs, (2.11) s'appelle une decomposition atomique en "atomes speciaux." Les operateurs bilineaires P que nous utiliserons dans la preuve du theoreme 1 seront definis par des symboles bilineaires 1f(~, lJ) verifiant les conditions

(2.12)

r(t 0)

L luAI1I1/IAlii <

suivantes:

et (2.7)

00

-00

lJ

i-

O.

(2.13) (2.14)

= 1f(lJ,~) 1f(~, 0) = 1f(0, lJ) = 0 1f(~, -~) = I

1f(~,

lJ)

si ~

i-

0 et lJ

si ~

i-

O.

i-

0

B?,I, un demier resultat doit etre mentionne.

Compte tenu des lemrnes I et 3, ces proprietes entrainent les conditions (1.4) a

Lemme 3. En conservant les notations du lemme 1, la condition (2.4) implique fa propriete de continuite faible suivante:

(1.7). En outre, si deux symboles bilineaires 1f1 et 1f2 possedent ces proprietes, alors 1f3 = 1f1 - 1f2 verifiera les conditions suffisantes du lemme 2. eela entrafne que, modulo B?,I, tous les choix de P que nous ferons seront

Avant de passer ala definition de

si Ii et gj sont deux suites bornees dans L2(lRn ) (2.8)

et si

Ii

->.

f et gj

->.

g, alors T(/j, gj)

->.

equivalents.

T(f, g). 3 RENORMALISATION ET PARAPRODUITS

Le lecteur trouvera dans I'appendice les preuves tres simples de ces deux lemmes. Rappelons maintenant la definition de I'espace de Besov homogene B?·I .

Le but de cette section est de donner une demonstration complete du theoreme 1 en utilisant un choix de l'operateur P qui se reliera naturellement la theorie du

a

paraproduit de J. M. Bony.

152

CHAPTER 7

Pour definir P on utilise la decomposition classique de l'espace de Fourier jRn \ {OJ en couronnes dyadiques ainsi que l'analyse de Littlewood-Paley-Stein

OPERATEURS BILlNEAlRES ET RENORMALlSATlON

Ona 00 Sex)

associee a cette decomposition. On designe pour cela par 1{! (x) une fonction a valeurs reelles, appartenant a la classe S(jRn) de Schwartz et dont la transformee de Fourier verifie les deux conditions (3.1)

;j,(~)

=0

si

I~I ~ 2/3 ou I~I ~ 8/3

et

L 1;j,(rj~)12 = -00

1

~

pour tout

¥ o.

= L(!i.jE). (!i.jB) -00

et nous allons, en designant par II . II la norme dans l'espace de Besov iJ?,I, demontrer Ie theoreme suivant:

Theoreme 2. En consefllant les notations precedentes et les hypotheses du theoreme 1, on a

00

(3.2)

153

1I(!i. j E) . (!i. j B)1I ~ C(n)lI!i. j EII211!i. j BII2

(3.8)

ou C (n) ne depend que de la dimension n.

Compte tenu de (3.1), la condition (3.2) se reduit a (3.3)

1;j,(~)12

+ 1;j,(2~)12 = 1

si

2 4 3 - lei ., < - -. 3

_ <

II resulteevidemment de (3.8) que Sex) E iJ?,I. Eneffet. on a L~oo lI!i. j EII~ II E II~ et L~oo 1I!i. j B II~ = II B II~ et cela entraine

On pose ensuite 1{!j(x) = 2nj 1{!(2 j x), j E Z, et ron designe par!i. j 1'0¢rateur de convolution avec 1{!j. La condition (3.2) signifie que

(3.4)

00 L

II (!i.jE) . (!i. j B)1I ~ C(n)IIEII2I1BII2·

-00

00 I = L !i.j!i.j -00

Pour etablir (3.8), nous utiliserons Ie lemme suivant:

Lemme 4. Soit lex) une /onction de/mie sur lRn , localement integrable, et possedant les deux proprietes suivantes:

ou !i. j est l'adjoint de !i. j'

On pose finalement, si / et g appartiennent a L 2(jRn),

00 (3.5)

V/=

(3.9)

P(j, g) = L(!i.J!)(!i.jg).

-00

II est trivial de verifier que P (j, g) appartient a L I (jRn) puisque l'on a 1I(!i.J!)(!i. j g)1I1 ~ 1I!i.J!11211!i.j gIl2

= aj{3j

etqueL~00lajl2 = IIf11~'L~001{3jI2 = IIgll~,gracea(3.2). Onadonc (3.6)

IIP(j. g)1I1 ~ 1If112I1g!l2.

Le symbole bilineaire de P est Jr(~, 11) = L~oo ;j, (2- j ~);j, (2- j 11). II verifie evidemment (2.12). La condition (2.13) provient de (3.1) tandis que (2.14) decoule de (3.2) et de ;j, (-~) = ;j, (~).

Pour demontrer Ie theoreme 1, on ecrit chaque produit E1(x)BI(x), ... , En(x)Bn(x) sous la forme P(Ej, B j ) + R(E j , B j ). On a R(E j , B j ) E HI et l'amelioration du theoreme 1 que nous avons en vue est Ie fait que (3.7)

Sex) = P(EI, B»

=

+ ... + peen, Bn)

.0 I E B '

I

(3.10)

/ ( - a f , ... , -a)

/ = div F

aXI

ou

1 n EL(lR)

aX n

F = (Flo ... , Fn ) ELI (lR n ).

Alors / (x) appartient a iJ?,1 (lRn ) et l' on a (3.11)

II/II

~

C(n)[II V /IIII1 F lllfl2.

On a pose IIF[1t = 11F11i1 + ... + IlFnlll et IIV/1I1 est definie de meme. La preuve elementaire du lemme 4 est renvoyee a la section suivante. Pour demontrer Ie theoreme 2, nous appliquerons Ie lemme 4 a la fonction hex) = (!i.jE) . (!i.jB). La transformee de Fourier de h est nulle hors de la 2j ce qui entraine boule I~ I ~

¥

(3.12)

16 . 16 . IIV/jll l ~ 32JllfJIII ~ 32JII!i.jEII211!i.jBII2.

grace au lemme de S. Bernstein dont nous rappelons l'enonce.

154

CHAPTER 7

Lemme 5. Si la transformee de Fourier f(~) d'unefonction f p ~ 00, est nu/le hors de la boule 1~1 ~ R, alors on a (3.13)

IIV flip

:s

E

LP(lRn), I <

Rllfll p

et si, en revanche, fest nu/le sur la boule I~I < R, l'inegalite inverse de (3.13) est verifiee au sens suivant: Rllfll p ~ C(n) IIV' flip

(3.14)

°

lII::J.j (u)1I2 ~ CTjllI::J.jBI12.

=

°

fjJ(~) =

(5.1)

°

fjJ(~) = I

si

I~I ~ 4/3

si

I~I ~ 2/3.

On pose alors lpj(x) = 2nj lp(2 j x) et l'on designe par Sj l'operateur de con­ volution avec lpj. On definit 1/I(x) par 1/I(x) = 2nlp(2x) - lp(x), on pose 1/1 j (x) = 2nj 1/1 (2 j x) et l'operateur de convolution avec 1/1jest Dj = Sj+ I - Sj. Soient f et g deux fonctions arbitraires, appartenant L 2(]Rn). On definit alors PI (f, g) par

a

PI(f, g) = L

(5.3)

entraine

fj(x)

(3.16)

Pour definir Ie paraproduit, on part d'une fonction radiale, lp(x), appartenant Ii la classe de Schwartz S(]Rn), a valeurs reelles et dont la transfonnee de Fourier fjJ(~) verifie les conditions

(5.2)

signifie B(x) = Vu(x) ou la fonction On observe ensuite que curl B(x) = (scalaire) u (x) est unique, modulo les fonctions constantes. Alors (3.14), appliquee Ii la fonction I::J.ju, s'ecrit

L'hypothese div E (x)

5 LE LIEN AVEC LE PARAPRODUIT DE J. M. BONY

et

ou C(n) ne depend qne de la dimension n.

(3.15)

155

OPERATEURS BILlNEAIRES ET RENORMALISATION

L

(Djf)(Djg).

lj-kl::::2

= div Fj(x)

Alors on a l'identite remarquable

j ou Fj(x) = (I::J.ju)(I::J.jE) et IIFjl1I ~ C2- llI::J. j BII211I::J. j EII2­ Le lemme 4 entraine donc Ie theoreme 2.

fg = PI(f, g)

(5.4)

+ R;(f, g) + R~(f, g)

ou 00

R;(f,g)

4 LA PREUVE DU LEMME 4 et

00

R~(f, g)

00

(4.2)

0, on a

Ces deux estimations resultent des deux assertions du lemme de Bernstein. Pour etablir(4.1), on observe que f = div FetqueI::J.j(f) = div(I::J.j(F». Onapplique alors (3.13) Ii chacune des composantes du vecteur I::J. j (F) et 1'on obtient (4.1). Pour verifier (4.2), on utilise l'inegalite inverse (3.14) que l'on applique I::J. j (f). Ceci tennine la preuve du lemme 4.

a

00

L fjJ(T(j-2)O~(Tj1/) L fjJ(T(j-2)1/)~(Tj~) -

-00

III::J. j (f) II I ~ CTjIlVfII1'

f)

sont les paraproduits definis par J. M. Bony. Le symbole bilineaire Jrl (~, 1/) de PI vaut

II I::J. j(f)111 ~ C2 j IIFIII ~

= L(Sj-2g)(Dd) = R; (g, -00

1-

tandis que, pour tout j

L(Sj_2!)(Dj g) -00

On observe tout d'abord que, pour tout a > 0, les fonctions f(x) et an f(ax) ont la meme norme dans iJ?·I. En choisissant convenablement a > 0, on peut se ramener au cas ou II V fill = IIF III . Pour demontrer Ie lemme 4, il suffit alors de verifier que, pour tout j ~ -I, on a (4.1)

=

-00

et verifie les conditions (2.12), (2.13), et (2.14). Cela signifie que l'operateur P que nous avons utilise dans la section 3 peut etre remplace par PI et que l'erreur commise appartiendra iJ?' I .

a

6 LE LEMME DE MURAT ET TARTAR Rappelons tout d'abord l'enonce classique du lemme du diY-curl [2].

- - - - - - - - - - - - - - - - - -.. ------.~.=~~----.----

~~~-.~~.-~.~

156

.. ~-==~= ......."""_"'"~~"""'-''''~''''~=~'T'"~','''..:-:."="''~''''.:;=_'''~':..~.z'"~~~~~""'='''''.~'~~=:''='-''._-=7=~-:.7"'''".:..;.:.~_=;~,~

CHAPTER 7

Soient E(m) et B(m), mEN, deux suites de champs de vecteurs verifiant

IIE(m)1I2 ~ Co,

(6.1)

E(m) -----"­ E,

(6.2)

(m ~ +(0).

B(m) -----"­ B

Supposons, d'autre part, que div E(m)

= 0 et curl B(m) = O. Alors on a

lim m~oo

f

E(m) . B(m)(x)u(x) dx

=

f

E(x) . B(x)u(x) dx

pour toute/onction de test u(x), continue et nulle a /'in/mi.

II suffit evidemment de demontrer (6.3) lorsque u(x) est une fonction de classe C I , a support compact. On utilise pour cela l'optSrateur P de la section 3. La propriete de continuite faible est automatiquement satisfaite pour les restes R(E(m), B(m), grace a (1.7), et il suffit donc de considerer p(E(m), B(m). On pose

f

It) =

et I'on a nj II ll j E(m)lIoo ~ 1I1/1jIl2I1 E (m)1I2 ~ C o2 / 2.

Cela permet de majorer IIt)1 par Cfj2nj lIu III. Pour demontrer (6.5), on observe que

(6.9)

(lljE(m) . (lljB(m)u(x) dx

(llju(m))(lljE(m) . Vu) dx.

On majore finalement cette integrale par IIllju(m)11211 IljE(m) 11211Vu1l 00 • comme nous l'avons deja note, II ll ju(m) 112 ~ CITj

et

IIlljE(m)1I2 ~

Or,

Co.

Les proprietes (6.5) et (6.6) etant demontrees, il reste a examiner (6.7). Or la convergence faible des E(m) implique la convergence simple des fonctions Il j E(m) (x). On a meme convergence unifonne sur tout compact. Puisque u (x) est a support compact, (6.7) en resulte. Le lemme de Murat et Tartar est donc demontre.

00

lim "I~m) = "

m~+oo~

~

J

-00

ou Ij =

f

I.

J

-00

(lljE) . (lljB)u(x) dx.

Pour cela, il suffit d'etablir l'existence de deux constantes C, et C2 telles que (6.5)

IIt)j ~ C,Tj

(6.6)

IIjm) I ~

C22nj

si si

j::: 0

j ~ O.

La condition (6.2) implique que, pour tout j fixe, on a

~

m~+oo

IJ~)=~

et (6.4) resultera de (6.5), (6.6), et (6.7). Pour etablir (6.6), on ecrit

(6.8)

f

7 LA RENORMALISATION PAR LES ONDELETTES 00

~n

Ijm) = -

(6.10)

et I'on doit verifier que

(6.4)

IljE(m) . IljB(m) = div[(llju(m), (lljE(m)]

et I'on a donc, apres integration par parties,

Lemme 6. Sous les hypotheses precedentes, (6.3)

157

OPERATEURS BILlNEAIRES ET RENORMALlSATJON

IIB(m)112 ~ Co

ainsi que

__=_.~.z...==:_~'''":..·~~._ ,,_.=;;;~.;~,,~~~.-_-"="=.=~"'''''':.'''~:=,~..:;::.:...:;~=_""~_-~

IljE(m)(x) =

f

l/tj(X - t)E(m)(t) dt

Considerons une base orthonormee, pour l'instant arbitraire, 1/IA(X), A E A, de L 2 (]Rn). Soit u(x) une fonction de L 2 (]Rn). On a alors u(x) = LAEA a A1/IA(x) et il en decoule que (7.1)

2

lu(x)1 =

L laAI211/1A(X)1 2 + AEA

w(x)

et f w(x) dx = O. On peut aussi considerer deux fonctions arbitraires u(x) et v(x) de L 2 (]Rn) , les decomposer dans la base orthononnee 1/IA' A E A, et ecrire OU w(x) E LI(]Rn)

(7.2)

u(x)v(x) =

L a APAI1/IA(X)!2 + w(x)

AEA

OU, la encore, w(x) ELI (]Rn) et f w(x) dx = O. En generalla fonction w(x) n'a pas d'autre propriete. Si I'on considere l'ex­ emple du sysreme trigonometrique (2Jr)-1/2e ikx sur l'intervalle [-Jr, Jr] au lieu de (1/IA) sur ]Rn, alors w(x) = u(x)v(x) - c ou c est une constante et w(x) est donc n'importe quelle fonction de L I [-Jr, Jr] d'integrale nulle.

_

~ ••• ~ __ c

,_._.

_

,,_

•• ~~'>;"~~~~==-====--"='~:~':=-";'-~~~~d"1..~~:::"~~~~~"~¥;:;:·~~j;."'-~~"'-.2,OE"s,;:""","?_"'·;i."""",~'"''''''''''~ _'"''-''~;;'''

_ __ " __,,_

158

CHAPTER 7

Soit maintenant VrA' A E A, une base orthononnee d'ondelettes reelles. On se reportera a [5] pour y trouver une description detaillee de ces bases. Excluons Ie cas particulier du systeme de Haar et supposons que la regularite r des fonetions VrA ne soit pas nulle (r ::: 1 dans les notations de [5]). Alors la fonction w(x) precedente appartient aHI des que u et v appartiennent

aL 2 (lR.n ). On se limite dans l'enonce qui suit a des fonctions sont deux fonetions de L 2(]Rn), on pose (u, v) =

f

avaleurs reelles et, si u et v

""-'ii£"'"

,""""'-'.',:""'''i'.i'''","''''''''''''.'''''<=~.'~

00

L

(8.1)

159

n?'\ etablir

:s

IIT(D j u, Sjv)1I

CIIull21lvII2

-00

et qu'it en est de meme pour les deux autres series. Les trois preuves etant semblables, nous nous limiterons a (8.1). Obser­ vons que T(Dju, Sjv) = T/u, v) ou Ie symbole bilineaire rj(~, 17) de T j est j j ~)fP(2- j 17)r(~, 17). Sur Ie support de rj, on a necessairement 1171 :s ~ 2 et

:s

:s

I~ 1

~ 2 j . Cela impIique

.

.

'3 2J :s I~ 1+ 1171 :s 4.2 J

si bien que l'on a

lara~rj(~, 17)1

(8.3)

regularite r ::: 1. Si u et v appartiennent a L 2 (]Rn), posons

:s Ca.tJr(\al+ltJl)j.

Les symboles r/~, 17) s'ecrivent donc rj (~, 17) = (}j (2- j ~, 2- j 17) ou les fonc­ tions () j (~, 17) sont portees par I~ 1 + 1171 :s 4 et verifient, ainsi que toutes leurs

~)u, VrA)(V, VrA)IVrA(X)1 2.

derivees, des estimations uniformes en j. Par hypothese r s'annule si ~ = 0 ou si 17 = 0 ou si ~

AEII

Alors eet operateur bilineaire P2 possede les propriites (1.4), (1.5), (1.6) et (1.7). .0 I 2 2 De plus P2(U, v) - PI (u, v) E B I ' si u E L et vEL. L'operateur P2 fournit donc la renormalisation la plus simple possible du produit UV, c'est-a-dire celIe ou 1'0n soustrait Ie moins de tennes possible. Cet operateur peut etre utilise pour demontrer la forme precisee du theoreme 1. Cependant eette demonstration, que 1'0n trouvera dans [3], est techniquement plus compliquee que celIe que nous venons de presenter. Les diffieultes techniques viennent du fait que 1'0perateur P2 ne verifie pas la fonnule de Leibniz qui est satisfaite par Ie produit usuel et par les operateurs P et PI. En d'autres tennes, P2 n'appartient pas la classe generale des operateurs bilineaires definis par (2.2).

a

8 APPENDICE (PREUVES DES LEMMES 2 ET 3) Reprenons, pour demontrer Ie lernme 2, les notations de la section 5. On a done 00

n

n

L(~k

I

+ 17k)17I(}?,I)(~, 17)

ou les symboles (}tl)(~, 17) sont portes par I~I leurs derivees, des estimations uniformes en j.

+ 1171 :s

00

= LT(Dju, Sjv) + LT(Sju, Djv) + L -00

T(Dju, Djv).

4 et verifient, ainsi que

Finalement on a (8.4)

rj(~' 17) =

n

4- j L

1

n

L(~k

+ 17k)17I(}tl)(rj~. rj17).

1

Pour tenniner cette analyse de rj(~' 17), on tient encore compte du support de rj(~, 17) pour ecrire rj(~' 17) = rj(~, 17)q(2-j~)p(2-j17) oil p et q sont deux fonctions de la classe V(lR.n ) de Schwartz et ou q est nul au voisinage de O. On a, en revenant al'operateur T j ,

(8.5)

j Tj(u. v) = 2- div Lj(uj, Vj)

Uj(~) = u(~)q(2-j~),

-00

= O. 11 en est de

1

OU

T(u, v) = L[T(Sj+lu, Sj+lv) - T(Sju, Sjv)]

+ 17

meme de rj et de (). On peut done ecrire . (}j(~' 17) = L

-00

.-~._'~-".='.=='-'--'-"-~"------'---""-~'-'-

que

12j

u(x)v(x) dx.

Theoreme 3. Soit VrA' A E A, une base orthonormee de L 2(W), d' ondelettes de

-00

__ C'

Nous allons, en designant par II . lila norme de l'espace de Banach

2

00

=-,",,'_C'

OPERATEURS B1LlNEAlRES ET RENORMALlSATJON

(8.2)

00

.-""'."-","","~"

Vt(2-

Alors on a [3].

P2(u, v) =

"""_=,,,"

Vj(17) = r

j

17IP(r j 17)v(17)

et ou l'operateur L j est defini par Ie symbole (}tl)(2-j~, 2- j 17). On a done, unifonnement en j, IlLj(f,

g)lh

:s CIIf112IlgI12'

160

CHAPTER 7

La preuve du lemme 2 se termine en appliquant Ie lemme 4, comme nous I'avons fait dans la section 3. n vient

(8.6)

IITj(u,

v) II

S Cllujll211vjll2

et I'on concluten observant que L lIujlj~ S Cllull~ etque L IIvjll~ S CIIvll~. Venons-en a la preuve du lemme 3. EIle repose sur 1'0bservation suivante:

Lemme 7. Soit R un nombre positij et K(x, y), x possMant les propriet~s suivantes: K(x, y)

(8.7)

=0

si

E JRIl,

Y

E JRIl,

un noyau

Ix - yl > R

pour tout Z E JRIl, on designe par B(z) la boule de JRIl X JRIl de centre (2, z) et de rayon 2R et l' on demande que (8.8)

[(

JJB(Z)

IK(x, Y)1 2 dx dy

tende vers 0 quand Izl ---+

+00.

Alors l' operateur T defini par Ie noyau K est compact. Pour demontrer ce lemme, on pose, pour tout m 2: I, Km(x, y) = K(x, y) si s met Iyl S met Km(x, y) = 0 sinon. Alors I'operateur Tm definit par ce noyau K m est un operateur de Hilbert-Schmidt. A ce titre Tm est compact. Par aiIleurs la nonne de T - Tm tend vers 0, grace ala condition (8.8), quand m tend vers l'infini. Donc T est compact. Revenons au lemme 3. n s'agit de verifier que Ixj

(8.9)

,lim (T(fj, gj), u) J-++OO

= (T(f, g), u)

pour une classe convenable de fonctions de test u. Nous supposerons, parexemple, que uappartient Ii la classe de Schwartz V(JR"). Alors (T(f, g), u) = (L(f, u), g) et Ie lemme 3 sera demontre si nous verifions que 1'0perateur defini par f Ho L(f, u) est compact. En passant aux transfonnees de Fourier, Ie noyau K(~, 1]) de cet operateur est u(-~ - 1])r(~, 1]). Ce noyau verifie evidemment (8.7). Puisque r(~, -~) = 0, on a, grace Ii (2.1), K (~, 1]) = O((l~ I + l17f)-l) quand I~ I + 1171 tend vers l'infini, ce qui entraine (8.8). Nous concluons cet appendice en reliant les algorithmes que nous avons utilises dans la section 3 au celebre theoreme T(l) de David et Journe [5]. La demonstration de ce theoreme repose sur l'existence, pour toute fonction b(x) E BM o (JRIl) , d'un op6rateur de Calder6n-Zygmund Tb tel que Tb(l) = b, (n(l) = 0 et qui soit borne sur L 2 (JRIl). Cette demiere condition fait partie de la definition des op6rateurs de Calder6n-Zygmund que ron trouvera dans [I] ou [5].

161

OptRATEURS BILlNtAlRES ET RENORMAUSATION

On a designe par (L Ie transpose de 1'0p6rateur L; Ie noyau-distribution de (L est L(y, x) si celui de Lest L(x, y). Considerons alors, en revenant aux notations de la section 3, 1'0perateur Sb defini, si b E BMO(JR"), par 00

(8.10)

Sb(f)

= L(~jb)(~j/). -00

Cet op6rateur est borne sur lorsque b appartient Ii 8M o (JRIl) , c'est un op6rateur de Calder6n-Zygmund et ron a Sb(l) = O. On a JJR' Sb(f)(X) dx = L~oo J(~jb)(~j/) dx = J bf dx puisque L~oo ~j~j = I. Cela signifie que (Sb(l) = bet que Sb est Ie transpose de l'op6rateur Tb que ron cherche Ii construire lorsqu'on demontre Ie theoreme T(I). Lorsque b(x) est une fonction de classe C 1, Ii support compact, on a j lI~j(b)lIoo S C2- j si j 2: 0 et lI~j(b)lIoo S C'2" si j S O. En fait C' = IIVrlloollulll. En outre, chaque tenne (~jb)(~j/) qui figure dans (8.10) 2 2 est un op6rateur compact de L 2 dans L 2 • 11 en resuIte que Sb: L ---+ L est compact. Puisque la nonne IISbll de Sb: L 2 ---+ L 2 est majoree par CIIbIl BM o, I'op6rateur Sb est encore compact lorsque la fonction b(x) appartient Ii V MO. Comme nous l'avions annonce dans I'introduction, Ie produit uv entre une fonction reguliere u et une fonction arbitraire vEL 2 (JRIl) n'a pas besoin d'etre renonnalise. Cette affinnation ne peut etre prise au sens strict car si P(u, v) etait nul chaque fois que u appartient Ii la classe de Schwartz V(JRIl) et que v appartient Ii L 2(JRIl), alors P(u, v) serait, par densite, nul si u et v appartiennent Ii L 2 (JRIl). Mais la regularite de u inftue sur l'importance de la renormalisation et Ie sens precis Ii donner Ii cette assertion est que 1'0p6rateur qui v associe P(u, v) est compact si u appartient Ii V MO. L 2 (JRIl)

a

Yale University University ofParis, Dauphine University ofParis, Dauphine REFERENCES [1] R. Coifman et Y. Meyer. Au dela des operateurs pseudo-dijferentiels. Asterisque 57,

2eme edition, Societe Mathematique de France (1978). [2] R. Coifman, P. L. Lions Y. Meyer, et S. Semmes. Compacite par compensation et espaces de Hardy. CRAS Paris, tome 309, serie I (1989), 945-949. [3] S. Dobyinsky. These du CEREMADE. [4] C. Fefferman and E. Stein. HP spaces of several variables. Acta Math. 129 (1972), 137-193. [5] Y. Meyer. Ondelettes et operateurs. Tomes I, net III. Hermann, 1990.

163

NUMERICAL HARMONIC ANALYSIS

8

Numerical Harmonic Analysis R. R. Coifman, Y Meyer, and V Wickerhauser

INTRODUCTION The purpose of this chapter is to describe recent developments involving the nu­ merical implementation of methods from classical harmonic analysis in signal processing and computational P.D.E. As an example, Littlewood-Paley theory, in which a function or a Fourier multi­ plier is analyzed by partitioning the frequency space in dyadic blocks, has recently been translated into a powerful numerical tool through expansions in orthonormal wavelet bases. (See [1], [2].) In this numerical setting one sees a general Calder6n-Zygmund operator or IJID.O. as given by an "almost" diagonal matrix having a simple analysis and being implementable by fast numerical algorithms (Le., algorithms of complexity eN log N, N = number of discretization points). Pseudo-differential calculus is translated into an efficient numerical calculus in which smoothing operators are represented by "small" matrices of low numerical rank (see [1]) permitting its use in explicit calculations of solutions to P.O.£. In particular, we can obtain a fast algorithm for the numerical computation of the Green's function for a variable coefficient Laplacian (with smooth coefficients). In this exercise of translation of methods and ideas from harmonic analysis into fast computational algorithms, one soon realizes that the ability to imple­ ment efficiently an integral operator applied to a function is equivalent to a good understanding of the interaction between geometry of the underlying space and cancellation properties of the operator. In the particular case ofCalder6n-Zygmund operators we see an efficient computational algorithm as being a translation of the method of proof of the T(l) theorem of David and Joume. For the case of fractional integrals and operators of potential theory, the need to come up with

efficient computations has led V. Rokhlin to the independent discovery of various versions of Calder6n-Zygmund theory as embodied in his multipole algorithms. As it turns out, in this case, the question of fast computation is more elementary than boundedness on L 2 or other spaces. It leads directly to issues of geometry of interactions and cancellations. This interaction between harmonic analysis and a number of concrete problems in applications, such as signal processing and computations, has opened a number of new fundamental questions in analysis. Our goal is to describe some of these problems on a few simple examples. We start with a fundamental question of signal processing, the question of compression of a signal. Stated simply, given a function (or more precisely, a vector which is a sampled function) one would like to represent the function with as few parameters as possible (here a representation is always assumed to have a given fixed preci­ sion). Such a representation could be given in terms of expansion coefficients, Fourier, Taylor, etc., or by stating that the function solves an equation which is easy to describe (say by giving coefficients of a differential equation). The ability to represent a function simply with few parameters is not only desirable in appli­ cations for storage purposes, it is also a test of our understanding of the structure of the function and its numerical complexity. Traditionally, the first attempt to represent a signal (or a function not described analytically) would be to expand the signal in a Fourier series, or in terms of some other orthogonal (or non orthogonal expansion). This leads to a variety of problems familiar to all analysts. Assume that a smooth function is supported on a number of disjoint intervals. It is "clear" that separate Fourier expansions restricted to these intervals will be much more "efficient" than a single expansion on the union. The actual answers are not so obvious since some intervals could be close to each other and the term efficient has not been defined. We see that we are confronted with the issue of selecting an optimal expansion inside a class of possible expansions. This leads naturally to the concept of a library of orthonormal bases, as well as to precise definitions of efficiency of an expansion.

DEFINITION OF MODULATED WAVE FORM LIBRARIES We start by observing that it is impossible to construct an orthogonal basis by localizing smoothly eikx . This is clear for the case of two adjacent win­ dows WI (x), W2(X) since the requirement of orthogonality between WI (x)e ikx and W2 (x )eijx implies that

f

WI (x)eikxw2(x)ei(k-j)xdx

= 0

164

CHAPTER 8

°

which implies WI (x )W2 (X) == (if it is supported in an interval of length smaller than 2:rr). Recently Daubechies, Jaffard, and Journe, as well as Malvar, observed that by taking equal windows and sines or cosines orthogonality can be maintained. It was observed in [3], [5] that the windows can be chosen to different sizes enabling adaptive constructions. (See Figures 5, 6.) We start by defining this library of trigonometric waveforms. These are localized sine transforms associated to covering by intervals of R (more generally, of a manifold). WeconsideracoverR = U~ooI;, 1= [Cli, Cl;+d,Cl; < Cl;+), writei; = Cl;+l­ Cl; = 11;\ and let p;(x) be a window function supported in [Cl; - ii-II2, Cl;+l + i;+1/2] such that

165

NUMERICAL HARMONIC ANALYSIS

o

-1

-1

00

L p7(x) = 1 -00

-1

-1

and pt(x)

=

I - pt(2a;+l - x)

x near Cl;+l

for

then the functions Sik(X)

Figure 1.

=

2

r,:;-;;-:- p;(x) sin[(2k

'v2ii

+

1) -

:rr

2l;

(x - Cl;)]

form an orthonormal basis of L 2(R) subordinate to the partition p;. The collection of such bases forms a library of orthonormal bases. It is easy to check that if HI, denotes the space of functions spanned by S;,k, k = 0, 1,2, ... then H li + Hli+' is spanned by the functions 2

P(x)

J2(i;

+ ii+l)

sin[(2k

+

1)

:rr 2(i;

+ ii+d

RELATION TO WAVELETS-WAVELET PACKETS WeconsiderthefrequencylineRsplitasR+ = (0, oo)unionR- = (-00,0). On L 2(0, 00) we introduce a window function p(~) such that L~-oo p2(2-k~) = I and p(~) is supported in (3/4,3). Clearly we can view p(2-k~) as a window function above the interval (2 k , 2k+l) and observe that

form 2

= p7(x)

+

1 (~~ - 2 ) ] p(Tk~) = Sk,j + 2'):rr k

sin (j [

(x - Cl;)]

where p

Local trigonometric waveforms.

P7+1 (x)

is a "window" function covering the interval I; U /;+1. This fundamental identity permits the useful implementation of the adapted window algorithm described in Figure l. (Other possible libraries can be constructed. The space of frequencies can be decomposed into pairs of symmetric windows around the origin, on which a smooth partition of unity is constructed. Higher dimensional libraries can also be easily constructed-as well as libraries on manifolds-leading to new and direct analysis methods for linear transformations.)

an orthonormal basis of L 2(R+).

Similarly,

(~~ - 2 ) ] p(Tk~) k

Ck.j = cos

[

(j

+

I 2'):rr

gives another basis. If we define Sk,j as an odd extension to R of Sk,j and Ck,j as an even extension, we find Sk,jJ..Ck',jl permitting us to write Ck,j ± iSk,j = e±;j1r~/2k 'if;(~ /2 j ) where 'if;(~) = e;1r/~ p(~) is the Fourier transform of the base wavelet \11 (see [4]). We therefore see that wavelet analysis corresponds to windowing frequency space in "octave" windows (2 k , 2k+1).

=~_='~=_~c

. _ . _ ~ ~ ~ ~ ~ ~ : ~ ~

166

CHAPTER 8

167

NUMERICAL HARMONIC ANALYSIS

15

1

5 -1

~o

1

-1

1

-1

I

·.1 Figure 2.

Local trigonometric wavefonns. Figure 3a.

A natural extension, therefore, is provided by allowing all dyadic windows in frequency space and adapted window choice. This sort of analysis is "equivalent" to wavelet packet analysis. The wavelet packet analysis algorithms permit us to perform an adapted Fourier windowing directly in the time domain by successive filtering of a function into different regions in frequency. The dual version of the window selection provides an adapted subband coding algorithm. This new library of orthonormal bases constructed in the time domain is called the wavelet packet library. This library contains the wavelet basis, Walsh functions, and smooth versions of Walsh functions called wavelet packets. (See Figure 7.) We'll use the notation and terminology of [5], whose results we shall assume. We are given an exact quadrature mirror filter h (n) satisfying the conditions of Theorem (3.6) in [5], p. 964, Le.,

L h(n -

2k)h(n - 2£) = lh,e.

Lh(n) =

n

We let gk (1.0)

Wavelet packet library.

= 2 L Skgk-2j·

Fdsd(i)

The map F(Sk) (1.1)

= FO(Sk)

EB F 1 (Sk)

E

£2(2Z) EB £2(2Z) is orthogonal and

F;Fo + FtFI = I.

h.

n

= h l- k(-ll and define the operations F; on £2(Z) into "£2(2Z)" Fo{sd(i) = 2

L Sk h k-2i

Figure 3b.

Schematic Description.

__

. . .,._u

_

__".,_,._.

... ~._,

.'_._,-

_.

..'_.

----

-

-".

168

"~·~jiy.'jf~~~"'~5..7atj~'AW;~f'...""·:;"',n·7't'~;''-._~;;,:,,,,,":,:~n~!;;l>,~'' 'ii';.';__;~!:;~\1;"':{i,"f,~'r"":·:'-~'"'·"·"'·\'~~,1·'(>:"o.;·'·';

CHAPTER 8

We now define the following sequence of functions.

_·"" ..1"",·.·"''''>·.;,...

,.":'=,'',·,:·,'-,,,...·.~·. _.~",''.,,·""~""",,,,"",", ..-~.".~,.~,.,~,=.,;,"~

169

NUMERICAL HARMONIC ANALYSIS

Proposition. Any collection ofindices (i, n) such that the intervals [2 1 n, 21 n + 1) form a disjoint cover of [0, (0) gives rise to an orthonormal basis of L 2.1

W2n(X) = hL:hkWn(2x - k) (1.2)

{

W2n+I(X) = hLgkWn(2X - k).

Clearly the function Wo(x) can be identified with the scaling function WI with the basic wavelet 1/1. L hke-ik~ and Let us define mo(~) =

f{J

in D and

h

ml (~)

= _ei~ mo(~ + Jr) = - 1 v'2

L

(These intervals correspond to the partition of frequency space alluded to in § 1.) Motivated by ideas from signal processing and communication theory we were led to measure the "distance" between a basis and a function in terms of the Shannon entropy of the expansion. More generally, let H be a Hilbert space. Let v E H, Ilvll = 1 and assume

gkeik~.

H = ffi

L Hi

an orthogonal direct sum. We define (Fo, FI) is

Remark. The quadrature mirror condition on the operation F equivalent to the unitarity of the matrix

M _ [mo(~) mo(~

-

ml(~)]

ml(~

+ Jr)

+ Jr)

2

e (v, {Hi}) = -

L IIvill enllvdl 2

2

as a measure of distance between v and the orthogonal decomposition. e2 is characterized by the Shannon equation which is a version of Pythagoras' theorem. Let

.

Taking the Fourier transform of (1.2) when n = 0 we get

H = ffi

Wo(~) = mo(~ /2) Wo(~ /2)

(L Hi) ffi (L H

j )

= H+ ffi H_.

i.e.,

n 00

Wo(~)

=

Hi and H j give orthogonal decompositions H+ =

mo(~ /2 j )

j=1

L

Hi, H_ =

L

H j • Then

e2 (v; {Hi, H j }) = e2 (v, {H+, H_})

and A

WI(~)

= ml(~/2)Wo(~/2) = ml(~/2)mo(~/4)mo(~/2 A

3

) ...

More generally, the relations (1.2) are equivalent to

n 00

Wn (~)

(1.3)

=

mej

(~/2j)

j=1

,",00

0-1

and n = L.j=1 e j 2J (ej = 0 or 1). The functions Wn(x - k) form an orthonormal basis of L 2 (R 1 ). We define a

library of wavelet packets to be the collection of functions of the form Wn (21 X - k) where e, k E Z, n E N. Here, each element of the library is determined by a scaling parameter i, a localization parameter k and an oscillation parameter n. (The function Wn (2 1x - k) is roughly centered at 2- 1 k, has support of size ~ 2-1 and oscillates ~ n times.) We have the following simple characterization of subsets forming orthonormal bases.

+ IIv+1I

2 2

e (11::11 ,{Hi})

2 2 +lIv-lIe

(

v_

IIv_II,{H j

)

}.

This is Shannon's equation for entropy (ifwe interpret II PH+ v 11 2 as the "probability" of v to be in the subspace H+, as in quantum mechanics). This equation enables us to search for a smallest entropy space decomposition of a given vector.

In fact, for the example of the first library restricted to covering by dyadic intervals we can start by calculating the entropy of an expansion relative to a local trigonometric basis for intervals of length one, then compare the entropy of an adjacent pair of intervals to the entropy of an expansion on their union. Pick the I We can Ibink of Ibis cover as an even covering of frequency space by windows roughly localized over Ibe corresponding intervalso

170

CHAPTER 8

171

NUMERICAL HARMONIC ANALYSIS

::.,'"? ,§'-:.":'..:<',

,k,··,·,,·;,·,·, •.•.,.·,,·;··,·,,·...·,.·;·,,·.

~::>., '~,.:~'

.

......

,,,.,,,.,,,·· __ .·v·.,·,·.,·····

'I,

!t""-----~-..~'~lMfYI~-......,..............--~-

----~--..........- .............~.....--.~~~,~~"I,.,.,~~

~: :;'1

~J..~_ _,.,.,.,.,.,...,.,...".._ _ . ~'AA:I:

.~

_

...

"''''

''''''''''''~'''''"''''''''''''''''''''''''''''''_ _ ''''·N'''''''''''''''''''·''W._''''_-:-''''''''''''·''~''

~~lm,,"'X;;';KiXo':r~.W<'#~~;:~-'~"f:i'i~i·_~-~~'~"" ·~·,%:···.~~~i-;::;.x.;olAA~~N*'~~"l<W~'''~l;$}'~'~~<>;;

Figure 4.

Wavelet coefficients of a function.

expansion of minimal entropy and continue until a minimum entropy expansion is achieved. (See Figure 4.) Of course, while entropy is a good measure of concentration or efficiency of an expansion, various other information cost functions are possible, permitting discrimination and choice among various expansions.

-i==:~:~~~;;:=~:;~~~:::~":'~~';:=:~=;=,;:~~;;;-;:;:~~;:

:::::~;;:~~:~;:~~:~:?@~:~-$J

Figure 5. A best-basis wavelet packet analysis. This analysis corresponds to selection of windows in frequency space, to minimize the entropy of the expansion.

We illustrate these points, as well as the effect of various analysis methods, in figures 5-7 in which the vertical axis represents the frequency axis and the horizontal is the time (or space) axis. The signals have 512 samples (and are wrapped around). Each rectangular box in this phase space corresponds to a coefficient obtained by correlating the signal with an element of the wavelet packet

172

:~~:~:~;,.:\\y.:~.t&·~-::::;.\· . -·~·~ ·"~/·;;:~:t.~):~,:,,

173

NUMERICAL HARMONIC ANALYSIS

CHAPTER 8

:L/;;::,. ~:,:. .~•.>.:.•:J.i.~. .

,.\>;,;<

Q>.::::::::.;'-;:,::.

:"};':",::;:--:,:/::'/~ ::;:,<:,;~

,:;:~:~:J;~~f.;.~,;~.:~.}:~ ........... "" .

:~c

...... ' '...- ........

:;,.·~~,::G:;;:;;;

........',1'

,,,

.:::.. ,:::: ;,::;,;:;;... :,;,

/

:~,

: /

•

........ ".-..

" .

.-"."

.;" ;;:;:":.. ;:,;;,~ ..'''''.''.

:,. ~~

~

~

::: .~

J :~

;~

~

t

~~

g

--; -~

:'<

; :'.'

t:

f

......

:~~:

~~;:.~:.:~~:".::.~.

::~:'~~':

:

<:. ':?:,::t\x:"; «< :';:-;"-:'; ~ ~.':~., ';:'(::"< :, :-:',:';

,

.. .

·

·

.

,I

f

~~

i!

t F

.,

f

:~

'j

W/tJh\\/i~'rl~~1

-.

's

.........---.....,.,...--..--.~~-.-JM~fJ'~~~[

;~

~:

if

;- ~~

~

it ,>; ij

Figure 6.

A two-windows expansion with no adaptation.

library whose time support lies "below" the box and whose frequency support is in the projection of the box on the vertical axis. Each box has area 512 pixels (Le., a cover of the discrete phase plane has 512 elements). The compression rate can be computed as the ratio of the visible gray area to the total area of the box (Le., the relative number of visible boxes). Of course we can try to characterize classes of functions which are well com­ pressible, i.e., for which we can estimate the number of coefficients needed for representing the function with a prescribed accuracy. Smooth functions are ob-

_ _ _ _ _. _ - .

Figure 7.

_.

- - . . . " ' ._ _..

~_,

."'O"""",,..<.............__ ,y... _.<.,..,'.... i<...

,·.~.'

A best-level expansion in which a fixed window size is chosen to minimize entropy.

vious candidates as well as functions which can be well approximated locally by trigonometric polynomials of short length. Various obvious definitions come to mind. At the moment it would seem that experiment will provide a better guide. The procedure for signal analysis described above is very similar to the usual methods of studying Fourier multipliers in which we break the multiplier by an appropriate partition of unity to simpler components whose spatial localization and structure are easier to understand. We can describe a similar procedure for

174

CHAPTER 8

integral operators T(f) =

L

k(x, y)f(y)dy

which are not necessarily convolutions. Our goal is to implement a fast discrete version of the operator. A procedure that is equivalent to PI QI decompositions of Calder6n-Zygmund operators (see [1]) can be obtained by trying to compress the k(x, y) viewed as an image, i.e., k(x, y) represents light intensity at pixel (xy). Here again the analysis consists in finding an 'optimal windowed expansion for k(x, y) (or k(t 1])) by selecting that combination of windows most efficient in capturing the kernel. Since the kernel is represented as a sum of products of functions of x and y it is easy to convert an efficient two-dimensional representation into a corresponding efficient computation. Observe also that each box selected represents an interaction between two windows on the line. It can be proved that for k(x, y) a single or double layer potential for Helmholtz on a curve or surface (with bounded curvature), this procedure leads to an order pN log N algorithm, where p is the number of decimals desired, and N is the number of discretization points (~ number of wave lengths on the surface). For more general curves or surfaces one has to develop specific, highly oscilla­ tory analogues to multipoles. (Smooth bases are useless.) This has been done by Rokhlin with a resulting description of local oscillatory interactions. Yale University University ofParis, Dauphine Washington University, St. Louis

REFERENCES Software for adapted waveform analysis is available by anonymous ftpfromceres.math.yale. edu, Internet address 130.132.23.22. [1] R. Coifrnan. Adapted multiresolution analysis. computation, signal processing and operator theory. ICM 90 (Kyoto). [2] R. Coifrnan, Y. Meyer, and V. Wickerhauser. Wavelet Analysis and Signal Processing. Proceedings of the Conference on Wavelets, Lowell, Mass. 1991. [3] R. Coifman and Y. Meyer. Remarques sur l'analyse de Fourier afenerre, serle I C. R. Acad. Sci. Paris 312 (1991), 259-261. [4] Y. Meyer. Principes d'incertidudes, bases hilbertiennes, et algebres d'Operateurs. Seminaire Bourbaki, 1985-86. Asterisque, 662. [5] I. Daubechies. Orthonormal bases ofcompactly supported wavelets. Comm. Pure Appl. Math. XLI (1988), 909-996. [6] E. Laeng. Une base orthonormale de L 2(R), dont les elements sont bien localises dans l' espace de phase et leurs supports adaptes a toute partition symetrique de l' espace desfrequences. serle I C. R. Acad. Sci. Paris 311 (1990), 677--680.

9

Some Topics from Harmonic Analysis and Partial Differential Equations Robert A. Fefferman

It is a pleasure to celebrate, together with many colleagues, the occasion of the sixtieth birthday of Elias M. Stein. To me, Professor Stein is not only an extremely great mathematician, but he is also my teacher and friend. He gives to each of his students his time and his energy very generously. Most of all, he provides us with such an inspiring example in the way that he does mathematics. I think this is the greatest gift that a teacher can give to a student, and I shall never forget this gift. lit this article, I would like to discuss some of the mathematical issues which have interested me and have guided much of my work. They are in large part taken directly from E. M. Stein's celebrated book, Singular Integrals andDifferentiability Properties ofFunctions [27] and will be motivated by the statement of three "model problems." These can be stated as follows:

Problem 1. If Q ~

jRn

is a "nice" region, and 1 < p <

00,

define TQ by

jfJ(~) = XQ(~)?<~)' When is it true that IITQfllu(lRn) .::: C p ,nllfllU(lRn ) where Cp,n is a constant depending only on p and n, but not on f?

Problem 2. Understand "Marcinkiewicz multipliers." By a Marcinkiewicz mul­ tiplier m(~, 1]), ~ E jRn, 1] E JRm, we shall mean that m(~, 1]) satisfies all the same estimates that ml (~)m2(1]) does, where ml and m2 are in some appropriate set of "classical" mUltipliers (such as Honnander multipliers). We shall not assume that m(~, 1]) takes the special fonn ml (~)m2(1]). This problem was first investigated by

~~{~"a-~~~_~~~~~~~~~~?i'.~"'~";"'='--~-~'''=~~~~~-'~;j~¥.'!2!\1j,'if~':_'::'~''.;;~-

176

CHAPTER 9

Marcinkiewicz (see Stein [27]) who obtained LP estimates for these multipliers. when I < p < 00. For many reasons it is natural to suspect that for the multiplier operators T corresponding to the Marcinkiewicz multipliers m, we should have estimates near L I of the form m{lT(f)1 > a} ~

C

-llfIILJog+ L(IR"xlR a

m)

and these estimates are not available by the (now standard) methods presented in [I] for the LP theory. In this article we shall be interested in the recent work in harmonic analysis which allows us to obtain finer estimates on these operators near Ll, Problem 3. It is a very important fact from the classical theory ([27]) that if B ~ IR" is the unit ball and we are given a function f(x) defined on S"-I which belongs to LP(da) (l < p < 00. and da is the surface area measure). then there exists a unique harmonic function u on B taking on the values f (x) as non-tangential boundary values and such that u* E LP(da), where u* denotes the non-tangential maximal function of u. Our third problem is to extend this classical theorem about harmonic functions to the solutions of a class of elliptic operators with bounded measurable coefficients. The solution to this problem provides at once a generalization of the theory of harmonic functions on Lipschitz domains. and also some interesting harmonic analysis involving classical weights and the Helson-Szego theorem.

Let M 9 denote the maximal operator defined by M9!(x) =

Theorem A (Charles Fefferman [14]). If S is a "nice region" in 1R 2 whose bound­ ary has a point ofnon-zero curvature then XS is not an LP (1R 2 ) multiplier for any p =f=. 2: Theorem B (A. Cordoba-R. Fefferman [8]). If S is a region whose bound­ ary consists of line segments, then the boundedness properties of the multiplier Xs are equivalent to the boundedness properties of a certain maximal operator corresponding to S. In Theorem B above, the maximal operator corresponding to the multiplier XS is very easily described. Suppose that B = {Bj }j=1,2.... denotes the sequence of directions of the normal vectors to the sides of the (infinite) polygonal region S.

sup xER RERn

(I )

m R

l

lf (y)! dy

R

n

where 9 is the family of all rectangles in 1R2 whose side lengths are arbitrary, but whose orientations are restricted to belong to one of the directions Bj E B. Then the behavior of the multiplier operator corresponding to Xs on LP is equivalent to the boundedness properties of M9 on the space L (p/2j'. For this reason in order to answer Problem 1. it is sufficient to gain an under­ standing of the operators M9 • To do this. we shall discuss a systematic approach which can often be applied to determine the action on (Orlicz) LP classes of vari­ ous maximal operators. We emphasize that just as in the case of M9 , the maximal operators we deal with are defined by a supremum over averages with respect to a family of sets whose geometry may be considerably more complicated than in the classical case of balls or cubes. Our method proceeds as follows. Suppose we are given a family of sets in IR" and we wish to analyze the operator M defined by

n

Mf(x) = sup _1_ ( If(y)ldy XER meR) JR RE'R

and obtain (for instance) the estimate mIx

Let us begin to consider each of these problems in some detail. Starting with the first problem, we quote two results:

177

SOME TOPICS FROM HARMONIC ANALYSIS

E

I

Q M(f)(x) > a} ~

C -lIf11LO>(Q)

a

(where Q is the unit cube and L denotes an Orlicz class corresponding to the convex increasing function <1». Then we shall describe a method which often makes it possible to prove a lemma along the following lines: Given a family of sets {R j }, Ri E Ri ~ Q. there exists a subfamily Rj such that (a) II L X'R;lIv"(Q) ~ C; and (b) m(U Ri ) ~ cm(U Rj ) for some pair of constants c, C > O. Here L 1jI is the Orlicz class which is dual to the class L which occurs in the estimates on M which we are trying to prove. That this covering lemma is sufficient to prove the weak type estimate of M on L above is rather trivial. In fact,settingE a = {x E QIM(f)(x) > a}, for each x E Ea.chooseanR x En so that m(k,) x !!(y)ldy > a. Now apply the covering lemma to the collection {R x }xEEa to get {R j } satisfying (a) and (b) of the lemma. It follows that

n,

JR

m(Ea )

~ (U RX) ~ ~ m (U R m

j)

XEE.

1

<

~

Lm(Rd j

iiWiiiiiiW;-----­

-----------

~)iiiiIiI---------------

178

::s -1

c

L

-1

i

Cl

1

1

C

1

--------·----------··i5iii:-------------;;iiiiiiirlzi------~---ilii~

1

_ If(y)ldy

= -1

R;

CCl

CHAPTER 9

1

If I LXii, dy

Q

~ -. c -lIf11L4>(Q)II"XR:IIL"'(Q) a ~ , ~

-c . Cl-llfIIL4>(Q)'

and this is the estimate on m(Ea ) we require. A similar argument shows immediately that if we desire to prove that M is of weak type (p, p) for some p with 1 < p < 00, then it is enough to prove a covering lemma 'exactly as above (where M is mapping into weak L I ) except that we must replace (a) by the estimate

IILXIi,IIU'(Rn)::S Cm (URi)'!P' To show how to prove covering lemmas for various types of sets, we present some concrete examples. The first of these involves proving the sharp covering lemma for rectangles in IRn with sides parallel to the coordinate axes. This covering lemma is exactly what is needed to prove the celebrated lessen-Marcinkiewicz­ Zygmund Strong Maximal Theorem [22]. m{x E Q Mn(f)(x) > a} ~ ~ IIfIlL(log+ L)n-l (Q) when M n is the strong maximal operator in IRn defined by

I

Mnf(x)

= sup XER

_1 [If(y)ldy,

IRI

JR

the sup being taken over all rectangles in IRn with sides parallel to the axes. We have the following:

Covering Lemma for the Strong Maximal Function [7]. Suppose {Rdi=I.2, .., is a given sequence of rectangles with sides parallel to the axes, wit.!! R i ~ Q, where Q is the unit cube in IRn . Then we may select a subsequence {R i }from the R i so that (a) II L X~ lIexp(L),/n-1 ::s Cn; and (b) m(U R i ) ::: cnm(U Ri) and where cn , Cn are positive constants depending only on the dimension n. Proof We order the rectangles R i so that their

X n side length decreases as n increases. Select the Ri according to the following rule: Set RI = R I. Assuming we have chosen R I , R2' ... , Rk-I, go down the list of R i following Rk-I, until we first have a rectangle R such that

m (R

n

[g

Ri]) /

meR) <

~.

,-

-----"--

~.;,.,·~~~!iT~i.ii~~u,:;,.·"iW'":';;:n·_·~-~··~·-~-iiiii!ii~_~~~-~~~~~

179

SOME TOPICS FROM HARMONtC ANALYStS

This rectangle R then becomes Rk • In this way we continue to obtain the selected rectangles Ri , i = 1,2,3, .... We now claim that the Ri have properties (a) and (b) as stated above. To prove estimate (a) we use duality, and fix rp ::: 0 such that II rpllL(1og+ L)n-I (Q) = 1 ~d estimate J L Xli, rp dx. To do this, set E i = R i - Uj <j R j , the disjoint part of R j • Then

f

LXli,rpdx = L i

j

(rpdx

hi

::s 2 Lm(Ei)~ m(Rj )

j

(rpdx.

hi

!

(Recall that the selection process assumes that m(Ej ) ::: meR;), and we have used this in our last inequality.) It is also clear that for each x E Ej , ----Lrpdx ::s Mn(rp)(x) so that m(R;) ,

h

L m(Ej )

[~ (rp dX] ::s [ meR,) hi J

Mn(rp)(x) dx.

Q

j

This estimate is not useful, because, first, if rp E L(log+ Ly-l, then JQ Mn(f)dx may diverge, and second, we are trying to prove a result concerning the operator M n • Both of these difficulties can be avoided by using the geometry of the rectan­ gles Rj • In the argument that follows, the R i will be assumed to be dyadic. The general case requires only minor modifications which a~ best left to the reader. The key point is that since the X n side lengths of the R j are decreasing, if we slice these rectangles with a hlperplane perpendicular to the X n axis to obtain n - I-dimensional rectangles T;, then

m. (

R, n [~RJ])

1 ­ < -m 2 n (R,) I

implies

m.-l

(T, n

[Wi])

<

~

m._l

(T,)

(The reason for this is thatmn_1 (T; n [U j
f

LXT,:rpdxI ... dXn-l

::s

2

f

Mn-1(rp)dxl ... dXn-l,

-'-.,,----_---

w

-,-

180

•••

CHAPTER 9

and integrating this in the X n variable yields

f

L

X'R;rp dx S 2

f

M n- I (rp) dx

_

••••, :

:

, _ " - " __ •

_ _ ••••, - : : , , : ' -

:.=::::';:':::::-===--=':::':::::::=-':""--=:::::,:::=:::::::='=:"'-=:::'::::==':.'=:==--==_':'=:::=:==;::5=-~"'-

=

Mlac(f)(x)

According to the theory of the strong maximal function M n - I (which could be assumed by induction), if IIrpIlL(log+ L)"-'(Q) = 1, then Q Mn-I(rp) dx S Cn and so this proves part (a) of the covering lemma. To prove part (b), suppose R is one of the R i which have not been selected. Then by the definitiol} of the selection criterion it follows that

1

URil)

>

2I meR)

where, in U iii, the union is taken only over those iii which precede R on the list of rectangles. Now, slice the rectangle R as above with a hyperplane perpendicular to the X n direction, calling slices of R an,.,? iii, T and T;, respectively. Then just as above, since the X n side length of each R i preceding R exceeds the X n side length of R, we have

mn-I (T

­ n [U Ii))

>

21 mn-I (T).

Mn-I(Xu'R;) >

2'

sup

k-I

By the theory of the n - I-dimensional strong maximal operator, M n -

SCm

which is part (b) of our covering lemma.

(I )

m R

r If(y)ldy.

JR

(1) II LXi lIiz S Cm(U iii) (2) m(U Ri ) ::: cm(U R i ),

(*)

1 {Mn-I(Xu'R;) > 2}'

(U Ri)

181

Then the first estimates for this operator were obtained using the method described above for the strong maximal operator. They were obtained by J. O. Stromberg [28], and involved weak type estimates on Mlac(f), for f belonging to a certain Orlicz class near L 2 • We shall present here an extension of the results in [28] where we obtain weak type estimates on Mlac(f) for f E L 2 (A. Cordoba-R. Fefferman [9)), and where the argument is considerably simpler than that in [28]. We wish to show that m{Mlac(f) > a} S ~ Ilflliz(R.z) for a > O. This will be done by proving that given a sequence of rectangles R i E 'lilac, there exists a subsequence iii satisfying

or equivalently that

m

...::'.

where c and C are positive constants. In order to do this, assume that the R i are arranged so that their longest side length is decreasing. We select the iii according to a slightly different rule from that given by rectangles with sides parallel to the axes: Choose iiI = R I and assume iii, ... , iik - I have already been chosen. We continue down the list of Ri until we first come upon a rectangle R so that

This clearly implies that on R we have

URi ~

__~._

oriented in anyone of a sequence of "lacunary directions".) Define

(dx is n dimensional measure here).

n[

'::_

SOME TOPICS FROM HARMONIC ANALYSIS

XER RE'R,oc

meR

.-._' -

L

meR

n iii)

i=1

I,

this gives

(U iii)

We then set R = iik • Now, we claim that the {iii} satisfy properties (1) and (2). To prove (1), we write

•

The moral of the story in the first example of our technique for proving covering lemmas is that by using the geometry of the relevant family of sets involved, we have controlled the maximal operator at hand (in this case M n ) by a simpler operator (Mn -I). This method will be employed in the next example ofthe lacunary directions maximal operator Mlac defined for functions in the plane. Suppose that Rlac denotes the family of all rectangles in ]R2 whose longest side makes an angle of2- k with the positive x-direction, for some positive integer k, but whose side lengths are arbitrary. (In other words R 1ac is made up of all rectangles

1 S - meR). 2

II LX'R;lIiz

=

i

IR

Z

LX'R;XRj

= 2 Lm(iii n iij ) + Lm(iii).

i,j

i<j

i

From the selection rule (*), we see that for each fixed j, j-I

2L

m(iii

n iij )

S m(iij

)

i=1

so that 2L i<j

m(iii

n iij )

S L j

m(iij ) S 2m(U iij ).

-

.,.. - ,._.

....

182

CHAPTER 9

M1

This gives

II

L

Xli; IIl2 S

4m(U R

j )

and establishes (1). To prove (2) we must use the geometry of the rectangles involved, in a way quite analogous to the previous example. Let R be a rectangle which is one of the R j that was not selected. Let S denote the smallest rectangle with sides parallel to the axes such that S contains R. Finally suppose Sdenotes the concentric double of S. Then, since R was not selected, meR) _

L

meR

n

R

~

>

j)

R before R

2'

-

i

denotes the two-dimensional strong maximal operator with respect to· where a pair of ~es oriented along the sides of R and where R) denotes the collection of those R j oriented in the same way as R. It follows that

URi

~ {M2(LXR;) 4~ }uulM1 (_L,XR;):::~} >

n

Ri )


meR)

m(S

so that

m(URi)SCII~XR;112 +~mIM1(_L,XR;):::~}

all,

L2

_L

n Rj)

meR

11."

[

m(S)

for all Ri whose longest side length exceeds that of R, and whose orientation is different from that of R. Now, there are two cases: 1 Case 1. meR)

L XR;

S C

n Ri ) A

-

:::

_ R;

L

meR

n

R i ) :::

1/4.

before R

II

In Case 1, from our geometric observation, it follows that m(S) i

(

> ­ 1

4C'

,) -

so that on R,

M2(L XR;) ::: 1/4C. In Case 2,

M1 (_L, XR;) R;E'R. J

>

~

2 ]

.

L'

L Xli; IIi2 S Cm(U R

i)

Xli;

2

S

Cm

L2

(_U, Ri) . R,E'R.J

Summing this on j gives

j

L m ~S n R·­

L L XR;

+

L II L

R, and R have same orientation

1 ------:x-

L ~,~ 2

j

R; E'R. J

meR)

R,E'R.J

Again, by the selection rule of the R j , and our estimates to show (1),

_L

from R

Case 2.

]

and for each j,

1/4

_ R; before R R, oriented differently

1

R; E'R. J

]

Very simple geometry shows that meR

183

SOME TOPICS FROM HARMONIC ANALYSIS

R;

Xli;lIi2 S Cm(U

E'R.J

Ri ),

all i

and this proves (2) and the desired covering lemma. We should point out that in [26], Nagel, Stein, and Wainger were able to extend our results on L 2 for I < p < 00, and their methods are entirely different. It would be quite interesting to obtain, by a covering lemma argument, the boundedness of the M1ac on the full range of LP spaces. The reader will notice that in the preceding two examples of covering lemmas, the rule for selecting rectangles in the sparse subcollection changed. In the case of the strong maximal operator, Mn , we chose a rectangle R if and only if meR

n[

U-

1

RiD S -meR).

i
2

.::_

__ •. :=',,",

""_= __ . ; ;,; ; .

."

.~.";;

.....::",,"'. .......

. ..__

- _" . ::-..

184

_.,.;;r"~~"=:::!~"'~£:"'~'~~~-''7",",~,~~"?

CHAPTER 9

In the second example of Mlac, a rectangle was selected if and only if "~ m(R

n

QIMzf(p) > a} ~

JR

(2) m(U Ri )

:::

2

m2(S) Js

Si of type I,and similarly for Ll/)

C

_ 1 ( [e EI t:;, _ 1] [e Ell t:;, - 1]

-lIf11L(log+L)(Q)'

m2(S) Js

a

JR([e E,=1 X'::"] I -

1] > ­ 1

-

S

(here Ll indicates that we sum over only the

To do this we order the R i so that the z side lengths of the R i are de~reasing. (We shall also assume in what follows that the Ri are dyadic.) Then set Rl = R h and, assuming that iii, i < k have been chosen, choose the first rectangle R on the list such that Ri

[e EX"::' 5i

z side lengths of the iii before

where again the sum only extends over those i for which Ri comes before R on our list of Ri • It is at this point that we make use of the fact that all of the rectangles in question (in JR.3) have side lengths of the form s, t, and 1/1 (s, t). It follows from this that each of the Si in the sum ~ove has eitherjts x side length greater than the x side length of S (call such an Si of type I) or Si has its y side length greater than the y side of S (type II). Then

1 + __ m2(S)

dx ~

(e E/X"::' 5i

1)

-

+ -1-

s

m2(S)

1[

i

R

[e EX':Ri:"

-

1

m2(S)

1

e EI X"::' 5i

({ [ EI X"::' JJ s e 5i

1

= mj(K)

I] > ­ I

2'

where the sum only extends over those iii before R on the list. Slice R and iii with a plane perpendicular to the z axis, and call the two-dimensional slice of R

s

(e EllX"::' 51

-

I).

-

1] [ eEll X"::'5i

-

1] dxdy > 1/6,

s

] [

1

-

e

JK( [ e E, X"::' 51

Ell X"::' 5i

]

I dxdy

-

1

]

-

1 dy ml(J)

f[

e

Ell X"::'] 5i

aIli

-Im(R)

1

since eEl t:;, - 1 is a function of the y variable only considered as a function on S, and eEllt:;, - 1 is a function of x only, we have (writing S = J x K)

2:'

It is a simple matter to verify that e E,X;; - 1 E L1(Q) and lI/ X;; - Illv(Q) ~ Cm (U iii) for some constant C. We need to verify now that m (U Ri) < Cm(U iii). To do this, suppose R is an unselected rectangle so that

1

Since the sum above exceeds 1/2, at least one of the terms must exceed 1/6. In case -1m2(S)

k 1 -

1

m2(S)

cm(U Ri ).

I

185

and iii, S and Si, respectively. Then, because the R exceed the z side length of R, we have

~ C, and

m(R)

..,,,,,:,,..,,,,,,,="""""=",'--",,,==,.,,,"""~_.'

SOME TOPICS FROM HARMONIC ANALYSIS

m2(S) Js

In 1978, A. Cordoba [6], using the methods we have described, proved this conjecture using a beautiful argument which we reproduce here. In fact, assuming thatJ Ri } is a sequence of rectangles in R z we prove that there exists a subcollection {R;} so that

II L Xi, lIexp(L)

=,;:,~",~""<:"",,,,-,,,,,,,,.~,,,,.,,,,

_1_ ([/t:;, _ 1] = _1_ ([e(EIt:;,+EIlt:;,) - 1]

then we get the estimate

(1)

.

·"'·~~~'"',,~""'.~"' ~_'''i'';''"''_

-1-

_1_ ( IfI

sup

pERER z m(R)

E

,-...

2

which is a s~ghtly stronger control of the overlap of R with the preceding selected rectangles Ri than in the case of Mn above. In our last example, the reader will notice a further strengthening of the selection rule. Consider, in JR.3, the collection of all rectangles R z with sides parallel to the coordinate axes, contl!ined in the unit cube, whose side lengths are of the fonn s, t, and 1/I(s, t) (in the x, y, and z directions, respectively) where 1/1 is increasing in each of the s and t variables separately. Then, A. Zygmund conjectured many years ago that if we set

m{p

'~O''''''=''''''""",''~''''·''"":.f",,,,,:",,-

- ~ -I m(R), Ri)

i
Mzf(p) =

."",,,,

Thus in this case at least one of the inequalities

I

ml(K)

or

1 ml(J)

is valid.

( [ EI X"::'] e 5, 1 dy >

JK

1J[e

Ell X"::'

1 5 -

I

1

J6 1

]dX> J6

-

1

1 dx > (5

186

CHAPTER 9

In case m2:

S)

it follows that

1 1[

~.

[eE1XS; -I]dxdy >

1 e EO:-::-] Sj I dy > -,

I-ml(K)

6

K

and a similar analysis holds in the remaining case where

- -I m2(S)

1[ S

e ElJX"::'] Sj 1 dxdy > -I 6

187

SOME TOPICS FROM HARMONIC ANALYSIS

for all multiindices a with lal ::: [~] + 1; here ((l(x) is any function COO(R 1) which satisfies ({l(x) = I for all x E [1,2] and ({l(x) = 0 for all x f/:. [~, 4]. It is a fundamental fact from the theory of singular integrals (see Stein [27]) that the multiplier operators Tm corresponding to the Honnander class are bounded on LP(~n), I < p < 00, and are of weak type (l,I). Now, to motivate the definition of the Marcinkiewicz multipliers, the reader should consider the estimates satisfied, on ~n X ~m, by multipliers of the fonn ml (~)m2('7), ~ E ~n, '7 E ~m, where ml and m2 are Honnander multipliers on ~n and ~m , respectively. These estimates on a function m defined on ~n X ~m are as follows: The functions m(81~, 02'7) satisfy (unifonnly in 81 and 82 ) the estimates

Putting all of this together yields

UR

j

~ {Mx (eEX;; -

Im(~, '7)1 :::

I) > 1/6} U{My (lX;; - I) > 1/6}.

C,

~ E ~n, '7 E ~m

for

IIDnm(~. '7)({l(I~I)]IIL2(lR") ::: C

(Here Mx and My denote one-dimensional maximal functions in the x and y directions,respectively.) Using the estimate IIlX;; - IIIL'(~) ::: Cm(U Ri ) and the weak type L 1 estimates for Mx and My, we see that m(U Ri ) ::: C(U Ri ) finishing the proof of the covering lemma. As the reader can see, it is far from routine to apply the covering lemma method to get infonnation about a particular maximal operator. Nevertheless, this method has had a number of deep applications, and can be expected to have more in the future. Before moving on to consider the second "model problem" it should also be pointed out that in order to do the operator theory of operators which are invariant with respect to certain actions ofl~n (certain groups of dilations, rotations, etc.) it is essential to understand the geometry of the relevant sets associated to these operators and for this, covering lemmas must be proved. It is not enough just to know the sharp estimates for the associated maximal operator. This will be illustrated quite clearly in the following part of this article dealing with the analysis of singular integrals on product spaces. Now, let us consider the second problem, namely obtaining sharp estimates for Marcinkiewicz-type operators. We shall begin by explaining the precise meaning of this class of operators, but the reader should be aware that the details of the definition could be altered considerably without making essential changes in their basic theory. Recall that a function m (~), ~ E ~n is called a "Honnander multiplier" provided all the dilates m(8~), 8 > 0 satisfy, unifonnly in 0, estimates of the fonn Im(~) I

:::

C,

~

E

~n

and

for each '7

E ~m

whenever

IID~[m(~, '7)({l(I'7I)]IIL2(1Rm) ::: C for each fixed ~

E ~n

m

whenever IIlI ::: [ "2 ] + I

and IIDf D~[m(~, '7)({l(I~I)({l(I'7I)]IIL2(lR"xlRm)

for all a,

n

m

I and IIlI ::: [ "2 ] + 1.

Il such that lal ::: ["2 ] +

Functions m (~, '7) satisfying these estimates are called here "Marcinkiewicz-type multipliers," and their corresponding multiplier operators, the "Marcinkiewicz operators." The problem we investigate now is that of obtaining the weak type inequality m{(x, y) E Q IITm(f)(x, y)1 >

C a

a} ::: -llfIILIog+ L(Q)

for all a > 0, where Q denotes the unit cube of ~n x ~m, and Tm is the Marcinkiewicz operator. This estimate is easily derived by iteration in case m(~, '7) takes the special fonn ml (~)m2('7). However the general case is much more difficult, and we shall explain why this is the case. One of the best ways to obtain the LP theory (I < p < 00) of Honnander operators is via the pointwise majorization of Stein:

a

II D [m(~)({l(I~ I)] 11L2(lR") ::: C

n

laI ::: [ "2 ] + I

(#)

g(Tmf)(x) ::: Cg;(f)(x)

~~~~""~.

_ _

__

••

--~_ ~-

-~- - - : " ' ~ ~ ~ ~ - - - ~_ ~ c = = - - ~ ~ ~

188

CHAPTER 9

gr

where g and are the classical Littlewood-Paley functions (Stein [27]). The quantity A > I in the inequality depends on how much smoothness the multiplier has. If the multiplier is minimally smooth ([ I ] + I derivatives in L 2 away from 0 and (0) then A is not much bigger than I. Unfortunately is not bounded on U(lRn) unless AP > 2, i.e., in this case is not of weak type (1,1). Thus when we are interested in the action of our multipliers near L 1 we must resort to something other than (#). The other way to handle T on LP(lRn ) is to realize

that T is essentially a classical Calderon-Zygmund operator. One then uses the decomposition into "good" and "bad" parts, or the atomic decomposition of the classical Hardy spaces HP(lRn) in order to obtain LP boundedness and the weak type estimates on L1(lRn). Our point here is that the classical Calderon-Zygmund decomposition and atomic decomposition cannot be used to analyze the more complicated Marcinkiewicz operators. The approach we take will be to study different spaces HP('f?n x IRm ) (we write HP('f?n+m) to denote the Charles Fefferman-E. M. Stein classical HP space, and HP('f?n x 'f?m) to denote our non-classical Hardy spaces) adapted to the study of the operators at hand. We construct the appropriate theory of BMO('f?n x 'f?m) and the atomic decomposition of H P('f?n x 'f?m), and study the interpolation properties of these spaces. Once all this is well understood, we will be in a good position to solve our second problem. Let us begin with the space HP(lR n x 'f?m), for p :s l. Suppose that f is a tempered distribution on IRn x IRm • Let ~I (x) E CO'('f?n), fP2(y) E CO'(lRm) with I~j = l,i = 1,2. Set~ol,02(x,y) = ~1(x/81)~(y/82)81n82m,for81,82 > O. We define the product non-tangential maximal function f* (x, y) by

gr

gr

J*(x, y) =

sup

If

* ~OI,02(X' y)1

__ ~

~ ~ ~ ~ _ -~ccc~~~~>~~

i~ f(.:c)xf(y)

189

f(x, y) = It(x)!z(y)

It

where

E

HP(jRn), !z

E

HP('f?m).

In particular, when p :s I if II = aj and !z = a2, where the aj are classical HP

atoms, then a (x, y) = al (x )a2 (y) is an H P(jRn X 'f?m) function which serves as a

model for what we shall call an H P('f?n x 'f?m) "rectangle atom." (The necessity

of inserting the word "rectangle" will be clarified shortly.) Clearly these a(x, y)

have the following properties.

(1) a(x, y) is supported on the rectangle R

=I

are cubes; a(x, Y)X U dx = 0 for all multiindices a with lal y E J, where Np(n) = [n( i - 1)]; (3) a(x, y)yfJdy = 0 for all multiindices P with IPI J x E I, where Np(m) = [m(i -l)];andfinally,

(2)

m

x J where I ~ 'f?n and J C IR

II

:s

Np(n) and for each

I

:s

Np(m) and for each

1

(4) lIaIlU(R)

1

:s m(RP-p.

Such a function a(x, y) on 'f?n X IRm is called an HP(lR n x IRm ) rectangle atom. m n For a while, it was thought that these rectangle atoms would span H P (IR x IR ) in a way that would immediately reduce the harmonic analysis of product spaces to the consideration of the action of the relevant operators on these very simple atoms. The conjecture was that every f E HP(jRn x IRm ) could be written as

= LAkak k

-

d8

If * lh,02(x, Y)I 2 d"X~n+11dyd8 ~m+l 01

2

02

where 1/1 (x, y) = 1/11 (x) 1/12 (y) and 1/11 E CO' (lRn), 1/12 E C8" ('f?m), I 1/Ii = 0 and 1/1; are suitably non-trivial. Then we have the following fundamental theorem. Theorem of Gundy-Stein [20]. For all p > 0, S(f) E U('f?n x IRm ) ifand only if f* E U('f?n x jRm). In this case the ratio IIS(f)IILP / 1If*IILP is bounded

above and below by strictly positive finite constants depending only on p, n. and

m.

_

This theorem forms the basis for the definition of product Hardy spaces: f E H P(IRn x 'f?m) if and only if f* E U (IR n x 'f?m) or equivalently S (f) E U ('f?n x 'f?m) for p > O. We define II!11HP(lRnxlRm) = II f*1I LP(lRnxlRm). There are certain obvious examples of functions in product H P, namely tensor products of H P functions of the x and y variables, respectively:

f

and the product Lusin area integral S(f)(x, y) by 2

__~ ~ _ ~

SOME TOPICS FROM HARMONIC ANALYSIS

01,02>0 Ix-.:cl
S (f)(x, y) =

~

where ak are rectangle atoms of HP(lRn x IRm ) and Ak are scalars satisfying Lk \AkI P :s CllfII~p(lRnXIR~)' In 1974, in [31], Lennart Carleson disproved this conjecture, suggesting that the harmonic analysis of product spaces had to proceed along lines very different from the ones of the classical theory. The fact that rectangle atoms did not properly span H P (IRn x IRm ) also suggested that the product theory could not be nearly as simple and satisfactory in its final form as the classical one. We now know that both of these suggestions are wrong, and to see this, the reader need only consider the following:

Theorem [15]. Let T be an L 2(lR n x IRm ) bounded linear operator. Suppose.

m further. that whenever a is a rectangle atom of HP(lR n x IR ) (0 < p :s 1)

..

~-=;;o===============~.~

190

CHAPTER 9

IT(a)IP dxdy

CRy

~

Cy- 8

for all

y::::

2 and some /) > O.

(Here R y denotes the concentric dilate of R by the factor y.) Then T is a bounded operatorfrom HPCWl,.n x ffi.m) to LP(Wl,.n x ffi.m).

The meaning of this is that, if T is to map H P boundedly to LP, then it is obviously necessary t1}at (t)

[

IT(aW dxdy

JR"xRm

~

C

for all rectangle atoms

(1) a is supported in Q*; 1

I

(2) Iia lIu(R" xR'") ~ m(Q)2 - P ; (3) a can be decomposed as a =

LReM(Q) CXR, where each CXR has the following properties: (i) CXR is supported in R; (ii) J cx R(X, y)x" dx = 0 for each y E ffi.m and for all multi-indices v such that Ivl ~ (n( ~ - 1)] = Np(n) and CXR(X, y)y" dy = 0 for each x E ffi.n and for all v such that Ivi ~ Np(m) = [me ~ - 1)];

J

[LREM(R)

lIaRII~]'!2 ~

m(Q)

4-

*.

Observe that (i) and (ii) describe aR as a rectangle atom except without any normalizing condition on liaR 112. This is, of course, compensated for. m n The atomic decomposition in [4] then states that every f E H P (lR x lR ) (here

o < p

~ 1) can be decomposed as

f = LAkak h

where Ak are scalars, ak are HP(ffi.n x ffi.m) atoms, and

a.

Carleson's counterexample shows that (t) is not sufficient for T to be bounded between H P and LP. Nevertheless, if we trivially strengthen the assumption above to say that not only is the integral (t) bounded, but the contribution to the integral from integrating over that part of space which is away from the support of the atom vanishes as we look farther and farther from this support, then this stronger condition is sufficient. Thus, it is not important that rectangle atoms fail to span HP(ffi.n x ffi.m). By examining the action of an operator on these atoms, we may still conclude its boundedness. Below, we shall give the main elements in the proof of this result. The first thing we must discuss is the correct notion of an H P (Wl,.n X ffi.m) atom. This atomic decomposition can be found in S. Y. Chang-R. Fefferman [4]. We set the following notation: If Q s:; IRn x ffi.m is an open set of finite measure, then the maximal dyadic subrectangles of Q will be denoted by M(Q). If R is a rectangle, then R denotes its concentric double, and if Q is an open set, then Q* denotes {Ms(Xn) > 2"~m J where M s is the strong maximal function in ffi.n x ffi.m defined by a supremum of averages over rectangles (products of cubes in ffi.n and ffi.m, respectively). Let 0 < p ~ 1 and suppose that Q s:; ffi.n X ffi.m is an open set of finite measure. A function a(x, y) on ffi.n X ffi.m is called an HP(~n x ffi.m) atom (with associated open set Q) if and only if:

191

SOME TOPICS FROM HARMONIC ANALYSIS

(iii)

supported on the rectangle R, we have

1-

.".-=,~

L !AkI

P

~ CIIfll~P(R"xRm)"

k

The second ingredient in the proof of our theorem is a geometric lemma due to J. L. Joume [23], which he used in order to obtain estimates on operators such as product commutators. We set the following notation in what follows: Let Q s:; IRn x ffi.m be an open set of finite measure. M 1(Q) will denote the collection of all dyadic subrectangles of Q which are maximal in the x coordinate. A similar definition applies to M2(Q). Let Q be a cube. We denote by Qy the y fold concentric dilate of Q. Then, given R, a dyadic subrectangle of~, R = I x J (l s:; ffi.n, J s:; ffi.m are cubes), we define Yl(R) = sup{t ~ lilt x J <; Q*}. Then we have:

Journe's Lemma. If Q s:; ffi.n then

L

X

ffi.rn is an open set offmite measure, and /) > 0,

m(R)Yl(R)-s ~ Csm(Q).

ReM2(n)

Of course, if Q is an open set in lR 1 , then the sum of the lengths of the maximal dyadic subintervals of Q equals the measure of Q. Although for Q open in ffi.2 it is not true that the sum of the areas of the maximal dyadic subrectangles of Q is finite (or less than Cm(Q», it is true if in the sum, we multiply the area of each maximal dyadic subrectangle by a suitable small constant depending (roughly speaking) on how much we may expand the rectangle without leaving Q (more precisely Q*). This is the meaning of Journe's result. Now let us prove the theorem. Let T be L 2(ffi.n suppose that

fCRy

IT(aW dxdy

~

X

lRm ) bounded and linear, and

Cy-

8

~~_

_~,__'!'!l'

.~if'",,,,~,,",!:.l"!~~_:,,-.,.=

...,~..,.,,,",",,, ....~-,,-,,,~~~~~~~~~,;;:"",,!~,,!!;,1'?';:'~":;"o,,'"

192

CHAPTER 9

for all Y ~ 2 whenever a is an HP('R,.n x JRm) rectangle atom supported in R. We claim that for f E HP(IR n x 'R,.m), p ::: 1, we have

II T fIIU(lll.nxlR m )

.,,,,,,,,,;,,~,,,.,~,,

dominated by

L

whenever a is an HP(lR n x JRm) atom. So, suppose a is such an atom with associated open set Q. Then IT(aW dxdy

REM(n)

+ In"

(

h..

~x~

.::: m(Q)

i

lIaRII~

REM(n)

::: C P

IT(a)IZdXdY)P/Z m(Q**)I/(*Y

i -I

(

IT(aW dxdy

=!
co."

co."

<

L

)

by 10ume's lemma.

L

(B) :::

p/Z (

( REM(R)

IjaR II~

IT (

L

aR)

REM(n)

L

I

REM(n)

C(l) xlR

+ (

/1

ReM(n)

L

m(R)Yz- 8 (R) /1

]

dxdy

Then

L

m(R)Yz- 8 (R) /1

=

ReM(n)

REM(n)

j:x.

m(R)(I-p/Z)(2/ p)' yz- 8
[ REM(n)

IT(aR)iP dxdy

(:x.:x.

)

) I/(Z/p)'

L

I/(Z/ p)'

P

L L

lIaRII~

Cpo

L

811 yz- (S)

SEM,(n')

L

meR).

REM(n)

R=S

IT(aRW dxdy

REM(n) J(lXJ)

<

m(R)(I-p/Z)(z/p)' y;8(Z/P)' (R)

REM(n)

ReM(n)

::: m(Q)p/Z-1

co."

L

)

In order to estimate (B) we proceed similarly.

-lm(Q**)I-p/Z :::

As for !co." IT(a)iP dxdy, we write a = LReM(n) aR as in the definition of an atom. Then we proceed as follows: Suppose R E M(Q), R = I x J. Let f denote the largest dyadic cube containing I so that f x J S; Q*. Then, let J denote the largest dyadic cube containing J so that f x J ~ Q**. We then estimate

!

) I/(Z/p)'

P/Z

IT(aW dxdy. ::: m(Q)

IT(aW dxdy ::: ( {

p/Z (

L

=

(A)

Clearly {

8 p m(R)1- /zllaRlliYz- (R)

We estimate (A) by HOlder's inequality:

IIT(a)IIU(ill.nXlRm) ::: C p

L"

L

+

= (A) + (B).

By the atomic decomposition this will follow once we prove that

lnXlll.mIT(aW dxdy =

m(R)I-p/zllaRllfYI- 8(R)

REM(n)

CpIlfIlHP(lll.nxlR m ).

:::

193

SOME TOPICS FROM HARMONIC ANALYSIS

Clearly any two dyadic R E M(Q) such that

L

\T(aR)iP dxdy

Ii = S must be disjoint, so

m(R)::: mrS),

REM(n)

m

R=S

:x.. IT(aRW dxdy.

llll.nxc(J)

We now use the fact that aRI(lIaRllzm(R)(\/P)-(I/Z») is an HP(JRn x JRm) rect­ angle atom, so that by the hypothesis of our theorem, the right-hand side above is

and so

L SEM,(n')

11 y;8 (S)

L ReM(n)

R=S

m(R):::

L SEM,(n')

811 yz- (S)m(S) ::: Cm(Q*)

===--;c---------------

.........

-----_~, -=----------~-------:==,"=-'='~~~-==__=_=_"'"c==._;=_--="~='"====_==,."'''~---

194

CHAPTER 9

-----~.«-~-,-.--~---~.-.---.----------------

195

SOME TOPICS FROM HARMONIC ANALYSIS

singular integrals T, that

by Joume's lemma. This proves that (B) ~ Cm(QV/ 2-'m(Q*)'/(2/ p )' ~ C'.

{

k.x.

~

IT(f)(x, YWw(x, y)dxdy when w

C (

~x.

This completes the proof of the boundedness of T from H P(Rn x ]Rm) to LP (]Rn x

for I < p <

]Rm).

the fact, from classical singular integrals, that

Once we understand that in order to check the boundedness of an operator on product H P, it is enough to check its action on rectangle atoms, it is not difficult to settle the question of the boundedness properties of Marcinkiewicz operators near L'. This proof of the weak type estimate

m{(x, y) E Q IITf(x, y)1 > o:} ~

1

w(x, y) dXdY) (m(IR)

L

P~' dxdy

r-' ~

AP(]Rn x ]Rm), then we need to have an analogue of

Here T is a Calderon-Zygmund singular integral, M is the Hardy-Littlewood maximal operator, and

C

w(x, y)-

E

(Tj)#(x) ~ C[M(fHe)(x)]'/l+e.

(##)

f#(x)

-llfIIL(log+ L)(Q) 0:

= sup

(I ) ( If(y) - fQldy,

Q

XEQ m

(Q denotes the unit cube of]Rn x ]Rm) where'T is a Marcinkiewicz operator proceeds much in the same spirit as the argument given in detail above for H P • In fact, it involves an atomic decomposition of L(log+ L)(Q) much like that of HP(]Rn x ]Rm). For the details, the reader should consult [19]. Again, we wish to stress that it is not the technical details that are impor­ tant here, but rather the philosophy that although the function spaces in the product theory are inherently more complicated than in the classical case, the operator theory is not. This philosophy has some very concrete applications be­ sides the one we have been discussing. As another example, we should mention the weighted inequalities for product singular integrals, which can be found in [15] and [16]. It is quite non-trivial to prove weighted norm inequalities for these singular integrals, because there are no known analogues of the classical Burkholder-Gundy good ,\ inequalities in this setting. One studies this problem by determining rather precisely the way that the strong maximal operator M s controls the singular integral. In what follows below we shall be a little more precise. To begin with, let us recall that a weight w (x, y) defined on]Rn x ]Rm is said to belong to product AP (written AP(]Rn x ]Rm» if and only if for all rectangles R (products of cubes in]Rn and in Rm, respectively) we have (m;R)

00

c.

This is equivalent to w satisfying a uniform classical AP condition in the x and y variables separately fixing the other variable. Then the way we prove weighted inequalities for general product singular integrals (not necessarily convolutions) in [15] and [16] is by studying a variant of the C. Fefferman-Stein sharp function, f# in the product setting. Thus, if we wish to prove, for very general classes of

If(x, YWw(x, y)dxdy

J

Q

J

(where fQ = mlQ) Q f and the sup is taken over all cubes Q ~ ]Rn containing x) is the C. Fefferman-Stein sharp function. In the classical theory, the value of the inequality (##) lies in the fact that f# E U(Rn) implies f E U(]Rn). Unfortunately, as we would expect, this is not true in product domains. To be more specific, suppose we define the mean oscillation of a function f (x, y) on ]Rn x ]Rm over the rectangle R by

oscR(f)

=

I -( fJJ1 meR) inf

1

2

If(x, y) - fl(X) - 12(y)1 dxdy

)1/2

R

where the inf is taken over all pairs of functions fl and 12 depending only on the x and y variables respectively. If we then define the (product) sharp function of

f, f#(x, y) by f#(x, y)

=

sup oscR(f) (x,y)ER

where the sup is taken over all rectangles containing (x, y), then it is simply not true that f# E LP(]Rn x ]Rm) implies that f E LP(]Rn x ]Rm). In order to use the product sharp function as a useful tool to prove weighted inequalities, we must transfer the definition to the context of operators, rather than functions. Intuitively, we would like to define the sharp operator S of an operator T by

Sf(x, y) =

sup OSCR T(f). (X,y)ER

As we stated above, the boundedness of Son LP(]Rn x ]Rm) does not imply the boundedness of T on this space. However, suppose, in addition, we require that whenever the support of f is far away from the rectangle R, i.e., if supp(f) ~ CRy then for some 8 > 0, and y ~ I

Sf(x, y) ~ yS oscR(Tj)

when

(x, y) E R.

196

CHAPTER 9

Then we say that S is a sharp operator for T. It is shown in [15] that if p ::: 2 then if S is bounded on LP(jRn x jRm). then so is T. Also, if T is a product singular integral, then M s (f2)1/2 is a sharp operator for T. This precise expression of the control of product singular integrals by the strong maximal operator can then be used in order to prove the product version ofthe Hunt-Muckenhoupt-Wheeden weighted inequalities for singular integrals, above. It is another graphic illustration that in product domains, one can use the philosophy mentioned above in order to bring things into a very simple and satisfactory form quite like that of the classical theory. Our last problem concerns elliptic differential equations, but the mathematical issues involved have quite a lot in common with model problems 1 and 2. In fact, we will see that just as in the first two problems, the theory of maximal functions, singular integrals, and especially the topic of weighted inequalities will play a central role. We shall begin to discuss the third problem by formulating it precisely. We consider elliptic operators in the unit ball B of jRn in divergence form. This means that Lu = div(AVu) where A(x), x E B, is a symmetric n x n matrix valued function satisfying

AI~12 ~ ajj(x)~i~j ~ A-ll~12,

A(x)

for all ~ E jRn and some fixed A > O. We will assume, in general, no regularity of the coefficients aij (x) except for measurability. The question we investigate is the solvability of the LP Dirichlet problem. To define this properly, we require several definitions. First, we say that a function u E H/oc(B) (Hl~(B) denotes those functions in B which are, together with their distributional first derivatives, locally in L 2 ) is a solution to Lu = 0 provided a(u, ({I) = 0 for all ({I E CO'(B), and where a(u, v) = fB aij(x)oiU(X)Ojv(x) dx. We denote by r(x) the right circular non-tangential cone with fixed aperture having vertex at x E 0B. Then, if u is- a function in B, we denote its non-tangential maximal function by u*(x) where u*(x)

=

sup lu(z)I,

for

x

E

The question we ask is: Given an elliptic operator L, when does there exist a solution u to the problem above given any f E LP(da)? Here p is a fixed exponent satisfying 1 < p < 00. The estimate on the non-tangential maximal function is included to insure uniqueness of solutions to the problem. Before giving a summary of the known results concerning this question, we shall discuss some general background information on the solutions to the elliptic equations we are considering. It is a celebrated fact from the classical theory that positive solutions of these equations satisfy the Harnack principle (Moser [25)). This means that if K S; B is a compact set, and u(z) > 0 is a solution of Lu = 0 in B, then there is a constant C depending only on K and A such that sup u(z) ZEK

a

= f(x)

for

x E oB,

and also the estimate

lIu* IILP(du)

~

CII fIILP(du)'

ZEK

== 0 in Band u(x) = f(x)

for all

x

There is also the maximum principle for solutions to Lu continuous on B and satisfies Lu = 0 in B then max u(z) ZEB

E aB.

=

0 in B: If u is

= max u(x). xEaB

By the previous results it is possible to define the harmonic measure wi associated to the operator L with respect to the point z in a manner exactly analogous to the familiar case of the Laplace operator: Suppose u f, f E C(aB) denotes the unique solution to Lu = 0 in B with uf(x) = f(x), x E aB. Then by the maximum principle, the map f --+ uf(Z) is a positive linear functional on C(aB). Hence, according to the Riesz Representation Theorem, there exists a unique probability measure wi so that

oB.

Then, by the LP Dirichlet problem for L in B we mean the following. Let there be given a function f E LP (da) (da is the surface area measure on B). Find a solution u(z), Z E B to Lu == 0 in B, satisfying lim u(z)

C inf u(z).

continuous on Ii so that

ZE['(X)

Z-"+X ZE['(X)

~

Another beautiful result from the classical theory of these equations is that the classical Dirichlet problem is always solvable for any elliptic L (Linman­ Stampacchia-Weinberger [24)): Given any continuous function f (x) on 0 B, there exists a unique function u (z)

Lu

= (ajj(x»

197

SOME TOPICS FROM HARMONIC ANALYSIS

u f(z)

=

r

f

dwi,

f

E C(oB).

JaB

This measure dwi is the harmonic measure. It is a trivial consequence of the Harnack principle that if Zl, Z2 E B, then the ratio wi' (E)jwi2 (E) is bounded above and below by a pair of finite, strictly positive constants for all Borel sets E S; 0 B. Thus, the exact choice of z E B makes virtually no difference when considering the properties of harmonic measure. In fact, if z = 0 we will write dw~ often simply as dwl, (or even dw if L is understood). Now the nature of the

198

CHAPTER 9

harmonic measure dWL is of critical importance. To be more exact about this, we recall the classical weights from the theory of singular integrals and maximal functions. We say that a measure J-L on aB is in the class BP(da) if and only if J-L == k da where

(_1 ill

C(_1 ill -

{kPda)IIP <

a(l1)

(kda)

a(l1)

for all surface balls l1. S; aB. A measure J-L == k da is said to belong to the Muckenhoupt AP class if and only if _1 ( a(l1)

f kda) (_I (k- p~, da)P-l C a(l1) ill ­ <

II

for all surface balls l1 s; aB. Then the key point for our purposes here is the following: The LP (da) Dirichlet problem is solvable for L if and only if the harmonic measure dWL belongs to BP' (da) where.!. + 1, == 1. P

P

Unfortunately, it is not true that dWL is always in some class BP(da) for an arbitrary elliptic operator L. In fact in Caffarelli, Fabes, and Kenig [I] an example of an elliptic operator L is given whose harmonic measure dWL is singular with respect to da. Thus, the nature of the solvability of the LP Dirichlet problem depends on the operator we are dealing with. Considerable progress has been made on this issue. In what follows, we shall give a quick overview of some of the highlights of the theory. One of the motivations for considering the class of elliptic operators with bounded measurable coefficients is to achieve a better understanding of the harmonic functions on Lipschitz domains. One strategy for solving the Dirichlet problem for the Laplacian on a Lipschitz domain is to change variables in such a way that a harmonic function on the Lipschitz domain is pulled back via the change of variables to a solution, in the upper half space, of an elliptic operator with bounded measurable coefficients. (The coefficients of this operator come by taking the derivative of the defining function of the Lipschitz domain; hence, they are merely in L 00.) The coefficients of the operator gotten in this way are independent of the X n coordinate in ffi.+.. It is a well-known result of B. Dahlberg [10] that the LP Dirichlet problem is solvable for the Laplacian in a Lipschitz domain when p 2: 2. Corresponding to this (stated for the unit ball rather than the upper half space) is a basic positive result for the theory we are now considering:

199

SOME TOPICS FROM HARMONIC ANALYSIS

Theorem of Jerison and Kenig [21]. Suppose Lu == div(A'V'u), is an elliptic operator whose coefficients A(x) are bounded measurable functions in B ~ ffi.n. Suppose further that A(x) is independent ofthe radifll variable, i.e., A(x) == A(y) 2 whenever x/lxl == y/lyl. Then the harmonic measure dWL E B (da). Thus, if the coefficients of the operator are "radially independent" then the LP Dirichlet problem is solvable when p 2: 2, and this can be thought of as an extension of Dahlberg's result on Lipschitz domains. A bit later, E. Fabes, D. Jerison, and C. Kenig [13] proved the following result: Suppose Lu == div(A'V'u) is elliptic, and that A(x) is continuous on B. Let w(o) == sup Ix-Yl:oo IA(x) - A(y)1 denote the modulus of continuity of A on B. x,YER

Assume that

w(o)2 - - do < o 0 1

1

00.

Then WL E BP(da) for every p < 00. This result holds only for continuous coefficients, and, in particular, does not cover the operators which arise via a change of variables from the Laplacian on Lipschitz domains. In order to remedy this, B. Dahlberg proved a perturbation theorem which applied equally well to the case of discontinuous coefficients. In order to state his result, we recall a definition. Suppose that J-L is a positive measure on B. Then we say that J-L is a Carleson measure provided, for any x E aB and

r > 0, J-L(B(x; r) n B) < C. r n-

1

­

We say that J-L is a Carleson measure of vanishing trace provided

J-L(B(x; r)

rn -

1

n B)

.

.

--+ 0 umformly 10 x as r --+ O.

Theorem of Dahlberg [11]. Suppose Lo and LI are elliptic operators with co­ efficients Ai (x) bounded and measurable in B. For each Z E B, set a (z) == 2 SUPYEB(z; '-;1'1) IAo(Y) - AI (y)l. Suppose that the measure a (z) o~~) is a Carleson measure of vanishing trace (here o(z) == 1 - Izl). Then WL o E BP(da) implies thatwL, E BP(da)forany I < p < 00. In Dahlberg's theorem, we have an easily-checked criterion that tells us that the coefficients of two operators are close enough near the boundary so that the solvability of the LP Dirichlet problem for one of the operators guarantees its solvability for the other operator. This result depends in a crucial way on the smallness of the Carleson measure on small balls. If the vanishing trace assumption

=,::",~-==--~--=-~~~--~:=~~""-~

--------

-'-""

-"--" --"-

- ~"',,="-"::::~~----=-~=-=~~='::""'~=::-~'~-==--~~':-----'''~::-_:::~~:-~--===.:::".::-----==--=-====:=:-=-

200

CHAPTER 9

is removed, then simple counterexamples show that the conclusion of the theorem is no longer valid. Nevertheless, in the absence of the vanishing trace condition, Dahlberg made the following conjecture:

Dahlberg's Conjecture. Assume that L o and L [ are elliptic in B with bounded measurable coefficients Ao and A I, respectively. As above, set a(z)

=

sup yEB(z;

¥)

harmonic measures of L t on A by Wt. Then this rescales everything so that c ::: Wt(~) ::: I,

where c depends only on the ellipticity constant of L t (see [2]). This means that to estimate Q{t) it suffices to estimate (using duality)

i

IAo(Y) - Al (y)1

This says that when we no longer assume smallness of the relevant Carleson measure, then what is preserved is the solvability of the LP Dirichlet problem in some range of P not necessarily the same for both operators. In what follows we wish to sketch the proof of this conjecture, which is given in [18] (R. Fefferman, C. Kenig, and J. Pipher). Let us begin by mentioning the method of proof of Dahlberg in his vanishing trace theorem. is a Carleson Assume that Lo and L I are elliptic operators, and that a 2 measure of vanishing trace. Dahlberg introduces the one-parameter family L t = (1 - t)Lo + tLI, and sets Q(t) equal to the BP(da) norm of the harmonic measure Wt of Lt. Then the argument proceeds by showing that for some large N we have \Q'(t)1 ::: CQ(t)N whereC ~ OastheCariesonnormofthemeasurea 2 ~ O. 2 Since we are assuming that a has vanishing trace, we may essentially assume that C in the differential inequality above is as small as we like; hence, Q{l) is controlled by Q(O), and the proof will be finished. What remains is the proof of the differential inequality, or more precisely, something essentially equivalent to it. Fixt E [O,l],andconsidertheBP(da)constantofwt = kt da,name1y Q(t). By definition, there exists a surface ball on aB, ~,so that

II f II u' ( ~) = 1 and f is a non-negative function on a(tq aA supported on ~. The next step in the proof is that if rp ::: 0 is a smooth bump function with compact support in A, which is suitably normalized, then by Harnack's principle,

i

¥

¥

a(~)

kf da)IIP/

_1_ i",[

i

so that

0=

i

L,(Ut)h, dz

Lt(Ut)htdz =

Suppose that ~ has radius r, and has center Xo E aBo Take A = B n B(xo; 2r). Then it is a simple consequence of the so-called "comparison theorem" (see Caf­ farelli, Fabes, Mortola and Salsa [2]) that if we restrict all our operators to the region A, then the resulting harmonic measure is equivalent on ~ to that of the operator considered on all of B. (Equivalent in this context means that the ratio of the two measures applied to any Borel set in ~ is bounded above and below by strictly positive constants.) We shall abuse notation by henceforth denoting all the

i

Utrp dz

i

== 0 V t

where Lth, = rp, htlaA = 0,

Lte(Ut)htdz

+

i

L,(Ut)htdz

Now the last of these three terms is 0, since LtUt

+

i

Lt(Ut)h t dz.

= 0, and integration by parts in

the second term transforms it into

i i =i

Ut,pdz.

Integrating by parts in the first term gives the identity

Ut,p dz

ktda.

a(M

fk t da ::: Co

where LtUt = 0 and Ut laA = f. It is therefore enough to estimate fA Utrp dz, and it is actually this quantity whose t derivative we shall bound. In fact, letting a dot stand for differentiation in t, we see that

¥

2(_1_ i",[

fktda

under the assumption that

¥

and assume only that ~2 is a Carleson measure. Then if WLo E BPO(da) (for some Po > 1) then it follows that WL, E BPI (da) for some PI > 1.

Q(t) :::

201

SOME TOPICS FROM HARMONIC ANALYSIS

where (sij)

=

S

maXi,j ISij(z)\, then

and s

i

=

sij(z)aiUt(z)ajht(z) dz

Al - Ao We must then prove that if s(z)

s\VUtllVhtldz ::: CQ(t)N.

It is quite easy to show that [ s\VutllVhtl dz ::: C

iA

1

xE'"

St U)(X) H(k,) da

=

202

CHAPTER 9

Then we show under assumption (A) that WI E AOO(da) (see R. Fefferman [17]). To do this we use a novel characterization of AOO(da): Observe that W = k da E A 00 (da) if and only if it satisfies

where S, denotes the Lusin area integral, i.e.,

=

S;(f)(x)

r

203

SOME TOPICS FROM HARMONIC ANALYSIS

IVutl Z8Z- n dz

Jr(X)

and where H is a classical Calder6n-Zygmund singular integral. Now to finish the proof, Dahlberg uses HOlder's inequality to arrive at

Ii

U,qidzl s

IIkllLlog+ L(Il'• E....) S Cll k ll v (Il', O'l6.) and we let L ,

CIIS,jIlU'(dcr)II H (k/)IIU(dcr,Il)'

By the boundedness of singular integrals on LP(da), we have

IIH(k r )IIU(dcr,ll) S CIIk / ll u

(dcr,ll)

S a(ll)llp.

= (l

- t)Lo

+ tL\

= sup

ilk/ilL

Q(t)

Ilk

Il

as before, but here we set

log+ L(Il; E....) II 0(6) / V (Il; O~~) )

(J"(~)

IQ(t)1

1

(A)

1 o

dt

a Z((1 - t)x) -

t

S C

L/h/

x

E

aBo

s(z)IVu/(z)IIVh/(z)1 dz

= 0 in A, = cp in A,

u/laA u/laA

= f and = O.

We estimate fA sIVu/IIVh/ldz by breaking this up into two parts A = AI U Az where we write II = B(xo; r) naB for some Xo E aB and r > 0, and then define Al by AI = B(xo; 3/2r) n B and Az = A - AI. The main estimate is the term

fA I slVutllVh/ldz which we carry out below. { sIVu/IIVh/1

S Ca

for all

Sci

L/u/

p>1

¥

CQ(t)

where

JAr

so our conjecture assumes that Wo E AOO(da), and we now show that WI E AOO(da) which proves it. To do this we must first single out an important special case: Assume rather a Carleson measure, the stronger condition than a Z

s

and this controls Q(l) in terms of Q(O) without any smallness of C. We do this as follows: We show that the quantity Q(t) can be bounded by fll fk, da for some choice of II and a non-negative f supported in II satisfying II f IIBMO + IIfllv(..!!!!-) S 1. Then an analysis similar to Dahlberg's shows that

¥

= U BP(da),

t E [0, 1].

class near L I, we actually have

00

AOO(da)

for

The idea is then to show that because Q(t) is now defined in terms of an Orlicz

IQ(t)1 As for II S,j lI u '(dcr) , according to a theorem of Dahlberg, Jerison, and Kenig [12] the area integral S, is bounded on all LP (dw,) for I < P < 00 for all elliptic operators. We need to estimate here the quantity II S,j II u' (dcr) , not IIS,jllu(dw,). However, by a good ).. argument one can show [12] that the operator S, is also bounded on all LP(dlL), where IL is any measure satisfying an Aoo condition with respectto w,, Thus II S,j II u' (dcr) sell f II U' (dcr) where the constant C is dominated by a constant depending only on P times the norm II d a II A (dw,) and this last quantity turns out to be less than C Q(t)N for a large power N. This competes the proof of the differential inequality and Dahlberg's theorem. Now, let us pass to the proof of the conjecture, where we only assume that is a Carleson measure in B, with no smallness assumption. the measure a Z We assume that the LP Dirichlet problem is solvable for the operator Lo when P = Po, and then seek to show that the LP Dirichlet problem can be solved for L I when P = PI, and where PI might be much larger than Po. In terms of the harmonic measure Wi, i = 0, I of the operators L;, this means wo E BPO(da) implies WI E BPI (da) for some I < Po, PI < 00. It will be important to think of this in terms of the so-called A 00 class. This class of weights can be defined in a great many different ways (see Coifman-C. Fefferman [5]). Here we recall that

E....) 0(6)

~

{ lXEIl

dz

(1

8I - n(z)s(z)IVu/(z)IIVh,(z)1 dZ) da(x).

fa(x)

Now we estimate

1

81-n(z)s(z)IVu/(z)IIVh/(z)! dz

fa (x)

S

ill o

B«(I-p)x;ap)

~ 8I - n(z)s(z)IVu/(z)IIVh/(z)1 dz­ P

---

~--

",.~~, .... _~~~":;~~-';;":~;;::'';'=:~~

204

I 10

S C (

pl-naa((l - p)x)

CHAPTER 9

(1

( lVur(z)12 dz B((I-p)x;ap)

)IP 1/2

(

l«(l-p)x;ap)

IVh r(z)1 2 dz

)

:;:.-:~,';~~~;';~~~::::::~~.~.-;:,-::~~<, '-;';.~.-;;,~

;.:~·::~':,".~:::.,~;:;:;;,~,..:--;;:~,.;=~_.~';"'-;;,"~.;;~:;.";':'7o,~-;:-.:"'-:::"·~;;';;;;"';;:-':::;;;;::;:;":7~~~~;:~~:~~';'~~~~;~~:';:;:;';;;;'~~~~~~~;~~~~:7.".;~;;:

Let h,w, = w)!1) It. f dWr. Then

(

lA

f2 dWr )

1/2

dp ­ p

S

(

i

h,w,1 2 dWr

If -

i

"c [(w)D.>

(here aa(Z) = sUPYEB(z;a6(z» 1£(z)1)

S C (

10

I

_

pl-naa((l - p)x)

(

)1/2

If -

+

i

(

fl,w dWr

)1/2

fA~12dW,r + IfA.~I].

We observe that

1

IVu r(z)1 2 dz

) 1/2

=

h,w,

B«(l-p)x;ap) p-lpn/2p2-nWr(ilx,p) dp

(*)

where ft. =

p

where ilx,p is the surface ball centered at x of radius p, and we have used Cacciopoli's inequality as well as the estimate

W)il)

rr/t.)

1

{

wr(il)

It.

i

fk r da S W)il)

(i

If - hlkrda

+

h)

It. fda S 1. Also

1

1

(

If - hlkr da = [w,(t.)] a(il) It. If - hlkrda rr(!1)

<

hr((l - p)x; ap) S Cap2-nwr(ilx,p).

II krllLlog+ L(drr/rr(t.»

Cllfll

-

Then (*)

205

SOME TOPICS FROM HARMONTC ANALYSTS

BMO(drr)

It. k r da/a(il)

S CQ(t).

s ( 1 aa((l

10

- p)x)

(

( IVur(Z)12c52-n dz B«(l-p)x;ap)

) 1/2

1

The term

pl-nwr(ilx,p) dp . p

Thus { £IVur(z)IIVhr(z)1 dz S ({ aa(Z)Szwr(il z ) d:

lA

llA

1

c5

1

1/2 2 2 where Sz = ( ffB(z;a6(Z» lVu r(y)1 c5 - n(y) dy ) and ilz = il( c5(z)). The last integral above is easily seen, by Fubini's theorem and the Cauchy-Schwarz inequality, to be bounded by

Vi;

C (

laA

1 ( ) 2 ( wr(il) 1t. If - ft.,w, 1 dWr

is dominated by a constant depending only on the doubling constant of Wr

(which, by [2] depends only on the eIlipticity constant of L r ) times the quan­

tity IIfIlBMO(dwl)' Now, we claim IIfIIBMo( dwl) is bounded by CQ(t). To see this

we estimate: For ilo ~ il

I -(A) Wr uo

(

S(f)(x)dwr(z) S CQ(t),

1 If - holdwr = - (1A)a(ilo)-(A) Wr uo a uo

1 t. o

If - ft.olkr da

II krllLlog+ L(drr/rr(t.o»

Cllfll BMO(drr)

[

(t. o) ] rr(t.o)

WI

This completes the proof of the conjecture under the more stringent hypothesis l thatio a 2«(l- t)x)~ E LOO(da(x)). In ordertoreduce the general case where a2 is a Carleson measure to this one we proceed as follows:

Choose a surface ball il = il(xo; r) ~ aB and a subset F ~ il with ~f~~ >

1 - ~. We will show that WI E AOO(da) by proving that ::f~~ > £ for some

£ > O.

¥

[lA S2(f)dwr(z) J

1/2

S(f)(x)dwr(x) S

t.o

S CQ(t).

We proceed as follows:

lA

1

S

S(f)(x)dwr(x).

Therefore our claim is reduced to showing that

laA

1/2

(

S C

lA

1/2

f2 dWr)

..;;::~~-..;.~.;;.-..;;,.-;,c_=~

206

CHAPTER 9

Observe that

if 11 > 0 is so small that by the A00 property of WM,

r (r

~

a (Ll) JXEt.

a 2((l _ t)x) dt) da(x) Jo t

~

C

a(H)

, 1

dt 2 a ((1 - t)x) - ~ o t

c' = 2C

for each

x

E

E.

Define r~(x) to be the right circular cone with vertex at x E aB, aperture ex (suitably small) and truncated at height r. Set R = UxEE r~ (x). Then define the intermediate operator Mu = div(AVu) by

A(z) =

(

fI.

Ao(z)

ifz

AI(z)

ifz E R

1.

.

wM(H)

- - > - 1ffiphes - - > 11 a (.1.) 4 WM (~)

by the Carleson condition we are assuming. This means that on a set E such that 4a (.1.) we have

aCE) ~

R

B(z) = IA(z) - Ao(z)l,

a(z) =

sup

a(x, t)

o

t

B(y)

for all

If (f

x

E

aBo

Thus, by the special case treated above, the harmonic measure of the operator M, AOO(da). Now the main question is "What is the relationship between WM and WL, ?" This is answered by a lemma from [12].

Lemma. Suppose that.1. ~ aB is a surface ball of radius r.

e

r~(x) for some set

~

agree on R we have

WL j (E)

-->C WL, (.1.) -

[WM(E)

-­ WM (.1.)

JII

C [WM(F n E)]II > wI(F n E) > > C11 WI (~) WI (.1.) WM (.1.) ­ --

~

* Vrt(x)! * CPt (x) .

* Vrt(x)\2 * CPt (x))2

J cP =

1. It is a special

dxdt -t

Let R

= f dx be a positive measure on JEtn which satisfies the doubling condition JL(B(x; 2r)) ~ CJL(B(x; r)) for all x E JEtn, and r > O. Then f E A 00 (JEtn) if and only if

= I

f f

* Vrt(X) * CPt (x)

/2 dxdt - is a Carleson measure on JEt~+l. t

It is interesting to note that the assumption that the measure f dx is doubling is crucial, and the theorem is incorrect if this assumption is deleted. Now how is this related to the Helson-Szego Theorem? The latter states:

Applying this lemma in our context gives WI (F)

~ ~.

Theorem [18]. Let JL

.1.. Then there exists a constant C > 0 and > 0 so that for all subsets E ~ S and operators M and L I whose coefficients

S

H

is a Carleson measure in JEtt, then f E AOO(dx). Actually this condition gives a complete characterization of Aoo(JEtn) for any dimension n.

WM E

UXES

if f

J

then for ex sufficiently small, we have

1

=

Here Vr, cP E Co (JEtI), with Vr dx = 0 and cP ~ 0 with case of Dahlberg's Conjecture that if

z E B and if

yEB(z;a8(z»

dt Q2((l - t)x) - ~ C'

for all

Thus, under the assumption that :~~j > 1 - 1510' we have proven that ::~~j > C11 11 and this shows that WI E AOO(da) completing the proof of Dahlberg's conjecture. Finally, we would like to discuss a beautiful connection between the Dahlberg Conjecture and the Helson-Szego Theorem from harmonic analysis. It is a simple matter to construct, given any doubling measure JL = f dx on JEtI, an elliptic operator L on JEt~, whose harmonic measure WL is JL. The difference (using the notation discussed above) between the coefficients of this L and the identity matrix a(x, t) for (x, t) E JEtt is essentially given by

A simple geometric argument shows that if

1

207

SOME TOPICS FROM HARMONIC ANALYSIS

Helson-Szego Theorem. A measure JL = f dx on JEtI is such that the Hilbert

II

transform is bounded on L 2 (d JL) log

f

= bl

+ H(b2)

where

if and only if bi E L OO (JEt 1), i = 1,2, and

IIb2 1100

< 77:/2.

208

CHAPTER 9

Taking into account the celebrated Hunt-Muckenhoupt-Wheeden Theorem on weighted norm inequalities for the Hilbert transform, as well as the C. Fefferman­ Stein Decomposition ofBMO, we see that except for the restriction II b 2 11 "" < Jr /2, the He1son-Szego Theorem would say that / E A 2 (~1) if and only if log / E BMO(lR 1). In a sense this says that log / E BMO is almost enough to guarantee that / is a weight in A 2 , and it tells us exactly the difference between these two conditions. What we have in our theorem above is an analogue of this for A 00 weights in all dimensions. To see this, let us write that log / E BMO using C. Fefferman's characterization of BMO in terms of Carleson measures: log / E BMO(1R 1)

if and only if j(1og f)

is a Carleson measure in 1R~. Suppose written

t (log f) I

* -ax

OlfJr

2 1

dxdt -t

Vr = qi.

dxdt

* VrtI 2 -t­

Then this last condition can be 2

a [(log j) * IfJr] 1 = t -ax I

dxdt - is a Carleson measure. t

Now suppose we commute the taking of the logarithm with the process of averaging (convolving with IfJr). We get (the inequivalent) condition

a [log(f * IfJr)] [2 ax

t -

I

dxdt t

2 / *Vrt 1 dxdt = I/

* 1fJ,

t

. a Carleson measure.

IS

But this is exactly the condition which we proved in [18] is equivalentto / E A"". This tells us that our Carleson condition, shown to characterize in all dimensions the space A 00 (IRn ) is nothing more than a commutation away from log / E BMO(lRn ). In this way we have a version of the Helson-Szego Theorem for the space A"". Not only does the Dahlberg Conjecture imply a special case of our character­ ization of A 00, but the characterization tells us that the Dahlberg conjecture is completely sharp. Thus, for instance, if we start with the Laplace operator and run through all elliptic operators whose coefficients satisfy

dz.IS a Carleson measure, a 2 8" then the class of harmonic measures which results is precisely the class A"". The conclusion of Dahlberg, given his assumption, is therefore absolutely sharp. Sim­ ilarly, the equivalence of the A 00 condition with the Carleson condition shows that under any assumption weaker than that made by Dahlberg, we cannot conclude that the resulting operator will have its harmonic measure in A"" (d a).

SOME TOPICS FROM HARMONIC ANALYSIS

REFERENCES

[1] L. Caffarelli, E. Fabes, and C. Kenig. "Completely Singular Elliptic Harmonic Measures." Indiana Univ. Math. J. 30 (1981). [2] L. Caffarelli, E. Fabes, S. Mortola, and S. Salsa. "Boundary Behavior of Non-Negative Solutions ofElliptic Operators in Divergence Fonn." Indiana Univ. Math. J. 30 (1981). [3] L. Carleson. "A Counterexample for Measures Bounded on HP for the Bidisc." Mittag­ Leffler Report No.7 (1974). [4] S. Y. Chang and R. Feffennan. "A Continuous Version ofthe Duality of HI and BMO on the Bi-Disc." Ann. of Math. 112 No.2 (1980). [5] R. Coifman and C. Fefferman. "Weighted Nonn Inequalities for Maximal Functions and Singular Integrals." Studia Math. 51 (1974). [6] A. Cordoba. "Maximal Functions, Covering Lemmas, and Fourier Multipliers." Proc.

Symp. Pure Math. 35, Part I. Amer. Math. Soc., 1978.

[7] A. Cordoba and R. Feffennan. "A Geometric Proof of the Strong Maximal Theorem."

Ann. of Math. 102 (1975).

[8] A. Cordoba and R. Feffennan. "On the Equivalence Between the Boundedness of Certain Classes of Maximal and Multiplier Operators in Fourier Analysis." Proc. Nat. Acad. Sci. 74 No.2 (1977). [9] A. Cordoba and R. Feffennan. "On Differentiation of Integrals." Proc. Nat. Acad. Sci. 74 No.6 (1977). [10] B. Dahlberg. "On Estimates of Harmonic Measure." Arch. Rat. Mech. Anal. 65 (1977). [11] B. Dahlberg. "On the Absolute Continuity of Elliptic Measures." Amer. J. Math. 108 (1986). [12] B. Dahlberg, D. Jerison, and C. Kenig. "Area Integral Estimates for Elliptic Differential Operators with Non-Smooth Coefficients." Arkiv. Math. 22 (1984). [13] E. Fabes, D. Jerison, and C. Kenig. "Necessary and Sufficient Conditions for Absolute Continuity of Elliptic Harmonic Measure." Ann. of Math. 119 (1984). [14] C. Feffennan. "The Multiplier Problem for the Ball." Ann. of Math. 94 (1971). [15] R. Feffennan. "Calderon-Zygmund Theory for Product Domains: H P Spaces." Proc. Nat. Acad. Sci. 83 (1986). [16] R. Feffennan.... AP Weights and Singular Integrals." Amer. J. Math. 110 (1988). [17] R. Feffennan. "A Criterion for the Absolute Continuity of the Harmonic Measure Associated with an Elliptic Operator." J. Amer. Math. Soc. 2 (1989). [18] R. Feffennan, C. Kenig, and J. Pipher. "The Theory of Weights and the Dirichlet Problem for Elliptic Equations." Ann. of Math. 134 (1991). [19] R. Feffennan and K. C. Lin. "A Sharp Marcinkiewicz Multiplier Theorem." To appear in Ann. Fourier Inst. Grenoble. [20] R. Gundy and E. M. Stein. "HP-Theory for the Polydisk." Proc. Nat. Acad. Sci. 76 (1979). [21] D. Jerison and C. Kenig. "The Dirichlet Problem in Nonsmooth Domains." Ann. of Math. 113 (1981). [22] B. Jessen, J. Marcinkiewicz, and A. Zygmund. "A Note on Differentiability ofMultiple Integrals." Fund. Math. 25 (1935). [23] J. L. Joume. "Calderon-Zygmund Operators on Product Spaces." Rev. Mat. Iber. 1 (1985).

University a/Chicago

209

---C;~';;~""~~,",,:::;-~~~;~~=;;;:W~_:~'~~':~"~~,~_... ;.;.;.;;,;:,~~~~~~:.:

210

__ ~,._.-:-_._.:~~~",

_:,~~~_~~.~~~~_

.._:::-:::::~,:"~-=-_"~-;;;;;~--:.:;::'~:;:,;;;:~~;;;;=

CHAPTER 9

[24] W. Littman, G. Stampacchia, and H. Weinberger. "Regular Points for Elliptic Equations With Discontinuous Coefficients." Ann. Scuola Norm. Sup. Pisa 17 (1963). [25J J. Moser. "On Harnack's Theorem for Elliptic Differential Equations." Comm. Pure Appl. Math. 14 (1961). [26] A. Nagel, E. M. Stein, and S. Wainger. "Differentiation in Lacunary Directions." Proc. Nat. Acad. Sci. 75 (1978). [27J E. M. Stein. Singular Integrals and Differentiability Properties ofFunctions. Princeton University Press, 1970. (28] 1. Stromberg. "Weak Estimates on Maximal Functions with Rectangles in Certain Directions." Arkiv. Math. 15 (1977).

10 Function Spaces on Spaces of Homogeneous Type Yongsheng Han and Guido Weiss

1 INTRODUCTION There are many topological vector spaces that arise naturally in Analysis. In the theory of partial differential equations and in harmonic analysis these spaces con­ sist of functions, or, more generally, distributions defined on the n-dimensional real Euclidean space ]R". Some of the best known examples are the Lebesgue spaces LP(]R"), I S p S 00, the Lipschitz spaces A a , 0 < CL < 1, the Hardy spaces H P, and the Sobolev spaces L~, CL > 0, I S p < 00. These are examples of a scale of spaces that are naturally defined by a method known . as Littlewood-Paley-Stein Theory. Our purpose is to describe how this theory can be extended so that this scale of spaces can be introduced in the gen­ eral setting of spaces of homogeneous type. There are, in fact, two scales: the Besov spaces and the Triebel-Lizorkin spaces. The former includes the Lipschitz spaces and the latter includes the Hardy, Lebesgue, and Sobolev spaces. We shall not present here a detailed description of the Littlewood-Paley theory in its various classical settings. We refer the reader to [FJW) for a presentation of this subject that is appropriate for our purposes. Another account of this theory that, in addition, explains E. M. Stein's considerable contribution to this subject can be found in [eWl). In order to explain our program, however, we do have to present those aspects of the classical Littlewood-Paley theory associated with ]R" that are needed for the extension of the spaces in question in the setting of spaces of homogeneous type.

212

CHAPTER 10

We begin with a version of the Calder6n reproducing formula. It is clear that we can construct a function ({J E S satisfying (1.1)

I~(~)I ~ c > 0

if

3/5 ~ I~I ~ 5/3.

The result that we call the Calderon reproducing fonnula then asserts that we can find two such functions ({J and 1/1 in S such that (1.2)

L 1/1k * iPko *f,

f =

kEZ

where 1/Ik(x) = 2kn 1/l(2kx), ((Jk(x) = 2kn ({J(2 kx) and iP(x) ~ ({J(-x) (see [PI]). The convergence of the series can be taken in several senses: for example, in L2 and, much more generally, S' /P (that is, the space of tempered distributions modulo the polynomials). For each dyadic cube Q of side length l(Q) = 2- k , Q

= Qkj =

{x E IR

n

:

where k E Z and j = (jl,

Xi

~

rk(jj

h. "', jn)

E

zn, let

2-

k

ji

({JQ

~

=

+

I),

The Triebel-Lizorkin space

i

= 1,2, ... ,n} ,

P;,q consists of all elements

~ ~[fu (2"1~, *

IIf11!;"

(1.5)

Supp~ C {~: 1/2 ~ I~I ~ 2},

213

FUNcnONSPACES ON SPACES OF HOMOGENEOUS TYPE

f in S'I'P such that

fI)'rl

< co.

The "dot" denotes the fact that we are considering the homogeneous version of these spaces. It is most important to show that these definitions are independent of the choice of the function ({J E S satisfying (1.1). This independence follows from the Calderon reproducing fonnula. To see that (1.4) defines the Besov space i3;.q independently of the choice of ({J, it suffices to show that there exists a constant C such that

(2ka lllh * flip) [L kEZ

(1.6)

qJl/q

~

C

[

L (2ka ll({Jk * flip)

q qJ 1/

kEZ

whenever 11 is another function satisfying (1.1). For simplicity we show this when -1 < a < 1 and 1 < p, q < 00. By interchanging the roles of ({J and iP in (1.2), however, we can find 1/1 satisfying (1.1) such that (1.7)

11k

2kn / 2 ({J(2 kx - j)

* f = 11k * L

iEZ

1/1 i

* ({Ji * f.

Thus, and, similarly,

1/1 Q = 2kn / 2 1/1(2kx

lI11k

- j).

iEZ

f =

L (j, ({JQ}1/I Q'

1('Ik

(1.8)

11

QEQ

where {g, h} = g(ii) when g E S' and h E S (this is the ordinary L 2 inner product when g E L 2 ), and Q denotes the collection of all dyadic cubes. The relatively simple argument establishing (1.3) is an extension of one that gives us the Shannon Sampling Theorem and can be found in [FJW]. The function ({J and the coefficients (j, ({J Q) can be used to characterize the spaces that are of interest to us. For the moment we shall restrict ourselves to the homogeneous versions of the Besov and Triebel-Lizorkin spaces. For a E IR, p "f: 00,0 < p, q ~ 00, the Besov space i3;.q consists of all elements f in S'I'P such that (1.4)

IIf11 n;,· =

[L (

2ka ll({Jk

kEl

* flip

rJ

II 11k * 1/I i

* ({Ji * flip ~

L lI11k *

where a vb

2- k

* .1,

V

< 00.

2- i

)(x)1 < crlk-il ~ _ _...,.. 'Yi (2-k V 2- i + Ixj)n+l'

= max{a, b}. This estimate clearly implies lI11k

* 1/I i lll

~ c r 1k ­ il

and, consequently, the left side of (1.6) does not exceed

c{ L(L kEl

2karlk-illl({Ji

* fIIp)q},/q

iEZ

= C{L(L {2iall({Ji * flip kel

[2(k-i l a- 1k-i l ]'/q}

iEl

1q /

1/Iilllll({Ji

iEZ

An easy calculation shows that

Then it is not hard to show, once (1.2) is established, that (1.3)

* flip ~ L

{2(k-i)a- 1k- i l} I/q,) q } I/q

* flip·

214

CHAPTER 10

S C{L{L2iaqllrpi kEZ

* fIIP(k-Oa-1k-i l }

and

lEZ

{L 2(k-i)a-lk-il }q/q,}

I s llj;,q ==

I/q

11(~[IQI-;;-tISQIXQry/q

L 2iaq

IIrpi

iEZ

= C

iL

* fII~ L 2(k-Oa-lk- i l }

The following theorem gives us this characterization (see [FJW]):

I/q

Theorem (1.9). Suppose a E R 0 < p, q S 00, and the functions rp and 1/1 are as in (1.1) and (1.2). Given f E S' /P let S~ = (f, rp Q} ,for each dyadic

kEZ

ziaqllrPt

* fII~

fEZ

}

I/q

cube Q (so that equality (1.3) holds). Then f E B;,q if and only if the sequence = {sQ} E b~,q;and f E P;,q if and only if the sequence s = {sQ} E i;,q,

,

S

where (1/q) + (l/q') = 1. HOlder's inequality was used to establish the first inequality, and C denotes a constant (depending on a and q) that varies appropriately. Similarly, the finiteness of the expression (1.5) that defines the norms for the Triebel-Lizorkin spaces is independent of the choice of the test function rp satis­ fying (1.1). Again, this can be shown with the aid of the Calderon reproducing formula: By (1.7) and (1.8) we have the pointwise estimate l17 k

* f(x)1

S C

L r1k-iIM(rpi * f)(x), iEZ

where M is the Hardy-Littlewood maximal function operator. Thus,

111~C2'"'",. fl)'

rt "C III~(~ 2'"T'HMC_,' fl)'j

11'11,

But, applying first HOlder's inequality and the assumption -1 < a < 1 and then the Fefferman-Stein vector valued maximal function inequality [FS] (valid here since 1 < p, q < 00), we see that the last expression is dominated by

c

III ~

(2'" MC_, •

f))f'll, "C III ~(2'" 1_, •f1lf'll,

The equivalence of the norms defined in terms of 17 and rp clearly follows from these inequalities. Let us now return to equality (1.3). We have stated that the coefficients sQ = (f, rp Q} can be used to characterize the spaces we have introduced. Toward this end we introduce the spaces b~,q and i;,q of sequences S = {sQ} indexed by the dyadic cubes Q and satisfying

Ilsllb;,q ==

{

L L (

kEl.

< 00.

P

iEZ

S C{

215

FUNCTION SPACES ON SPACES OF HOMOGENEOUS TYPE

i( Q)=2-'

I

[

a

I

IQIP -;; - ilsQI

JP)q/P}I/q

< 00

provided p <

00.

When this is the case we have the norm equivalences

II s II jU,q Jp

;::::

II fII F"q p

and

lis II ba ,q p

;::::

II fII Ba ,., p

There are several reasons why these spaces are "natural" and why Theorem (1.9) is useful. The space L 2 is rather special in many ways; for example, it is endowed with an inner product with respect to which the Fourier transform is a unitary operator, and the bounded convolution operators on it are characterized by the fact that their kernels have bounded Fourier transforms. These properties can be used to give rather simple proofs of the boundedness of many operators on L 2 • While these properties are no longer true for LP, P # 2, several of these boundedness results still hold when 1 < p < 00, but they fail to be true for either L I or LOa. There are, however, appropriate replacements for these two spaces on which such boundedness properties are valid. In the one dimensional case, the Hilbert transform provides an example of this situation, where the Hardy space H I replaces L I and BMO replaces LOa. The scale of spaces P;,q includes the spaces LP for 1 < p < 00 (p~,2 is LP in this case) and it also includes HI (= p~,2) and BMO (= p~2). In addition to this the Hilbert space techniques that simplify these problems in the case p = 2 have somewhat successful extensions to the scale of spaces we are describing. This is a consequence of Theorem (1.9) and the "almost orthogonal" expansion (1.3). In a well defined sense the operators of interest correspond to matricial operators on the sequence spaces b~,q and i;,q that are "almost diagonal" (see [FJW] and [T] where many of these features of the Triebel-Lizorkin and Besov spaces are discussed). As stated at the beginning, our purpose is to describe how one can define Triebel­ Lizorkin and Besov spaces in the setting of spaces of homogeneous type. We are now in a position to be more specific about our goal. In this general setting we do not have an underlying group that gives us a convolution which can be used in the definitions of the norms given in (1.4) and (1.5). The lack of a convolution structure is also an obstacle for obtaining the reproducing formula (1.2). Nevertheless, the

216

CHAPTER 10

original motivation for the introduction of spaces of homogeneous type was to introduce a most general setting in which the ideas of hannonic analysis can be extended. Moreover, many important convolution operators, such as the Calderon­ Zygmund Singular Integral Operators, have been extended to operators defined by more general kernels, such as the Calderon-Zygmund Operators (see [CM] and [J]). Thus, it is natural to try to extend more general operators and their properties when the underlying space is of homogeneous type. Such extensions can be found in the literature (see [DJS], for example); however, these results do not include the scale of spaces that we are considering. Very recently considerable progress has been made in this dire~tion. In this paper we describe this effort. We shall not prove all the results we announce; however, we will give complete references of the work now in progress. This paper should be considered to be a companion to [HS], where the details that are not contained here are given. On the other hand, much of the motivation for the material developed in [HS] is presented here.

2 MOTIVATION FOR A CALDERON REPRODUCING FORMULA ON SPACES OF HOMOGENEOUS TYPE

A quasi-metric on a set X is a function p: X x X ---+ JR+

=

{r E JR: r ::: O}

satisfying

+

p(z, y)] for some K < 00 and all x, y, and z in X.

That is, a quasi-metric satisfies the properties of a metric except that the triangle inequality is replaced by the more general condition (c) (clearly K must be at least 1 if X contains more than one point). A space of homogeneous type (which we will denote as an HT space) consists of a measure space X that, in addition to a measure J-L, is endowed with a quasi-metric p; moreover, J-L and p are related in the following way: > 0, where B(x, r) = (y X: p(x, y) < r}, (ii) there exists a positive constant c such that for all x E X and r > 0 (i)

J-L(B(x, r)) < 00 for all x E X and r

reproducing formula that has the same basic features as the one involved in equality (1.2), that does not depend on a convolution. We claim that this can be done in JRn by using certain families of Calder6n-Zygmund operators. We begin by describing a version of this program that can be found in [HITW]. In this context the operators we consider are linear and map the space of test functions V(JRn ) continuously into the space of distributions V'(JRn). By the Schwartz kernel theorem each such operator T is associated with a distribution

K

E

V' (JRn x JRn) such that (Te, 1J)

= (K,

1J 0 e),

for all e, 1J E V. K is called a Calder6n-Zygmund kernel if its restriction to the set of all (x, y) E JRn x JRn such that x i- y is a continuous function K (x, y) that satisfies: 1

(2.1)

IK(x, y)1 ~ c

(2.2)

IK(x, y) - K(x, y')1 ~ c

I

(2.3)

lK(x, y) - K(x ' , y)1 ~ c

I

for all x

Ix - yin Iy - y'l' x-y

.

,_

Ix - xii' x-y

., _

i-

y,

whenever21y - Y'l ~ Ix - YI, whenever21x - xii ~ Ix - YI,

for some constant c > 0 and E E (0, 1]. In this case T is called a Calder6n­ Zygmund operator and we write T E C Z 0 or, if we wish to emphasize the "Lipschitz condition of order E," we write T E CZO(E). If T E CZO, e E V,

(a) p(x, y) = 0 if and only if x = y, (b) p(x, y) = p(y, x) for all x, y E X, (c) p(x, y) ~ K[p(x, z)

217

FUNcnONSPACES ON SPACES OF HOMOGENEOUS TYPE

and x ¢ Supp

e, then Te(x)

(2.4)

(2.5)

If we want to introduce norms that are analogous to the ones in (1.4) and (1.5), we must find appropriate substitutes for the convolution operators f ---+ ({Jk f. Moreover, in order to develop the theory of the spaces defined by these new norms along the lines we indicated above, we also need a version of the Calderon

*

JR'

K(x, y)(j(y)dy.

The family of operators that can be used to give a more general Calderon repro­ ducing formula consists of such integral operators, where the above equality holds without the support restriction. We call such a family, {SklkEZ, an approximation to the identity. It consists of a sequence of integral operators with kernels Sk(X, y), k E Z, satisfying the following conditions:

E

J-L(B(x, 2r)) ~ cJ-L(B(x, r)).

= {

ISk(X, y)1

2-kf ~ c (2- k + Ix _ yl)n+, ,

ISk(x, y) - Sk(X ' , y)1 :::: c (2-k

for Ix - xii ~ ~ (2-k

+

+

2-

k '

Ix _ yl)n+'

E

JRn;

[ I X - xii ]' 2- k + Ix - yl

Ix - yl);

2-kf

(2.6)

for all x, y

ISk(x, y) - Sk(X, Y')I :::: c (2-k

+ Ix _

YI)n+,

[2-

Iy - y k

+

'I

Ix ­ yl

]'

-----------

._--,_...__.-.-_._--------,_._....__._­

218

CHAPTER

for Iy - y'l

::s

(2.7)

! (2-k + Ix -

(2.8)

yi);

{

Sk(X, y)dy = 1 for all x

{

Sk(X, y)dx = 1 for all y E JRn and k E Z.

JIRn JRn

10

E

JRn and k

E

Z;

Christ and Ioume [CI] considered families such as these that satisfied conditions (2.4) and (2.5). When the last three properties are added one can use (Ski to obtain norms that are equival<:nt to the ones given in (1.5) and (1.6). More precisely, the following result is shown in [HJTW]: Theorem (2.9). Suppose that {Sk}. k E Z, is an approximation to the identity and Dk = Sk - Sk-l· Thenfor -E < ex < E and I ::s p, q ::s 00, l/q (2.10) IlfIlB;'. ~ [ L(2ka IlDdllp)q ]

FUNCTION SPACES ON SPACES OF HOMOGENEOUS TYPE

The operators R N , on the other hand, belong to CZO(E') for 0 < E' < E and their kernels RN (x, y) satisfy the following conditions: There exist e, .5 > 0 such that

IRN(X, y)1

(i)

IRN(x, y) - RN(x', y)1

11111;.;'

~ II[fu(Z""Ddll'J'l

jRN(x, y) - RN(x, y')1

l/q

::s

e { L(2kaIlDdllp)q kEZ

.

}

We can do this by making use of an idea due to R. R. Coifman. He observed that it follows from the properties (2.4)-(2.7) that (2.13)

1= LDk. kEZ

From this equality (which is easily established when, say, the operators involved are applied to test functions), we obtain

=

{L Dk } {L Dt} kEZ

=

eEZ

L DkDt Ik-lI>N

+

Ik-el~N

where N is a fixed positive integer. In [DIS] it is shown that the operators TN are bounded on L 2 (R n ) and converge to I, as N -+ 00, in the strong operator topology. Moreover, it is shown in [HITW] that I/q

(2.15)

IITNfIIB;.

::s

e { L(2kaIlDdllp)q } kEZ

x

,

1

I ::s 2: Ix - YI;

Iy - y'l

I

::s 2: 'x

- YI;

moreover, these operators satisfy the following version of the weak boundedness property

I(RNf,

g)1 :s cr N8 t n(lIf11oo + tliV flloo)(llglioo + tllVgll oo )

RN I

= R~ I = 0

t

> 0, as well as

modulo constants.

We can then apply the "Tl theorem" for Besov and Triebel-Lizorkin spaces that is proved in [FTW], [T], or [HITW] to conclude that (2.16)

IIR N fIIB pu,. -< er N8 I1fIlB pu,.,

(2.17)

IIRNfllpu,. p

:s e2- N8 I1f11pu,., p

when -ex < E < ex and I ::s p, q ::s 00. These estimates applied to TNl (/ - RN) -I = L~=o R~ show that this operator is bounded on iJ~·q and on i;·q for these parameters ex, p, and q. We thus obtain another Calderon-type reproducing formula:

DkDe == R N + TN,

L

Ix -

_ yl-(n+€')

er N8 1Y - y'j€'lx _ yl-(n+€') for

(2.18) (2.14) I

::s

(iii)

(v)

IIf11B;'.

er N8 1x - x'I€'lx

for all f, g E V having support in a cube Q whose diameter is the David-Iourne condition

The role played by a more general Calderon reproducing formula in establishing these equivalences is the following. Suppose we want to show that (2.12)

::s

for

(iv)

(2.11)

::s er N8 1x _ yl-n;

(ii)

kEZ

and

219

f = TN1TNf = TN 1 LDfDd, kEZ

where Df = LIiI~N Dk+j (so that TN = Llk-e\~N DeDk = LkEZ Df Dk). We can now give a simple proof of (2.12): Since TN is bounded on iJ~·q ,

IIfIlB;.

= II TN 1 TN fIIB;'. ':S ell TN fIIB;·. ::s

ka e [ L(2 II Ddllp)q ] kEZ

. where the last inequality follows from (2.15).

I/q

,

220

CHAPTER 10

The inequality in the opposite direction to the one in (2.12), (2.19) {

L(2

ka

kE'Z

l/q

llDk/ll p )q }

::5 cllfllB~,q,

is easier to establish. In fact, it can be done by exactly the same argument used for proving (1.6) by replacing the operator g --+ 1]k g with the operator g --+ Dkg. All that is needed is to observe that inequality (l.8) remains valid if 1]k * 1/1i is replaced by D k 1/1 i" This, then, gives us (2.10). Completely analogous estimates give us (2. I I). We are now ready to extend these ideas and results to HT spaces.

*

Our task, then, is to carry out the program described in the previous two sections. We want to establish an analog of the Calderon reproducing formulas described in §1 and in §2. Moreover, we have to show how this analog can be used to introduce a class of Besov and Triebel-Lizorkin spaces associated with HT spaces. Guided by the material in §2 the definition the spaces iJ;,q and p;.q should be, simply, that these are the spaces of all f such that (3.2)

IIfllB~q == [ L(2ka llDk/ll p )q ]

When working on HT spaces it is often quite useful to introduce the measure distance m(x, y), equal to the measure of the smallest ball containing x and y. Without loss of great generality we can assume that m (x, y) is the quasi-metric metric p (x, y) and that it satisfies a Lipschitz smoothness condition !p(x, y) - p(x' , y)1 ::5 cp(x, x,)p[p(x, y)

+ p(x ' , y)]1-tl

(see [CW2] and [MS]). For example, replacing the usual Euclidean distance in n lR by its nth power gives us such a measure distance which does not change the topology of the underlying space. The estimates we shall present require a smoothness property of this type (though much of what we present is also true when the HT space is discrete) and their fonn and derivations are simpler if we are dealing with a measure distance. There are many ways of introducing approximations to the identity in HT spaces.

A construction by Coifman gives us a "continuous approximate identity"

Prf(x) =

l

Pr(X, y)f(y)dJ-t(y),

where Pr(x, y) = 0 for p(x, y) > cr, Pr(x, y) = Prey, x), IPr(x, y) Pr(x, y')1 ::5 p(y, y')p jrl+fJ , 0 ::5 Pr(x, y) ::5 ~, and f Pr(x, y)dJ-t(y) = 1. It can be easily shown that lim,....o Pr f = fin L 2(X) (but we shall be interested in other types of convergence, which will be discussed in a moment). Let the "vari­ ation" of Pr, Dr, be defined by either Dr f = (Pr - Pr/2) f or by Dr f = r ~ Pr f. A discrete version is then obtained by letting r = 2- k , k E Z. Then, as in (2.13), we have (3.1)

(3.3)

< 00

11/11;;·'

=~ [~(2kaIDdl)'

rl

<

00.

We must, however, give meaning to these inequalities. In a general HT space we do not have a theory of distributions; thus, it is not clear what we mean by f. Moreover, we do not have a class of test functions available. Finally, and most importantly, we must show that the definition of these spaces iJ;·q and P;,q we have just given in terms of the norms (3.2) and (3.3) is independent of the choice of the family {Dk}. In this section we give the properties of the class of operator families {Dk} that can be used in these definitions and describe in detail the program that we are following. As stated before, the technical details are contained in the companion paper [HS]. We begin by introducing the classes of test functions that are needed for our purposes. All this is in strong analogy with the theory of Hardy spaces associ­ ated with HT spaces that we developed in [CW2]. There the test functions were elements of Lipschitz spaces that turned out to be the duals of the Hardy spaces. In the present situation the test functions are "localized smooth molecules." We say that a function f on X is a strong smooth molecule of type (f3, y), centered at Xo EX, and having width d > 0 if and only if (i)

(ii)

dY

!f(x)1 ::5 c (d If(x) - f(x ' )! ::5 c { d

(iii)

l

+ p(x, XO»l+y

P (x, x I)}P + p(x, xo) [d

for

f= LDk/. kE'Z

l/q

kE'Z

and

3 THE CALDERON REPRODUCING FORMULA ON SPACES OF HOMOGENEOUS TYPE

221

FUNCTION SPACES ON SPACES OF HOMOGENEOUS TYPE

p (x, x') ::5

f(x)dJL(x) = O.

d

,

dY

+ p(x, xo)]l+y + p(x, xo) 2K

,

222

CHAPTER 10

Denote by !JJ1(P,Y) (xo, d) the collection of all such molecules and let the "norm" of such an f be the infimum of all constants c for which (i) and (ii) hold. If we choose a fixed Xo EX, then the space !JJ1(P, y) (xo, d) is our space of test functions. The theory of Calderon-Zygmund operators has been extended to HT spaces by several authors. (For a good account of this see the monograph by Christ [Ch].) In [HS] it is shown that they are bounded on the spaces !JJ1(P,Y), where the indices f3 and y are adapted to the degree of smoothness of the kernels of these operators. The analog of (2.18) can be derived from the following result in [HS]:

223

FUNCTION SPACES ON SPACES OF HOMOGENEOUS TYPE

Theorem (3.5). Suppose {Sd, k E Z. is an approximation to the identity on a space of homogeneoul type and Dk = Sk - Sk-l' Then there exist families of operators

{Dd and {Dd. k

E

Z, such that f = :LDkDd kEZ

and

f = LDJjd, kEZ

Theorem (3.4). Suppose {Sd, k E Z, is an approximation to the identity on an HT space and Dk = Sk - Sk-I. Set TN = LkEZ Df Db where Df = Llil:SN Dk+ j and N is a positive integer. Thenfor N large enough, TN l is a Calder6n-Zygmund operator, TN! (1) = (TN !)*(1) = 0 and TN! has the weak boundedness property.

where these series converge in the topologies of !JJ1(P,y). Moreover, the kernels

In the setting of JRn this theorem follows from results of David and Journ6 [DJ] and techniques described to us by Tchamitchian that involve aspects of the theory of wavelets that are not available to us for the HT space case. The method used in [HS] parallels the approach we described in §2 (see the estimates (i), (ii), (iii), (iv), and (v) that follow inequality (2.15)). More precisely, Theorem (3.4) is obtained from the following estimates on the operators R N = L\ll>N LkEZ De+kDk: if o < € ' < € then there exists a constant c such that for all m = 1, 2, 3, ...,

_ [ p(x x') ] (ii) IDk(x, y) - Dk(x', y)1 ::: c 2-k + ~(x, y)

IR~(x, y)1 ::: cmTN~mp(x, y)-l;

(ii)

IR~(x, y) - R~(x', y)1 ::: cmTN~mp(x, X')f'p(X, y)-O+f')

(iii)

(iv)

p(x, y) p(x, x') ::: ~;

I (R~f, g} I :::

p (y, y'):::

p(x, y) 2K

cmT N8m /l-(B(xo, r»

for all f and g having support within B(xo, r), with Lipschitz norms not exceeding r-T/ and L oo norms bounded by 1;

= (R~)*(1) = 0, where R~ (x, y) is the kernel of R M, fJ > 0 and 0 < TJ < 1. These estimates and the fact that TN 1 = L~=o R'N are the basic tools needed to

v

IDk(x, y)1 ::: c (2- k

(i)

+ p(x, y»l+f'

fior (iii)

l

Dk(X, y)d/l-(x)

=0

p (x, x

')

for all

f'

; k '

2- f

(2-k + p(x, y))l+f'

< 2-

k

-

+ p(x, y) 2K

y E X and k E Z

replaced by Dk(X, y') and p(x, x') is replaced by p(y, y').

IR~(x, y) - R~(x, y')1 ::: cmTN~mp(y, y,/p(x, y)-(l+f')

for

2- kf '

_

The kernel}, Dk(X, y) satisfy the same conditions except that in (ii) Dk(x', y) is

(i)

for

Dk (x, y) of these operators satisfy

R~(l)

prove Theorem (3.4). We can now announce a version of the Calderon reproducing formula for HT spaces:

These are the basic definitions and results that are needed to carry out the known program on JRn to the new setting of HT spaces. As mentioned above, the details are carried out in [HS]. There one can also find the atomic and molecular de­ compositions of these general Besov and Triebel-Lizorkin spaces. The theory of interpolation of operators acting on these spaces is also extended to this general setting. We hope that this exposition furnishes sufficient motivation to make it easier to go through the rather complicated technical details that are needed for the thorough presentation contained in [HS]. University ofWindsor, Windsor, Ontario Washington University, St. Louis

REFERENCES [Ch) M. Christ. Lectures on singular integral operators. CBMS Regional Conference Series, Number 77 (1991), 1-132.

----------_..

224

-_._.

__

._-

,.~~~~;;;-~-~

CHAPTER 10

[CW] M. Christ and J.-L. Journe. "A boundedness criterion for generalized Calderon­ Zygmund operators." Acta Math. 159 (1987),51-80. [C] R. R. Coifman. "Multiresolution analysis in non-homogeneous media." In Wavelets. Time-Frequency Methods and Phase Space. Combes, Grossman, and Tchamitchian, eds. Springer Verlag, 1989. [CM] R. R. Coifman and Y. Meyer. "Au dela des operateurs pseudo-differentiels." Asterisque 57 (1978), 1-185. [CWI] R. R. Coifman and Guido Weiss. "Littlewood-Paley and Multiplier Theory, by Edwards and Gaudry." Bull. Amer. Math. Soc. 84, Number 2 (1978), 242-50. [CW2] R. R. Coifman and Guido Weiss. "Extensions of Hardy spaces and their use in analysis." Bull. Amer. Math. Soc. 83 (1977), 569--645. [DJS] G. David, J.-L. Journe, and S. Semmes. "Calderon-Zygmund operators, para­ accretive functions and interpolation." Rev. Mat. Iber. I (1985), I-56. [FJ] M. Frazier and B. Jawerth. "A discrete transform and decomposition ofdistribution spaces." J. Funct. Anal. 93 (1990), 34-170. [FJW] M. Frazier, B. Jawerth, and Guido Weiss. Littlewood-Paley theory and the study of function spaces. CBMS Regional Conference Series, Number 79 (1991), 1-132. [FS] C. Fefferman and E. M. Stein. "H P spaces of several variables." Acta Math. 129 (1972),137-193. [HITW] Y.-S. Han, B. Jawerth, M. Taib1eson, and Guido Weiss. "Littlewood-Paley theory and €-families of operators." Colloq. Math. LX/LXI (1990), 321-59. [HS] Y.-S. Han. and E. T. Sawyer. "Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces." Memoirs of the AMS lIO (July 1994), 1-126. [J] J.-L. Journe. Calder6n-Zygmund operators, pseudo-differential operators and the Cauchy integral of Calder6n. Lecture Notes in Mathematics, no. 994. Springer Verlag, 1983. [MS] R. Macias and C. Segovia. "Lipschitz functions on spaces of homogeneous type." Adv. Math. 33 (1979), 257-70. [T] R. Torres. "Boundedness results for operators with singular kernels on distribution spaces." Mem. Arner. Math. Soc. 442 (1991),1-172.

11

The First Nodal· Set of a Convex Domain David Jerison*

INTRODUCTION Our pwpose in this paper is to prove that the first nodal set of a long, thin convex domain touches the boundary. Let B(z, r) = {y E Rn

inradius(Q)

= max{r:

:

Iy - zl

< r}

B(z, r) C Q for some z}.

Theorem. Let Q be a convex, open subset of R n . There is a dimensional constant C such that if diameter(Q)/ inradius(Q) ~ C, then the nodal setfor the second Dirichlet eigenfunctionfor Q touches the boundary. In other words, if /).u = -AU in Q, u = 0 on aQ, and A is the second eigenvalue for the Dirichlet problem in Q, then A = {z E Q: u(z) = O} satisfies A n aQ =I- 0. In the case n = 2, An aQ consists of exactly two points.

This theorem was announced in [J] in the case of planar domains. A few months later, A. Melas [M] proved the result for all smooth convex planar domains, not just ones with large eccentricity. (He also showed by a limiting argument that even if the domain is not smooth, the nodal line touches the boundary in at least one point.) Our method of proof is much more crude, but it has some advantages. First of all, it applies in higher dimensions, as we show here. Secondly, it applies 'This research was supported in part by a Presidential Young Investigator Award and National Science Foundation grants DMS-9106507 at the Massachusetts Institute of Technology and DMS­ 9100383 at the Institute for Advanced Study.

v

-

=

.

~

~

;

.

~

,

=

~

=

=

-

=

.

;

.

=

;

.

;

;

;

~

~

_,•. ;;.;;-~.;";:~,;;;;;;,,,;,~.~.;;;=~;~,;;.;.;.:..;.;;...~-,;.~~~;~~~~~;;~~,.;:;,;;.._

·•

226

CHAPTER 11

directly to all convex domains, with no smoothness hypothesis. Finally, it is more quantitative than the one given by Melas, in that it gives some indication as to where the nodal set is. We hope to return to this in the future. L. E. Payne made the conjecture that the first nodal line touches the boundary for any planar domain [P]. S.-T. Yau made the conjecture for all convex planar domains [Y]. The best previous results on the first nodal line are as follows. L. E. Payne [PI] proved that the nodal line touches the boundary of a convex domain in the plane which is symmetric under a reflection. This result was extended by C.-S. Lin [L] to the case of conveJLdomains which are symmetric with respect to a rational rotation instead of a reflection. A basic notion underlying our approach is that eigenfunctions behave like har­ monic functions. This was motivated in part by results of [KP]. Another result that motivated this work is the well-known lower bound for the first eigenvalue in terms of the inradius due to W. K. Hayman [H]. We would like to thank Carlos Kenig for pointing out the generalized maximum principle (Proposition I). The use of this maximum principle simplified our original proof which was based on comparisons with harmonic functions using a Neumann series. We would also like to thank Charles Fefferman for pointing out an error in Remark 2 of the original manuscript.

OUTLINE OF THE PROOF F. John's theorem [dG, p. 139] says that any convex domain is comparable to a rectangle. More precisely, there is a dimensional constant C 1 such that Q can be rotated and translated so that (1)

{Z E R n : IZkl

:s

L k}

c

Q C {Z ERn: IZkl

:s

C1Ld.

We may assume without loss of generality that the sidelengths Lk are increasing with k. After a dilation we may assume that L I = I. Let n = p + q. We will consistently split the variables into Z = (x, y) E RP x Rq. The theorem follows from Claim. For any M there exists N (depending on M and n) such that if L P and L p+ I ~ N. then the nodal set for Q touches the boundary.

:s

M

We prove by induction that the claim implies the theorem. Let N I be the value given in the claim for M = I. If L 2 ~ N I, then the nodal set touches. If not, then since L 2 < N I we can choose N2 according to the proposition corresponding to M = N I • If L 3 ~ N2, then the nodal set touches. If not, then we may continue in this fashion until finally the nodal set touches the boundary provided L n ~ N n- I .

o~~_".-~;,_c_-",;,.;;;.",~=._.,~~

.. ,,~~.~;-";;;","·~;;~,,"~ .. ~<-;,.,.;-.,

~~

_.~.~"=.~~~,,,_~~,"

__=.'._~_,~=·~~ •.__ ~<__ .'~'-'~"O='~"~~_,-~~~~._~="~,~-~.",-.'~~;~""~.~"-~~~~'"--' ",,_._~'~-'''~-'-'-'-''---'''---

THE FIRST NODAL SET OF A CONVEX DOMAIN

227

Because Nn-] is a dimensional constant and because Ln = Lnl L] is comparable to the ratio of the diameter to the inradius of Q. the theorem is proved. We will prove the claim by contradiction. Assume that the nodal set does not touch the boundary. Denote Q- = {z E Q: u(z) < O} and Q+ = {z E Q: u(z) > OJ. Without loss of generality we may assume that Q- is the component that is relatively compact in Q. Denote u_(z) = -u(z) for z E Q_ and u_(z) = 0 elsewhere. Denote u+(z) = u(z) for z E Q+ and u+(z) = 0 elsewhere. Denote Q(y) = {x E RP: (x, y) E Q}, Q_(y) = {x E RP: (x. y) E Q_}, and similarly for Q+(y). For any domain DIet )\.j (D) denote the first eigenvalue of D, and let ).dD) denote the second eigenvalue. Let

AQ

= yERq inf A](Q(y)).

The proof goes as follows: 1. When N is large, A2(Q) is very close to AO (Lemma 1). 2. It follows from Step 1 that for "most" points y in a large cube Q', Al (Q- (y)) is very close to Al (Q (y)) and AQ (Lemma 4). 3. It follows from Step 2 that Q_(y) is "most" of Q(y), that is, the Lebesgue measure, m(Q+(y)), of Q+(y) is very small (Lemma 3). In other words, Q+ is very thin in some places. 4. By the Hopf lemma (Lemma 5) and Carleson lemma (Proposition 3) max u_ dominates max u+ on the thin portion of Q+. (See (12).) 5. u+(z) decays exponentially as z approaches the center of the thin portion of Q+. On the other hand, because we have assumed that Q- is relatively compact in Q, Q_(y) is relatively compact in Q(y). In particular, Q+(y) is nonempty, and, using the Hopflemma again, max u + on the central part of the thin portion of Q+ dominates max u_ nearby. Thus we have come around in a circle, and max u_ on a certain subset is bounded by a factor times itself. The exponential decay (the factor 2- w for some c > 0) dominates all of the earlier bounds (which grow at most as fast as powers of N) so that for large N the factor is less than 1, and we obtain a contradiction. The reader may also wish to consult the paper [J] for the very similar argument in the case n = 2. The main differences between the two-dimensional argument and the one in higher dimensions are these. In R 2 , Step 3 is trivial because the cross-sections Q (y) are one-dimensional. In Rn. n ~ 3, Step 4 requires an extra technical lemma (Lemma 6) to show that very thin filaments of Q- do not prevent the control of max u+ by max u_ via the Hopf lemma and Carleson lemma. The rest of the paper consists of one long section giving the details of the argument for n ~ 2. Then in the final section we indicate the modifications

228

CHAPTER 11

needed to prove that when n points.

=

2 the nodal line touches the boundary in two

Throughout the proof we will use the convention that the constants c > 0, and C < 00 are either dimensional constants or depend on dimension and M and may vary from line to line. Lower case c will be used for lower bound constants and upper case C for upper bounds. The numbered constants c\, C2, CI, etc., do not change from line to line, which should help the reader to follow their interdependence.

THE PROOF

Lemma 1. There is a constant C2 depending only on n such that the second eigenvalue A = A2(0) satisfies

+ C2 N- 2/ 3 .

Ao S A SAo

229

THE FIRST NODAL SET OF A CONVEX DOMAIN

Suppose that u and ware continuous in D. Finally, suppose that u > 0 in D and w > 0 in D. Then

max u/w S max u/w.

D

aD

To prove this, observe that v = u/w satisfies the inequality li.v + 2V log w . V v 2: O. The inequality then follows from the usual maximum principle. (See [PW].)

Remark 1. Let B = B(x, r), a ball of radius r S 1. The boundary problem li.u + u = 0 in B and u = f on aB can be solved for every continuous function f and

= cn(r) a(;B)

u(x)

iB

fda,

and cn(r) S 2. (Here, and elsewhere s will denote surface measure.)

Proof It is easy to see that Al (0) 2: Ao. In fact, if v is the first eigenfunction for

o then

In{ IVvl

2

2: {

(

JRq In(y)

2: Ao {

(

IVyv(x, y)1 2dxdy

JRq In(y)

= Ao

l

v

2

Iv(x, y)!2dxdy

•

Moreover, A 2: Al (0), so the lower bound follows. Choose D = O(y) so that AI(D) SAo. By (1), (0, y') E 0 provided IY~I S

N, k = I, ... , q. Therefore, by convexity, «(1 - N-2/3)x, (1 _ N-2/3)y +

N-2/3 y ') E 0 for all xED. In other words, D' x Q c 0 where D' = (1 _

2 N- /3)DandQ = (1-N- 2/3)y+{y': IY~I S N I/ 3},acubeofsidelength2N I /3. Separation of variables shows that A2(D' x Q) = Ao(1 _ N- 2/ 3)-2 + rr2(q + 3)(2N 1/3)-2. (1) also implies that at least one cross section O(y) contains a cube of sidelength 2 so thatAo S rr 2p/4,adimensionalconstant. ThusA2(D' x Q) S Ao + C2N-2/3 for some dimensional constant C 2. The min-max principle implies • that A == A2(0) S A2(D' x Q), so the lemma is proved. Let us recall the generalized maximum principle for solutions to eigenfunction equations. (See [PW] for a more detailed discussion.) ProPOSition 1 (Generalized Maximum Principle). Let h be a bounded function. Let u and w satisfy li.u + hu = 0 and Ii. w + h w S 0 in a bounded domain D.

Proof The existence of a solution follows from the generalized maximum prin­ ciple and the existence of a solution to the equation that is positive in the closed ball B. Such a solution exists for r S 1, and even for r less than the first zero of an appropriate Bessel function. We will need a few facts about Bessel functions later on anyway, so let us write down the solution in detail. Denote Fk(r) =

i

l

cos(sr)(1 - s 2l-I/2ds.

The relationship of Fk with the Bessel functions is 1k (r) = Ckrk Fk (r) for some constant Ck ([SW] pp. 153-154). If k = (n - 2)/2, then Bessel's equation implies (Ii. + I)Fk (lxl) = O. Because 1/2 < cos(t) S 1 for 0 S t S 1, we have Fk(O) S 2Fk(r) for all r, 0 S r S 1. In particular, Fk(r) > 0 for 0 S r S 1. Now, we can confirm the formula for u(x). By symmetry we see that u(x) is some constant multiple cn(r) of the average of f over the boundary. The constant can be evaluated using the radial solution Fk(!Z - xl) for k = (n - 2)/2: cn(r) = Fk(O)/ Fk(r) S 2.

•

Proposition 2 (Harnack Inequality). There is a dimensional constant C3 such that if 0 < r < 1/(max h)I/2, li.u + hu = 0 and u > 0 in B(x, 2r), then max u S C3 min u. B(x.r)

B(x,r)

(If h S 0, then the inequality holds for all r > 0.)

230

CHAPTER 1 I

Proposition 2 can be proved by using the Bessel function w(z) = F(n-2)/2(lz ­ xl) and the Harnack inequality due to Serrin [Se] for the function v = ulw. (See [PW].) In Lemma 3 below we will use Propositions 1 and 2 for a function h which is a constant plus the characteristic function of a ball. However in all other cases we will have h = A, a constant. In that case there is another way to prove Propositions 1 and 2, namely, to deduce them from their counterparts for parabolic equations using the observation that if (~ + A)U = 0, then the function e-Atu satisfies the heat equation (at - ~)(e-At u) = O. (For other applications of this idea, see [KP].) The parabolic analogues of Propositions 1 and 2 are well known. (See [PW].) Proposition 3 (Carleson lemma). Let ¢: R n - 1 -+ R satisfy the Lipschitz condi­ tion W¢I s M and ¢(O) = O. Denote D = {x E R n : Xn > ¢(Xl, ... , xn-d. Let 0 < r S 1/0.anduisapositivesolutionto(~+A)u = OinB(0,2r)nD such that u vanishes on B(O, 2r) n aD. Let z = (0, r). There is a constant C4 depending only on M and n such that max u S C4 u(z).

Proof Forn

Proposition 3 was proved by Carleson [C] and Hunt and Wheeden [HW] in the case of harmonic functions. (In that case, A = 0, and the upper bound on r is superfluous.) There are many subsequent versions. An examination of the proofs in [C] and [CFMS] shows that the result only depends on the scale-invariant Harnack inequality (Proposition 2) and a uniform HOlder continuity for solutions at the boundary, which is obtained by a majorization of solutions by an explicit supersolution (barrier) on a cone, that can be constructed using Bessel functions. We will not carry out the proof, since another one is available. But we will need the barriers in annuli given by the next lemma.

Lemma2. ForanyE > OandO < p < 1/4 there exists a function w.(r) defined for p S r S 2p satisfying (a) w.(r) > Ofor p S r S 2p, (b) (~ + l)w.(lxl) S Ofor p S Ixl s 2p (x ERn), (c) w.(r) = 1 + Efor r = 2p, (d) As E tends to zero, w. tends uniformly to afunction Wo such that

for some dimensional constant C.

3, define

and

awo

a;:- (p) sCI p

•

Then ~w.(lxl) = -( ~ - 1)2 Ixl -2 w.(lxl) and the properties listed are easy to check. In particular, w~ (p) = 2n / 2- 1I p log 2. For n = 2, define

w r = Fo(r) [lOg rip •( ) Fo(2p) log 2 Then (~+ l)w.(lxl) = for Ix I < 1 because

F~(lxl)/lxlFo(2p)

F~(r) =

-1

+

E] .

log 2. This last expression is negative

1

s(sinrs)(l - s2)-1/2ds sO

for 0 S r S 7r since the integrand is positive. Once again, the other properties are easy to check. For example, because Fo is decreasing on the interval under consideration, 2 (p) = Fo(p) < Fo(O) < ­ o Fo(2p)p log 2 - pFo(/2) log 2 - P

W'

The parabolic version ofthe Carleson lemma was proved by Salsa [S], and Propo­ sition 3 is an immediate consequence, as in the case of constant h for Propositions 1 and 2.

wo(2p) = 1

~

w.(r) = (rI2p)l- 2" [lOg logr2i P + E]

B(O,r)nD

wo(p) = 0,

231

THE FIRST NODAL SET OF A CONVEX DOMAIN

•

Lemma 3. Let D be an open convex subset of RP such that B(O, 1) CDC B(O, M). There exist constants Cl and K depending only on M and p such that for any open subset V of D, Al(D\V) ~ Al(D)

+ Clm(V)K.

Proof Suppose that w is the first eigenfunction for D, with f-L = Al (D). Nor­ malize w so that max w = 1. Let B = B(O, 1/2). Consider a dilation of the function w. of Lemma 2 in dimension p so that it satisfies (~ + A)W. S O. This changes the bound on the derivative in part (d) by only a bounded amount because A is bounded above by a dimensional constant. Now translate w. so that the inner circle is tangent to aD and contained in the complement of D. This is possible because D is convex. The generalized maximum principle for the intersection of D with the annulus implies that w S w., where W. is the translated and dilated w•. Taking the limit, we find that w S WOo Thus, there is a dimensional constant C such that w(x) S C dist(x, aD). It follows that w attains its maximum, 1, at a point whose distance to aD is greater than l/C. It then follows from Harnack's inequality that there is a constant C2 > 0 depending only on M such that (2)

min w B

~

C2.

Now consider the first eigenfunction 1/1 for the operator Lu = -~u + XBU. Thus L1/I = f-Ll1/1 in D and 1/1 = 0 on aD. Normalize 1/1 so that max 1/1 = 1. Since

=,-==-=,,==~-"=,--===

_,_:~'":::'::::~:-=--=~=-~~:-:::'::::::::::=""-:::':=:::=::=::::::-:::"-==::='::=.==='.:::,:::::.:==-=~~..:::':::::~::~.:::::'::::::::::::_-:=-=::~':;~~=-'

232

CHAPTER 11

'--::::-;;::-'::::-::.::::"-

--,----~~--~~'--,- ~---"

w. in the proof above is only required to be a supersolution, the proof applies to > 0 depending only on M for which

Therefore,

L i IVJ/rlz +

Hence, JLI ~ (JL

J/rz

C3

L

IVvi +

(3)

i

v

Z

L

(JL

L

J/rz.

+ C3)

L

vZ

¢

=t

on au, ¢

lDI

vw

= ( lDI

v(t:.w) - (t:.v)w

=- { laD,

(

v¢.

lD,

S

n U)

~c

w( )

m(U) >-­ - m(D)

(a(s) )K2 a(aD')

for some constants c > 0 and Kz depending on M, and for every Borel subset S of aD'. But if S = aD' n U, then by the maximum principle, ¢(x) ~ tw(S). Therefore, min ¢ ~

C4t

K3

B

+

Kz8. Extend v by zero outside D I • It follows that

(4)

v¢

{

~

C4tK3

lD I

v.

{

lB

{ v

w av da an

=a {

lB

v.

lD I

We consider two cases. Case 1.

= 0 on aD.

Denote by v the first eigenfunction in D" with v > 0 and t:.v = -vv in D I . Because D I is a Lipschitz domain the following calculation based on Green's formula is valid. (Here, a/an denotes the outer normal derivative of aD I • Note that av/an :::: 0.) (v - JL) {

=v

Define a by the equation

ED: w(x) > t}) ~ 2Ct li .

Choose an open subset U of V n {x ED: w(x) > t} with a smooth' boundary and such that m(U) ~ Ctli. Since Al (D\ V) ~ Al (D\U) we will confine our attention to D I = D \ U. The boundary of D I consists of a smooth part, au, and a convex part, aD. In particular, it is a Lipschitz domain (without any control on the Lipschitz constant). Thus we can solve

= 0 in D I ;

(a¢ - v - ¢ -av) da an an

for some r, 0 < r < 1. On the other hand, the A oo estimate for hannonic measure in Lipschitz domains due to Dahlberg [D,CF] implies that the hannonic measure w for D' at x satisfies

with K3 = I

for some constants C and 8 > 0 depending on M. Choose t by the equation = 3Ct li . Then

t:.¢

aD,

a(aD')

{x ED: w(x) < t}) :::: Ctli,

n {x

1

In order to obtain a lower bound for the last integral, consider any point x E B. By Fubini's theorem there is a dilate D' == r(D - x) + x of D, with respect to x as the origin, for which

m(V)

m(V

av -da =

(t:.¢)v - (t:.v)¢

a(aD'

The function w is superhannonic. We can bound from below the rate of van­ ishing of w at the boundary of D by comparing it with the rate of vanishing of a positive hannonic function that vanishes on the boundary of a cone. The hannonic function vanishes at most as fast as a power of the distance to the vertex, and the power is bounded above in terms of a lower bound on the aperture of the cone, that is, in terms of M. So we deduce from (2) that there is K I depending only on Mforwhichw(x) ~ cdist(x, aD)K , . It follows that

n

233

lDI

for any function v that vanishes on aD.

m(D

_.-.:::::.":"~' ~_'.C:::,-:: -':"_'::=':':::':::_:.':::':.":':::"_':-':':"'-:'::'::::::::::=~'-'~::~-:::".'-:::::~:::,:::='::"'-:::::.~~-~~.::::::,,::".;:':';'-:-._'=~:':-.'::7:~::::::::-'-:

au an

= {

J/rz.

~ (JL + C3) ~

1

~ -t

+ C3) and we have z

..~:':_:::.:~v:_:::--_:"":;.

THE FIRST NODAL SET OF A CONVEX DOMAIN

J/r, so that (2) holds with J/r in place of w. In particular, there is a constant C3

Is J/rz ~

-,--~

a<

c for a constant c, depending on M, to be specified later.

Let us first verify that there are dimensional constants A > 0 and C5 such that (5)

i

fZ :::: C5E

i IV

fl

z + C5 E -

A

(i IfI)

Z

for all E > O. Let p > 2. HOlder's inequality and the fact that A I A 2 < A:+a + Ai+a for positive numbers AI, Az,anda = (p - 2)/p > oyields

i

gZ ::::

(i

gp) I/(p-I)

(i

g) (p-Z)/(p-l)

PEB_

!,!;f~

;''!'.

234

CHAPTER I I

S

E

(l

gP) 2!p

(1

+ E- P!(p-2)

If - alP y!P S C

'"""._-:~-.t";'\.;~..".!'!'1':;~~~w.~~:;t~~~~~~,.~~i~~~\WF..;~~~~~~£~~~:r,n ~",~....,t,=.~~~"''';;''~J;~~~~~~~~\~'f{3i~';'~.~~~~~

Remark 2. Let Q be a cube in Rq of sidelength s. Let E C Q. There are dimensional constants C6 and K (q) such that

1

IV fl 2

2 If - al S CE

S CE

1 1

IV fl 2 + E- P!(p-2)

(1 (1 I) If -

IV fl 2 + 4E- P!(p-2)

2

1 1

If - al

i

2

+a

~ IV f(y)1 2dy + r

2

If

/I

2

2

IV fl 2 + 4E- P!(p-2)

~ (l -

CSE)

l,

~ (l -

CSE)(JL

~ (I -

C")(I"

(1

If I) 2

+ 2m(B)

(i

If I) 2,

IVvl 2 -

+ C3)

+

CsE-

II v 2

A

(1 v) + i v (i v)

A - CsE-

2

2

2

c,) In, v' - c,,-'o (10, v)'

U')2 /

+r

I

( f2

JE

~

=

1. For q

f2 S 5 /U')2

=

1 we let I

= [0,

f(y)2dy. 1]. It is easy to

+ 5 min f2,

I

I

rm(E) /U')2 I

m(E)

=

1

m(E(x'»dx'

1B v ~ 1D C

1

vw and consequently

v - JL ~ CC4vtK3 ~ clm(V)K3!J,

I

+ (

Q'\F

rm(E) / f2. 5 I

~m(E) + m(F).

m(E(x'»dx' S

10

JF

Thus m(F) > m(E)/2. If V' denotes the gradient in the first q - 1 variables, then using the inequality for q = 1 and the induction hypothesis we have

~ IV fl 2 + rXEf 2 = ~ IV' fl 2

1(/ Q'

We can choose E depending only on M so that (l - CSE)(JL + C3) > JL + C3/ 2 . Then there is a constant C depending only on M such that if () < c then JL + C3/ 2 _ A CE- () > JL + C3/ 4 . It follows that Al (D\ V) ~ v > JL + C3/4 and the lemma is proved in this case.

+ rm(E) min f2 ~

For the induction step, let Q = Q' x I where Q' is the unit cube in Rq-I. Let E(x') = {tEl: (x', t) E E} and F = {x' E Q': m(E(x'» > m(E)/IO}. Then

+

2. () ~ c. Then

~ c(q)(rm(E»K(q) ~

f(y)2dy

Hence,

Now, we apply (5) and then (3) to the function v extended by 0 outside D I to get

IVvl

1

The remark is the special case r show that

and (5) is proved.

l,

Proof. Without loss of generality we may assume that s = 1. We will prove by induction on q that for 0 < r S I,

a l)

Therefore,

S 2CE

f(y)2 d Y ) .

6

•.

f2 S 2

1

~ f(y)2dy S C (:~~;) K(q) (s2 ~ IV f(y)1 2dy +

mlB) 1B

where a = f, the average value of f, and C is a dimensional constant. Applying this inequality and the previous one for g = If - a I, we have

1

235

THE FIRST NODAL SET OF A CONVEX DOMAIN

for some constant CI depending only on M. This concludes the proof of Lemma 3.•

g) 2

for any non-negative function g. Recall the Sobolev estimate

(1

....

2 lar!(x', t)1 dt

+r

I

f(x', t)2 dt ) dx'

( JE(X 1 )

~ ~ IV'fl 2 + ~, ~m(E(X'» /

f(x', t)2dtdx'

~ ~ IV' fl 2 + ~, ~ mi~) XF(X')

: f (10,

IV' f(x', 1)1' +

1

f(x', t)2dtdx'

'~~E) XF(x')f(x', t)2d

X)

dt

~

-,_.

~~~~====~.~~=====~T=-~=====~~=~==~~=~=~~~~~'~==TT

237

~.,-

236

CHAPTER 11

!

~

c(q-l)

~

1[

r~m(F) 2

+ C6 N1 /5

f(x',t)2dx'dt

50

Q'

I

]K(q-O

(E)

THE FIRST NODAL SET OF A CONVEX DOMAIN

]K(q-I)

c(q - 1) r m (:; [ l

~

fa f2,

•

(6)

[1

1r Q

r u2 = Qla(y) 1

We consider three subfamilies of F.

11 11

Q\E n(y)

AO

AO

Q\E Q(y)

~ max

(

(Ao

F: €(Q) > 5C2N-2/3},

for all Q in Fl.

Next, we clearly have

F3 = {Q

E

F: (c) fails for Q}.

(8)

1[ Q

In(y)

IVyu_(x, y)1

+

IVx u_1 2

+

(1

f r u~ 1EnQ In(y)

+ 2tC6 N 1/5) ),,0 + N-l/3

11 Q

n(y)

IVxu_1 2

IVxu-I 2 .

(Ao

11

1r

1r

r r IV(x,y)u_1 lQ lQ(y)

+ 5C2N- 2/3)

E

~

IVx u_1 2 + (1 + 2tC6 N I / 5 )

o

F 2 = {Q

u_(x, y)2dxdy

u_(x, y)2dxdy.

n(y)

Lett = lOC2 A I N- 2/ 3 • Then, noting that Ao 2: 1/M 2 , we find that for sufficiently large N, .

F: E(Q) ~ 5C2N-2/3 and (b)failsfor Q\,

1Q lo(y)

1

(1 - t) (1 + 2tC6Nl/5») --, 1/3 Ao ),,0 + N-

E

2

r

1EnQ

r r

F] = (Q

C 6s 2N I / 5

2C6NI/5

lEnQ In(y)

(7)

For QinFl wehavem(Q)/m(EnQ) ~ N 2a . Fixx,anddefinef(Y) = u_(x, y) if (x, y) belongs to Q_ and zero otherwise. If we apply Remark 2 with E replaced by E n Q and integrate in x we find

[[

(1 - t)

-

Proof Choose a tiling of Rq by cubes Q of measure N a , that is, a family F of cubes that overlap only on their boundaries and whose union is all of Rq. Let s denote the sidelength of Q. Thus s = Na/q. Define E(Q) by

lQh M

~

2 2 2 r 1 u +t r 1 u + r 1 u 1Q\E O(y) 1Q\E n(y) 1EnQ n(y)

< (1 - t)

the eccentricity of Q(y) is controlled: the inradius is bounded below by a dimensional constant and the diameter is bounded by a constant depending onM.

Q aM

(1 - t)

~

(d) If Q' denotes the cube concentric with Q ofhalfthe diameter, thenfor y E Q'

11

[ [ u_(x, y)2dxdy lEnQ In(y)

On the other hand, making use of (6) and bounds in terms of Vx, we have

11

u_(x, y)2dxdy.

u_(x, y)2dxdy

In(y)

u_(x, y)2dxdy ~ u_(x, y)2dxdy ly€B X€a(y) y€Q XEn(y)

IVyU_(x, y)1 2dxdy = E(Q) [ [

C6S2E(Q)Nl/5j [ u_(x, y)2dxdy Q In(y)

For N sufficiently large, C6s2 N 1/5E(Q) < ~,and, hence,

We can now show that ~L is very wide (in the x variable) for many values of y.

Lemma 4. Let ex = l/lOK(q). There is a constant C depending only on M and n such that for any N 2: C, there exists a cube Q in Rq satisfying (a) m(Q) = N a ; (b) m(E n Q) ~ N-a. where E = {y E Rq : Al (~L(y» ~ AO + N- 1/3}; (c) If B denotes the unit cube with the same center as Q, then

u_(x, y)2dxdy

EnQ In(y)

+ C6 NI /5

which proves the remark.

N

1[

+ 5C2 N- 2/ 3 )

u~ ~ n(y)

Q

u~:: Qla(y)

Qlo(y)

IVxu_1

2

for all Q in F2.

For Q in F3 we use Remark 2 again but with E replaced by B:

11 Q

Q(y)

u2 = (1 - t)

11 Q

Q(y)

u

2

+t

11

Q Q(y)

u

2

2

238

CHAPTER II

< (1 - t) { { lV'xu_IZ Ao 1Q In(y)

+

tC6NI/LO ( { lB

< max (1 , - t

I

In(y)

+ tC6sZNI/1O ( {

lV'yu_I Z

Q

EZ

n(y)

IV'~,~ u_I Z

lQ In(y)

u~

U~.

To prove (9), let

S' =

(x, y) E R'. By Lemma 4(c) and Fubini's theorem there is

1 a(aB(S', r»

S 1 - t [[ AO lQ In(y)

IV'(x,y)u_l z.

o

+ 5CzN- z/ 3 )

uZ

a dimensional constant C such that for some r, I/2JI S r S 1/ ..II,

Forlarge N, (l - t)/(l - C6tN-9/LO) < 1 - lOCzA l N- z/3, and, consequently, (8) holds for all Q in :F3 as well. Now if every cube in :F is in one of the three families, then we can sum over all cubes and obtain (Ao

({

1B In_(y)

"1~xu- S CgNI/zE.

(9)

Therefore, if t = N- I / z and N is sufficiently large, (l - tC6N-9/1O) [ [

-

Let Q" be the cube with the same center as Q' but with half the diameter. Define

R' = {(x, y) E Q: y E Q'} and R" = {(x, y) E Q: y E Q"}. We claim that

11

tC6SZN I / LO )

11.0

+tC6N-9/LO [ [ lQ In(y)

Let B and Q be given by Lemma 4. Denote

1 Q In(y)

u~

239

THE FIRST NODAL SET OF A CONVEX DOMAIN

Lu~ s L

lV'u_l

z

=A

Lu~

which contradicts Lemma 1. So there must be a cube that does not belong to any of the three families, in other words, a cube for which (a), (b), and (c) of Lemma 4 hold. Finally, let us deduce (d) from (b). The theorem of F. John (1) implies that the diameter of Q (y) is bounded by a constant times M for any y. AO is bounded above by a dimensional constant. Therefore there is a dimensional constant cp such that every domain contained in a rectangle with side of length less than cp has first eigenvalue greater than AO + 1. By the John theorem applied to the convex set Q (y), there is a dimensional constant c' such that if the inradius of Q (y) is less than c', then Q (y) is contained in a rectangle with a side smaller than cp, so that AI(Q(y)) > AO + 1. Now suppose that there is a point yO in Q' such that the inradius is smaller than c'. Then AI(Q_(yo)) ~ Al(Q(yO)) > AO + 1. In particular, yO belongs to E. The set of all y for which the inradius is larger than c' is convex. Therefore, there is an entire halfspace containing yO for which the inradius is smaller than c'. But this half-space intersects Q in a set of large measure, and all of its points belong to E. This contradicts (b). (The proof of (d) can be simplified using a theorem of E. Lieb that Al (Q (y» is a convex function of y.)

Consider the function (~+ A)1/I

1

aB(~,r)

­

1/1 that solves

= 0 in B(S', r);

s

= u_(z) + E for z E aB(S', r). S 1/1 in B( S', r) n Q_, and by Remark I,

1/I(z)

The maximum principle implies that u_

u_(S) S 1/I(S)

u_da < CNI/ZE.

2 a(aB(S', r»

[

laB(~,r)

(u_

+ E)da

S CgNI/Z E.

•

LemmaS. Suppose that 0 < r < I/4JI, B(S', r) C Q+ and aB(S', r) nA =f:- 0.

Then u(O S C9 max u_. B(~ ,2r)

Proof Let S = max B({,2r) u_. Let w€(z) = wf(JI(z - S')) where w€ is the function given in Lemma 2 with p = rJI < 1/4. Then (~ + A)w€ = 0 in B(S', 2r)\B(S', r), and W€ is positive in the closed annulus. Therefore, by the maximum principle for the region Q_ n B(S', 2r)\B(S', r), u_(z) S Sw€(z). Taking the limit as E tends to 0, we have u_(z) S SWo(z). Let 11 belong to aB(S', r) n A. Harnack's inequality implies that there is a dimensional constant

C such that

u(S) S C min u. B(VI2J

The function u is superharmonic in the ball B(S', r). Therefore, u is larger than a Zn dimensional constanttimes the harmonicfunction G(z) = u(O(lx/ rl - - 1) on B(S', r)\B(S', r/2). (When n = 2, we put G(z) = log Iz/rl.) Since both G and u vanish at 11, it follows that IV'G(I1)1 S CjV'u(I1)I, and hence.

u(S) S CrlV'u(I1)I·

.,.".."'."-""""""'-.. ."', ....._-' ..'" -

-~- ·,.,.....,"-at--·~~·- ~~~';~~~.....r-""'===-~~~~~~~:::"~=:::::~-~~."."~~~~
~

240

CHAPTER 11

In particular, VU(TJ) i= 0, so that, near TJ, A is a smooth hypersurface tangent to aB(~, r). Because bothu and iVo vanish at 1/, it follows that IVu(TJ)1 s ISVwo(1])I. Thus, the upper bound from Lemma 2(d) implies u(O S C9 S, as desired. •

°

Remark 3. Suppose that < v S 1 and (~+A)V = Oin D. LetO < r S 1/..rA. If ~ E D, m(B(~, r) n D) S m(B(~, r»/4, and v = on B(~, r) n aD, then

°

v(O ::: 1/2.

°

Proof By Fubini's theorem, there exists s, < s ::: r, such that a(aB(~, s) n D) ::: a(aB(~, s»/4. For any E ~ 0, let v. satisfy (~ + A)V. = in B(~, s). v. = v + E on aB(~, s) n D and v. = Eon aB(e s)\D. By the generalized maximum principle in D n B(~, s), v ::: v. provided E ~ 0. Passing to the limit as E tends to 0, we find v S vo. Remark 1 implies v(O

s

. vo(O ::::

2 a(aB(~,

s)

°

1

aB({,s)

Vo

I da < -. -

2

•

Lemma 6. Denote Y = maxw u_. Choose Zo E aQ such that Izo - ~ I = r = dist(~, aQ). There is a constant CIO depending on M such that if ~ belongs to R ' , B(~, lOr) n Q c R ' , and (10)

u(O > ClOy

then B(zo, 2r)

nA

= 0.

Proof If B (~, r) is not contained in n+, then it intersects A. Choose s < r such that B(~, s) is contained in Q+ and aB(~, s) intersects A. Then by Lemma 5 and Harnack's inequality the opposite inequality to (10) holds. So (10) implies B(~, r) C n+.

Denote Bo = B(zo, 4r) and

Bk = B(zo, rk)

with

rk = 2(1

+ rk)r.

Thus the intersection of Bk for all k is B(zo, 2r), and rk+1 = rk - 2- kr. Denote the convex hull of z with B(~, r) by fez). We claim that

0)

sup u_ ::: C 1l 2- Zk - y 1

Bk

and (il)

An fez) c

B(z, (n-Zkr)

forall

z

E

Bk

n aQ,

Where 0 > 0 will be specified later. We prove the claim by induction. For k = 0, 1, ... 5 part (i) is true by the definition of Y, with a suitable choice of C II. let B be the ball of largest radius in fez) centered on the axis of fez) such that

:-

'!<:~~~.c=_=""'="-,~~_~~_

241

THE FJRST NODAL SET OF A CONVEX DOMAlN

Be Q+,butaB intersects A. ItfollowsfromLemma5thatu(zz) S C9 Y.where Zz is the center of B. If we assume that part (ii) fails for some k S 5. then the radius of B is greater than 2- 20 0r. On the other hand. by following the cone as it expands, this ball can be connected by a chain of balls which overlap in a large fraction of their volume to the ball B(~, r). The number of such balls is at most a dimensional constant times log l/e. Therefore. the Harnack inequality applied log l/e times shows that u(O ::: Ae-Au(zz) for some dimensional constant A. To assure that (ii) is true for k = 0, 1, ... 5 we assume C IO ~ AO-A. Next, for the induction step, suppose that (i) and (ii) hold for some k ~ 5. Suppose that Zl E Q_ n B(zo, rk - 10· 2- Zk r). We now show that (11)

m(B(ZI,

r

2k

r)

n Q_)

< C 12 0m(B(ZI,

r

2k r»,

by showing that every z E B(ZI, 2- Zk r) n Q_ is within e2- 2k r of an. The ray from ~ to z meets an at a point Z3 and because IZ3 - ~ I ::: 5r and the convex set n contains the ball B(~, r), the angle of the ray with a plane tangent to an at Zz is at least 1/5. It follows that Z3 E Bk n aQ and by part (ii) of the induction hypothesis, z E B(Z2, OrZkr). In particular, z is within erZkr of aQ. Furthermore. the fact that B(~, r) is contained in n implies that the Lipschitz constant of an n Bk is less than 5. so that (11) holds for some dimensional constant C 12. If we choosee so that CIZO < 1/4. we can apply Remark 3 repeatedly to obtain for j = 1,2, ... u_(z) ::: rjy

for all

z

E B(zo, rk - (10

+ j)r 2k r).

In particular. if 10 + j = 2k we have u_(z) ::: 2 10 r

Zk

y

for all

z

E

Bk + l ,

which is the induction step for part (i). Now consider any z E Bk+1 n an. If there is a point of fez) n A at a distance greater than e2- Zk - Zr from z, then there is a ball B in f(z) of radius at least a dimensional constant times er2k - 2 r which is tangent to A and such that the convex hull of B with B(~, r) is contained in n+. By Lemma 5, the value of u at the center of B is less than C92 10 2- zk Y. Moreover, B can be connected by a chain of overlapping balls to the ball B(C r) as above. The number of balls in the chain is less than a dimensional constant times log(2 zk +ZIe). Therefore, by Harnack's inequality, u(~) ::: A(2- 2k - zO) -AC92 IO r

Zk

Y.

Choosing CIO sufficiently large. we have a contradiction. This proves the induction step for part (ii). Now, take the intersection over all k to obtain B(zo, 2r) n A = 0, which proves • Lemma 6.

CHAPTER II

We can now deduce that if R" denotes the subset of Q that projects onto a cube Q" concentric with Q' with half the diameter, then (12)

max u+ R"

:s

C l3 max u_.

En Q'} U (x, y): x

E

y E Q'\E}. Lemma 4 implies m«((x, y): y E En Q'j)

:s

E Q+(y) and (CIM)P N-1/1O.

Lemma 4(d) and Lemma 3 imply that for y E Q'\E,

+ clm(Q+(y»K :s Al(Q_(y» It follows that m(Q+(y» :s C;I/K N- /3K. Let Z5 E AI(Q(y»

< AO

Q project onto the center of the unit cube B given by Lemma 4. All the points of B(Z5, T), where T = C5Na/q, project onto Q". Choose a dimensional constant C7 < minO/../)., 1/1O}. The estimates above for the measure of E and for Q+(y) when y is in the complement of E show that if N is sufficiently large, then for any z E B(Z5, T),

n Q+)

:s

I

4" m(B(z, C7).

Therefore, we can apply Remark 3 repeatedly to conclude that for j = I, 2, ... u+(z)

:s 2- j

maxu+ RO

Z E B(Z5, T - j C7)'

forall

In particular, u+(z)

:s r

T

max RO u+

for all

z E B(Z5, T /2).

Now take any point z E Q- that projects onto B of Lemma 4. Because we assumed that Q- is relatively compact in Q, there is an open ball centered at z contained in Q- whose boundary intersects A. By the same reasoning as in Lemma 5 (with the roles of u+ and u_ reversed), we find that u_(z)

:s

C92-T max u+. R'

Integrating and using (9) and (12), we have E

:s

T

C14 2- max u+ R'

:s

THE CASE n = 2 We would now like to show that in R 2 the nodal line must touch the boundary in exactly two points. The Courant nodal domain theorem says that the sets Q+ and Q_ are connected. Therefore, A cannot intersect aQ in more than two points. In the very last stage of the proof above we used the fact that any ball in Q_ can be expanded until it touches A. In the case n = 2, we prove a variant of Lemma 4 which will imply that Q+ must touch both sides of aQ somewhere in the region that projects onto Q. We will make a slightly different normalization of the domain Q for notational convenience. We claim that we can assume that

+ N- 1/ 3.

J

m(B(z, C7)

If N is sufficiently large, the 2- T term dominates, and this inequality cannot hold unless E = O. But if E vanishes, then u vanishes on an open subset of Q, a contradiction. This concludes the proof that the nodal set touches the boundary.

R'

For any ~ in Q+ n R", either u(O :s ClOy and we are done, or else the opposite inequality holds and the hypothesis ofLemma 6 is satisfied. With Zo as in Lemma 6, consider the ball B(zo, s) disjoint from A but such that aB(zo, s) intersects A. Thus s ~ 2r, but because of the estimates ofLemma 4, s must be smaller than some negative power of N. In particular, s < I/../).. Consider a point Z4 E B(zo, s) at a distance, say s / M from a(Q n B(zo, s». By Lemma 6 and Harnack's inequality, U(Z4) :s CCIOY, for some constant C depending on M. Now by Proposition 3, uCO :s CC4 C lO Y, which proves (12). Note that Q+ n R' c {(x, y): y

243

THE FIRST NODAL SET OF A CONVEX DOMAIN

C I4 C l3 2- T CS N 1/ 2 E.

Q

c

(x, y) E R 2 : 0 < x < I},

the diameter of Q is N and the inradius is at least I - N- I • In fact, suppose first • that the inradius is 1/2. Choose a disk D of radius 1/2 in Q. If aD touches aQ in two points on the same diameter of D, then Q is contained between the two tangent lines perpendicular to the diameter, which are a distance I apart. If no such pair of points exists, then aD n a Q contains at least three points and the tangent lines at those points fonn a triangle containing Q. The longest side of the triangle has length at least N and it is easy to check that the altitude perpendicular to that side has length at most I + N -I. Thus, n is contained in a rectangle of length N parallel to the longest side and width I + N- I • If we dilate by the factor (1 + N- I ) -I and rotate so that the rectangle is horizontal, we have our normalization. As in the case of Lemma I, an easy consequence of this normalization is

Lemma I'. rr 2

:s A :s rr 2 + C2 N-2/3.

Lemma 4'. Let Q be a convex subset of R2 with the normalization above. There is a constant C IS such that If N ~ CIS, then there exists an interval J of length comparable to N 1/ 4 such that (a)

the rectangle I x J C

n_,

where I = (x: N-l/ 8 < x < I - N- I / s ). Moreover, J ,the square

QI = {(x, y): Ix - 1/21

:s

if Yo denotes the center of

1/8, Iy - Yol

:s

1/8}

w.

--j"-

244

.=----­

CHAPTERlI

satisfies

","''--=-'-'~f~~",,~-io.,",_\'. .....i;;~-.~~'j;i,~~~;;;;'~.d~AA;~~-i';;i;;;~:ij:;::..;;;.,''f,,~~~~.:;.,.--i;~aiit."Y~c;;;r-¢;;i;-""_,,,,~,;i:i~""'~:'~~;.;i;.:~ •.:i.i'~~:~T';';;~~~~~,~';,;:o,

245

THE FIRST NODAL SET OF A CONVEX DOMAIN

the curve exits V. Let

1

(b)

~L

U2 :::: N 2 { U2.

'1

J

=

'11

and

min y

(X,y)Ey

= (X,y)EY max y.

Q1

If '1 :::: y :::: '11 then there exists x such that Ix - fl < f/4 and u_(x, y) = O.

The differences between Lemma 4' and Lemma 4 are that we show here that an entire rectangle is contained in ~L, instead of showing that all but a subset of small measure of a rectangle is contained in ~L and also that we can control the integral of u 2 on a large set by the integral on a "unit-sized" set Q I whose double is contained in ~L. . Suppose that 11. intersects aQ in at most one point. Then, since aQ_ is a simple closed curve that encloses Rio either for every y in J there exists x such that o < x < N- I/8 and (x, y) E 11. or else for every y in J there exists x such that 1- N- I/ 8 < X < 1 and (x, y) E 11.. In either case, one can use the same argument as above to control u+ by u_. Then one finds exponential decay of u+ along a thin strip of Q+ either in 0 < x < N- I / 8 or 1 - N- I/ 8 < X < 1. Then one can control u_ on Q I in terms of u+ in either one of the strips by translating a disk of radius 118 around a point of Q I to the left (decreasing x) or to the right (increasing x) until its boundary touches 11.. This yields the same type of contradiction as above, and concludes the proof that the nodal line touches in two points. It remains to prove Lemma 4'. Lemma 7. Suppose that 0 < f < 1/2 and (f, 0) ¢ Q_. There exists '1. I'll < f such that if E = {y E R: '1 < Y < '1 + f/8}, then rr 2

{{

JE JO(y)

u~dxdy:::: (l

- f) ( {

IVu_1 2dxdy.

JE JO(y)

The proof makes use of the inequality (13)

rr2iL j(s)2ds :::: L2iL j'(s)2ds,

for all functions j such that j(O) = j(L) sin(rr s I L).) Also, (14)

=

O. (The minimizer is j(s)

rr2iL j(s)2ds :::: 4L2iL !,(s)2ds,

Therefore, rr2i1 u_(s, y)2ds = rr2ix u_(a, y)2ds

+ rr 2

:::: x2ix (aIU_(S, y»2ds

+

i

l

u_(a, y)2ds

(l - X)2

i

l

(aIU_(S, y»2ds

1

:::: (l - 3f / 4)21 (a1u-(s, y»2ds,

wherealu-(x, y)isthepartialderivativeofu- with respectto x. If'1I-'1::: f/8, then we are done. If not, then y exits V on one of its vertical sides. Suppose that the side is the one for which x = 3f 14. (The case in which y exits V on the side x = 5f 14 is similar.) We divide the rectangle R = {(x, y): '1 < Y < '1 + f/8,0 < x < l}intofourparts. Denoteh = '1 -'11 +f/8, P = (f -h, '1+ f / 8 ), PI = (f, '11), and P2 = (f - 2h, '1 + f 18). Let SI be the segment from P to P[, and let S2 be the segment from P to P2. The region RI is defined as the u,nion of horizontal segments with left endpoint on SI and right endpoint on the line x = 1 with the shortest vertical segments with upper endpoint on SI and lower endpoint on y. R2 is union of horizontal segments with right endpoint on the line x = 0 and left endpoint on S2 with the shortest vertical segments with upper endpoint on S2 and lower endpoint on y. R3 is the union of the shortest horizontal line segments with right endpoints (1, y) for all '1 < y < '11 and left endpoint on y. Finally, R4 is the rest of the rectangle R. It is easy to see that R4 is a disjoint union of vertical line segments at least one of whose endpoints belongs to y. The region RI is foliated by curves that start at x = 1, move horizontally until they reach S[, and then drop vertically to y. These curves have length at most

1 - 5f 18. We claim that this implies that (15)

rr 2

{

JRI

u~::::

(1 - 5f/8)2 {

JRI

IVu_1 2 •

In fact, we can make an area preserving change of variable of the form forallfunctionsjsuchthatj(O) =0. (Theminimizerisj(s) = sin(rrsI2L).) Because Q_ is simply--connected, there is a continuous curve in the complement ofQ_ that joins (f, 0) to the real axis. Denote V = {(x, y): Ix - fl < f/4, Iyl < f/8}. Let y denote the component of the curve from (f, 0) to the first time that

(s, t)

F(s,t)= { (-t,s+2t)

s+t:::O s+t::::o

""""'''''''''''''''''''......... ._--_._._­ 246

CHAPTER 11

Note that

a v(F(s, t» as

!

=

We can now proceed with the proof of Lemma 4' in much the same way as we proved Lemma 4. Let J be an interval of R of length N 1/ 4 • Define E(J) by 1 l E(J) u_(x, y)2dxdy = 1 (a2U-(X, y»2dxdy.

s+t>O

alv(F(s, t)) (hv(F(s, t»

fi

s+t
if v is defined on a set U and v vanishes on au. For any numbers a < b let W = un {F(s, t): a < t < band s E R}. Let L = max length(W a:9:::,b

n {F(s, t):

s

E

Jw

~v(F(s, t)) 2dsdt as

F-l(W)

f

RD,

:::; L 2 (

Jw

JRz

u~:::;

(9E/8)2 (

J~

(Jr 2 +

(17)

lV'vI 2.

lV'u_1 2.

u~:::;

JR 3

(1 - 3E/4)2 (

Jr2111 [f :~

'.

JR3

lV'u_1 2.

Finally, R4 is foliated by vertical line segments of length at most E /4, so that by

JR

u~

JR

4

(16)

Jr2 .

711 8

j(s)2ds :::; -

0

0

:::;

2 lV'u_1 dxdy.

t

1 N- 1/ S ) 1 1 lV'u_1 2dxdy.'

j(y)2dy :::; C6 {N 1/ 2

1

J'(y)2dy

+

8N

3

/

s

1

j(y)2} ,

and, hence,

III u~dxdy

:::; C6! N

12 /

1

1 1 (a2 U_)2dxdy

+ 8N 3 / s

1 12 = C6! E(J)N / 1 1 (u_)2dxdy

+

8N

11 u~dxdy 1 11 u~dxdy I·

3

1

/

1

s

4

Now, since all of the coefficients on the right-hand integrals over R 1, R 2 , R 3 , and R 4 are less than 1 - E, Lemma 7 is proved. We also need the inequality

11

t

u~dxdy :::; (I -

1

lV'u_1 2.

:::; 4(E/4)2 (

1\alu_)2dXdY,

1JJo u~dxdy 1JJo

Remark 2 implies

(4),

Jr2 {

1 1/2)

2C6N

f

If dJ) :::; 1/2C6NI/2 and (a) fails for an interval l' with the same center as J, but of length N 1/ 4 - 1, then by Lemma 7 with E = N- 1/ S, there is an interval E C J of length N-I/s /8 for which

The region R 3 is foliated by horizontal line segments of length at most 1 - 3E /4, so that by (3),

Jr2 {

u~dxdy :::;

we have

The segment SI has slope -1, so a translate of this change of variables yields (5). The region R2 is also foliated by right angles with comer on S2. These curves have length at most 9E /8, so that by the same argument as for (15),

Jr2 {

f

If E(J) > 1/2C6NI/2, then since 1 Jr2 1

the length of the longest "right angle" curve in W. Using (3) and the change of variable F we find

Jr2 { v 2 :::; L21

247

THE FIRST NODAL SET OF A CONVEX DOMAIN

j'(s)2ds

+ 1~

It follows that

1

:::; 16c6 N

Now j(s)2ds,

3/S

for all functions j for which j(O) = jO) = O. (The essential feature here is that 7/8 < 1, so that there is an improvement on the eigenvalue for the Dirichlet problem on the interval if one adds the extra integral on the right-hand side.) Inequality (16) can be proved by direct computation of the lowest eigenfunction; the calculation is left to the reader.

11 u~dxdy u~dxdy 1J\E Jo u~ Jo

1E

t

= (1 - t)

< I

t

3

/

s

u~dxdy.

+ tit

J\E Jo

u~ +

{

t

JE Jo

u~

1

~ t i t (aIU_)2 + (l + 16C6N 3/ St) t u~ J\E Jo EJo

Jr

---=;Jr

< I

1

11

t

1J\E 1 1

(aIU_)2

0

+ Jr-2(1 + 16C6N 3/ St).

~~~_"'!""'

..._ - - - - ' - - - - - - - - - - - - - - - - - -- - - - - ' - - . - - - - - - - . - . - - -•.." - - - . - - - - - - . - - - - - . , , ' -•.- . -•. --- . - - . - - . - - - - - - - - -•.:--...~- - . - - , - . - - - - .. --.-.------.--.-"-.-'-~ -"' •• --,.;.~~,~0iG'~"""':,;;;~~,,~~~~"'>:F~~,~,,~ ...~~~;,,~~~

248 . (1 - N-

Choose t

=

N- 5/ 8 • Then for large N,

(1

+

1 8 / )

Ie i

CHAPTER 11

..

;"',~;;.'

....

THE FIRST NODAL SET OF A CONVEX DOMAIN

~~~ ~;';';~Y_"""''''';;;'~~

249

1

lV'u_1 2•

16C6N3/8t)(1 - N- 1/ 8 ) S 1 - N- 1/ 8/2 S 1 _ N- 5/ 8 ,

[CF] R. R. Coifman and C. Feffennan. "Weighted nonn inequalities for maximal functions and singular integrals." Studia Math. 51 (1974),214-250. [CFMS] L. A. Caffarelli, E. B. Fabes, S. Mortola, S. Salsa. "Boundary behavior of nonneg­ ative solutions of elliptic operators in divergence form." Indiana Univ. J. Math.

30 (1981), 621-640. [0] B. E. J. Dahlberg. "Estimates for hannonic measure." Arch. Rat. Mech. Anal. 65

and, hence,

(1977),275-283.

rr2111 u~ S (1 -

(18)

[dG] M. deGuzman. Differentiation ofIntegrals in Rn. Lecture Notes in Mathematics, no. 481. Springer Verlag, 1975. [H] W. K. Hayman. "Some bounds for principal frequency." Appl. Anal. 17 (1978),

111 lV'u_1 2.

N- 5/ 8 )

Finally, consider the case in which E(J) S 1/2C6 N 1/ 2 and (b) fails for J. Then the interval E = [Yo - 1/8, Yo + 1/8] C J, and, by (16),

rr

2

t u~dxdy S (1 1E 10 {

t Uh u _)2dxdy + 1E{ 13/8 t/ 1E 10

8

1/8) {

u~.

The same argument as the one leading to (18) yields

rr j 1t u~ s 2

J

(1 -

N- 1/ 2 )

0

j 1t J

lV'u_1 + 2

0

1E13/8 u~. {5/8

And since (b) fails,

rr2111 u~ S

(19)

(1 - N- 1/ 2 )

111 lV'u_1 2+ N- 1u~. 2

Now if Lemma 4' were false there would be a disjoint covering of the projection onto the y-axis of n by N 3/ 4 intervals J satisfying one of (17), (18), or (19). Taking the sum, we obtain

rr

2

{

1n u~ s

5

34 8 {N / 1n lV'u-f + N2

(1 - N- / )

{

1 u~. n

Therefore,

(rr

2

-

5 4

5 8

N- / )(1 - N- / )

1u~ s

1,V'u- ,2

= A

1u~,

which contradicts Lemma I' when N is sufficiently large. Institute for Advanced Study Massachusetts Institute of Technology REFERENCES [C] L. Carleson. "On existence of boundary values for hannonic functions in several variables." Arkiv Math. 4 (1961),393-399.

247-254. [HW] R. Hunt and R. Wheeden. "On the boundary values ofhannonic functions." Trans. Amer. Math. Soc. 132 (1968), 307-322. [J] D. Jerison. "The first nodal line of a convex planar domain." Int. Math. Res. Not.

1 (1991),1-5. [KP] C. E. Kenig and J. Pipher. "The h-path distribution of the lifetime of conditioned Brownian motion for non-smooth domains." Prob. Theo. Rei. Fields 82 (1989),

615-624. [L] C.-S. Lin. "On the second eigenfunction of the Laplacian in R2." Comm. Math. Phys.ll1 (1987),161-166. [M] A. Melas. "On the nodal line of the second eigenfunction of the Laplacian in R 2 ." J. Difj. Geom. To appear. [P] L. E. Payne. "lsoperimetric inequalities and their applications." S./ AM. Rev. 9 (1967),453-488 (Conjecture 5). [PI] L. E. Payne. "On two conjectures in the fixed membrane eigenvalue problem." J. Appl. Math. and Phys. (lAMP) 24 (1973), 721-729. [PW] M. H. Protter and H. F. Weinberger. Maximum Principles in Differential Equations. Springer Verlag, 1984. [S] S. Salsa. "Some properties of nonnegative solutions of parabolic differential operators." Ann. Mat. PuraAppl.128 (1981),193-206. [Se] J. B. Serrin. "On the Harnack inequality for linear elliptic equations." J. d' Anal. Math. 4 (1954-56), 292-308. [SW] E. M. Stein and G. Weiss. Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, 1971. [y] S.-T. Yau. "Problem Section." Seminar on Differential Geometry, S.-T. Yau, ed. Annals of Mathematics Studies 102. Princeton University Press, 1982.

~

-- ...

'1" -

~~-

. _­ ~

ON REMOVABLE SETS FOR SOBOLEV SPACES IN THE PLANE

. •. \, 4a"

i1; c~

12

On Removable Sets for

Sobolev Spaces in the Plane

Peter W Jones*

1 INTRODUCTION Let K be a compact subset of C = R 2 and let K C denote its complement. We say K E H R, K is holomorphically removable, if whenever F: C -+ C is a homeomorphism and F is holomorphic off K, then F is a Mobius transformation. By composing with a Mobius transform, we may assume F(oo) = 00. The con­ tribution of this paper is to show that a large class of sets are HR. Our motivation for these results is that these sets occur naturally (e.g., as certain Julia sets) in dynamical systems, and the property of being H R plays an important role in the Douady-Hubbard description of their structure. (See [4].) To prove that the sets in question are H R we establish what may be a stronger result. A compact set K is said to be removable for W I •2 if every f which is 2 continuous on R and in the Sobolev space WI,2(K C) (one derivative in L 2 on KC) I 2 2 is also in W • (R ). It is a fact that if K is removable for W1,2, K is HR. We do not know the answer to the following question: If K is H R, is K removable for W I ,2?

251

W' ,2 (K'), F ¢ W ',2 (R 2). Now us;ng the fact that IK I 0, we deduce K E HR. Take a homeomorphism F with F'(oo) = 1. Thenf(z) = F(z)-z E W 1•2 (K C ) because integrating IF'1 2 gives the area of the image. Now f E W I ,2(R 2) and f = 0 except on a set of measure zero implies (Weyl's lemma) f is holomorphic. Therefore, F(z) = z + a. We recall some elementary facts concerning HR. If IK I > 0, it follows from the "measurable Riemann mapping theorem" (see [1]) that there is a nontrivial quasiconformal mapping F which is holomorphic off K. (Thus K ¢. HR.) If K has Hausdorff dimension less than 1, Dim(K) < 1, the fact that K E H R follows from the Cauchy integral formula (Painleve's theorem). Similarly, if K is a rectifiable curve, Morera's theorem implies K E HR. Kaufman [7] has produced examples of curves where Dim(K) = 1 but K ¢. HR. The "difficult" case is the one that occurs in conformal dynamics: K is connected and has some "fractal" properties. (The case of "pure" Cantor-type sets is easy; they are HR. By a pure Cantor set, we mean, e.g., one arising from a Cantor construction with a constant ratio of disection, or the Julia set for Z2 + c where c is not in the Mandelbrot set.) We also point out that the case where K is a quasicircle seems to be folklore-again, K E HR. That the property of being H R is related to quasiconformal mappings is seen from the following:

a

Remark. K is H R if and only if whenever F is a homeomorphism of C which is M quasiconformal on K C , F is globally quasiconformal (and hence M

quasiconformal). (See [8], page 200.) To prove the remark, first assume that K is HR. By the measurable Riemann mapping theorem there is a globally quasiconformal mapping G such that G 0 F is holomorphic off K. Since G 0 F is a Mobius transformation and IK I = 0, F is globally (M) quasiconformal. For the other direction, standard LP estimates (see [1]) show that necessarily IK I = O. If F is a homeomorphism which is analytic off K, F is globally quasiconformal and hence (I K I = 0) a Mobius transformation. Let n be a domain on the Riemann sphere and let Zo E n. Then n is a John domain (with center zo) if there is s > 0 such that for all Zl E n there is an arc yen which connects zo to Zl and has the property that

To prove the fact we first show that the two dimensional Lebesgue measure of K, IKI,iszero.lfnot,letFn = 7<1,"- * (ein(x+Y)XKY))· Thenlimn -+ oo IlFnIlLoo(R2) = 0 and Fn is continuous. Since laFnl = Xn, lIaFnllu(R2) = IKI 1/ 2 . On the other hand, Fn (z) -+ 0 for z f/. K, so L 4 bounds on convolution with ~ when combined with Holder's inequality show IIF~IIL2(K') -+ O. (See [11], Chapter 1.) Taking a sum of functions like the Fn , we obtain a globally continuous F E

d(z) ~ sd(z, Zt),

Z E

y.

Hered(z, Zl) is the chordal distance from z to Zl and d(z) is the chordal distance of

z to an. We call such an arc y a John arc. In this paper we will choose coordinates so that Zo = 00, and this allows us to replace d(z), d(z, Zl) by the corresponding Euclidean distances. The property of being a John domain is preserved under globally quasiconformal mappings. If n is a simply connected John domain, it is

'This research was supported by the National Science Foundation under grant DMS-89l6968.

1 "

I i ..

1

-

~=~;;';;;;::::;;~~==--='--=-"-=:;~~';':"'~~~~;;;;;;;;;;;';:;;;:~=~=;'=':::"-::

'--,-'-:_"

--.:i.o~~-:;:,~~:;.~,,:,~,z;;~~,,:"=::=::,=

252

....7.i'-~~""""''',,,=c,,,,,,-=,,,.=,,,,,,,,,=~,,

CHAPTER 12

n is a John domain and K

=

an, then K is removable for w1. 2 .

J

and 6 are not needed.) The proof of Theorem 1 starts by proving it in the case where n is simply Le., K is connected. The general case then follows from connected on

Notice that the hypothesis demands that K = an, but says nothing about the other components of C\K. This is because the hypothesis will be seen to force some geometry on those.other components. (For example, the interior of a cardioid is a John domain while the exterior is not. The parabolic basin for Z2 + is also a John domain. while the basin for oo-the exterior domain-is not.) It is of some philosophical interest to note the similarities between Theorem 1 and the results of [6] on extension problems for Sobolev spaces. Since the John condition is quasiconformally invariant, we obtain directly (see also "Remark")

t,

i

Theorem 2. If

d

+ et.

Here Pn(z) = Po··· 0 P (z) is the nth iterate of P. (This definition may be a bit restrictive, but it is all we will need for this paper.) The following question is open: If J is the Julia set for a polynomial. is J

E

H R?

It is proven in [3] that whenever a polynomial P (z) is subhyperbolic on its Julia set J, then A oo , the basin of attraction at 00 for P, is a John domain. Since J = aA oo , we obtain Corollary 2. If P(z) is subhyperbolic on its Julia set J, then J

E

n is an

n' with

an Can'. While the proof of Theorem 2 is perhaps not immediately obvious, it turns out to follow from a simple construction with planar graphs. Section 2 contains background material, and sections 3-7 are devoted to the proof of Theorem 1. The idea is to redefine F near K so that it is Coo near K and so that the Sobolev norm does not change much. Theorem 2 is proven in section 8. ~{

2 BACKGROUND MATERIAL Let F E W I ,2(K C ) be continuous on C. An easy argument with the Dirichlet principle shows that to prove F E W1,2(R 2) it is sufficient to treat the case we now assume, where F is harmonic near K. We also assume the reader is familiar with elementary properties of logarithmic capacity, which we denote by Cap(o). See, e.g., [10] for the first two of the next three lemmata. Let f: D --+ C be univalent, f(O) = 0, 1'(0) = 1. Then f has a Fatou extension to T = aD and this extension is always defined except on a set of capacity (and hence Lebesgue measure) zero. In our applications, all image domains will have locally connected boundaries, and, hence, f will be continuous on D. The following results are due to Beurling. The values of c below are various universal constants. Lemma 2.1. If E C T,

Cap(f(E» ~ cCap(E)2.

HR.

The corollary answers a question of A. Douady and J. Hubbard and was the starting point of this investigation. Douady posed the question to the author for the particular (subhyperbolic) case where P(z) = Z2 + c has the (Misiurewicz)

(e) John domain, there is a (c(e» John domain

n' simply connected and

Corollary 1. If n is a John domain and K = an, any global homeomorphism which is quasiconjormal off K is globally quasiconjormal (with the same constant of quasiconformality).

A(Z)-IA(Pn(z))l dz Pn(z)1 ~ c(l

253

property that the origin is preperiodic but not periodic (e.g., Z2 + i). This case is not fundamentally different for the general case of subhyperoblic polynomials. An amusing feature of our proof is that the Julia set for a Misiurewicz point (from the family Z2 + c) is actually easier to deal with than those arising from the hyperbolic case. (When K C = n, our argument is a bit simpler. The arguments of sections 5

I,

We say that a polynomial P(z) is subhyperbolic on its Julia set J if there is a metric A(z)ldzl such thatA(z) - Lj Iz - zjl-a j is Coo for some numbers aj < 1, and P(z) is hyperbolic on J in the metric A. In other words there are c, e > 0 such that for all n ~ 1.

••;"''''''''';..'''''''''''''''''''''''=''"'''"''''';,.,'".;:;;,;: ";,,,••;,.~'""

.,:...-~"'04,· ... ";"",.~'",=-=a·"""""'~·==,,~""."":;:.:,·::--"'=;;;:=~'"..;m;~"

ON REMOVABLE SETS FOR SOBOLEV SPACES IN THE PLANE

easy to show that the are y may be taken to be the hyperbolic geodesic from Zo to 00. (See [9] for an exposition of properties of John domains.) The main result of this paper is Theorem 1. If

. .,,,':=.;'""'=-."...-......·

Lemma 2.2. Let go

=

f({re io : 0 ~ r < I}). Then

If £0 denotes arclength,

Cap({ej(l : £(g(l) > A}) ~ CA -1/2

254

CHAPTER 12

Lemma 2.3. Suppose H is harmonic and continuous in D, and (IV' HI 2 2

I H y I ),

Ii

=

IHx I2

+

t

while by our last estimate, d(Z3) :::: CG(Z3)I/a '" CI~1

j~

2

IV' HI dxdy = 1.

,,·v l!~

Cap({e

IH(e ) - H(O)I ~ iil

:

-

~211/a.

''Ii

:,'1~l

In other words,

Then

iO

255

ON REMOVABLE SETS FOR SOBOLEV SPACES IN THE PLANE

An : : ce-rrJ.

I~l

2

-

~21 ~

CIZI -

a

z2l ,

•

and it follows from the definition of logarithmic capacity that

This last lemma can be found on page 30 of [2]. We next require some elementary geometric facts about simply connected John domains. For the next result see [5].

Cap(E, zo, Q) ~ M- 1 Cap(E)M.

•

The other direction of the lemma follows from Lemma 2.1. Lemma 2.4. If g is a Poincare geodesic from

00 to Zo E aQ where

Q is an (e)

Lemma 2.6. Suppose Q j are (e) John domains with centers Zj, j = 1,2, and

John domain, then g is an arc ofa K (e) quasicircle.

suppose d(ZI), d(Z2) '" 1. Suppose also that F is harmonic on Q 1 U Q2 and

Suppose now Q is a bounded (e) John domain and suppose the John center Zo satisfies d(zo) = 1, where

continuous on

f.!l

U

f.!2.

Then

if

E C aQ I

Cap(E)

= distance(z, aQ). D -+ Q, f(O) = Zo be any choice of Riemann

d(z)

Then diameter(Q) '" 1. Let f: mapping, and define for E C aQ, Cap(E, zo, Q)

==

jil

Cap({e iil : f(e ) E ED.

~

n aQ2 satisfies

8 > 0,

there are geodesics gj C Q j from Zj to aQ j such that gl and g2 terminate at the same point ~ E aQ I

W(n -

n aQ2, and

F(zj)1 :::: A(e,

ll)(fl'V'

2 FI dxdy)I/2,

j = 1,2.

}

Proof Let

Lemma 2.5. For any Borel set E C aQ,

E j = {z E aQ j : IF(z) - F(zj)1

Cap(E, zo, Q) '" Cap(E).

~

A(fllV'

2 FI dxdy)I/2}.

}

In the last line we mean that A '" B if there is a constant M = M (e) such that M- 1A M

::::

B :::: MA 1/ M •

Proof Let G(z) = G(z, zo) be Green's function for Q with pole at zoo Then it follows from the John condition and the Koebe theorem that

!

G(z) ~ cd(z)a,

= aCe),

a

whenever Iz - zol ~ ~. Suppose now that Zj E E, Zj = f(~j), j = 1,2. Fix a point ~3 E D such that (l - 1~31)

'"

1~1

-

~31

'"

1~2

-

~31

and let Z3 = f(~3)' Then by the John condition

IZI -

z21 :::: Cd(Z3),

'"

1~1

-

~21

If A is large enough, Lemmata 2.3 and 2.5 show Cap(E I U E 2 ) < 8. Then E\(E 1 U E2) =f:. rp, so we may select ~ from that set. •

3 QUASICIRCLES We now give a quick outline of our proof for the case where K is a quasicircle. This represents the only idea of the paper. The rest of the sections contain only technical arguments which make the same philosophy work for the general case. Let Q+ and Q_ denote, respectively, the unbounded and bounded components of C\K. Fix two points Z± E Q± satisfying 8(z+) '" 8(z_) '" Iz+ - z-I '" 8,

and build domains V± C Q± which are bounded by quasicircles and such that av+ n av_ is a subarc of K with diameter'" 8. The points z± are made to be

256

CHAPTER 12

IF(z+) - F(z_)1 ::: A(

rr

Lemma4.1. 11 - rjl "-' 11 - rj+i1 "-' distance(L j , Lj+I).

IV Fl2dxdy) 1/2.

JJv+uv_

Proof. We show that l(}j+1 - (}jl ::: C(I - rj). The proof that /(}j+1 - (}jl < C(I - rj+l) is the same. The lemma will then follow from (4.1). Let I {e ill : (}j ::: () ::: (}j + Jr}. By symmetry

Standard smoothing techniques now show there is F E W 1•2 (R 2 ) such that F = F outside of K8 = (z: d(z) ::: 8}, F is Coo near K, and

rr

JJK~

IV FI 2dxdy

Sending 8 to zero we see that F

E

::: c

rr

JJK

257

ON REMOVABLE SETS FOR SOBOLEV SPACES IN THE PLANE

the "centers" of D±. Then by Lemma 2.6 there is A such that

w(~j,

IV Fl 2dxdy.

I I, .D* ) = 2

where ~j == rje iOj . Here w(z, E, V) denotes the harmonic measure at z of E C aVin V. By Beurling's so-called! theorem [10], if we set I j = {e ill : (}j ::: () :::

e,

WI,2(R2) and

(}j+1 }, 2 fl2lVF,2dXdy = fie IVFl dxdy.

If F is M quasiconforrnal on K C , Lemma 2.2 and an argult:lent similar to the one above show that F is globally quasiconforrnal. The point of this vague remark is that, whatever argument we use, it should show that F being M quasiconforrnal C on K implies F is globally quasiconforrnal. (See the "Remark" in Section 1.)

w(~j,

I/(e ) - l(e

iOj

)! :::

1,

t'

~' 'i

illj

) - I (e

illj

J

Now fix M :::: 1 and let rj < 1 be the largest value of r so that

= M.

Setting L j = {re iOj , rj ::: r < I} we see that (4.1)

distance(Lj,L t ):::: c(l-rj),

nj

= I(V j ) is an (8') John domain with John center Zj = I('i j ).

E V j and let L = [~, 'i j ] be the line segment from ~ to'i j . Then L C V j and if ( E L, distance(~', aV j ) :::: cl~' (This follows from the elementary geometry of V j .) Now if L' is the geodesic from ~ to 00 in D*, L' = {R~: R :::: I}, p(~', L') ::: C for all ( E L, where p is the hyperbolic metric on D*. The lemma now follows from the John property on the arc I(L') and the distortion theorem for I. The details are left to the reader. Lemma 4.2 is actually a special case of the following fact: If I: D ~ n, 1(0) = zo, and n is an (8) John domain with John center Zoo and if V c D is a (8) John domain with John center the origin, then I (V) is an 7/(8,8) John domain with John center Zoo We leave a proof of this statement as an exercise for the reader. •

)1 :::: ~.

llj illj I/(ri ) ­ I(e ) I

Let V j be the domain bounded by T, L j , Lj+l, and the line segment [~j, ~j+l] and let 'i j = Rjei"'j where R j - 1 = ! min(rj - 1, rj+1 - 1), and flJj = ! «(}j + (}j+d. Then since we are assuming diameter (K) » 1, each V j looks like a quadrilateral (in D* with one side on T) with bounded geometry.

Proof. Let ~

(}j::: () ::: OJ+l,

+

if M is large enough, and the lemma follows from simple estimates on the Poisson kernel. •

Lemma 4.2.

and I/(e

I

l(lj), n) ::: CM- 2,

w(r. 1\1·J' D * ) > ­4I '>J' -

In this section we construct certain domains related to a point Xo E K and a scale r. Since the John condition is scale invariant, we may assume Xo = 0 and r = 1. We will add to K certain curves to obtain a new set K so that, in a certain sense, C\K looks like a union of quasidisks of diameter about 1 (near K). Let I: D* = {lzi > I} ~ n be univalent with 1(00) = 00. Since K = an is locally connected, I is continuous up to T. Select angles 0 = 00 < 0 1 < (h < ... < ON = 2Jr so that ill

w(f(~j),

because diameter (f(lj)) ::: 1 and distance (f(~j), l(lj)) :::: M - 1. Thus

II:

4 SOME GEOMETRY

I j , D) =

n

j ;:f=k,

At this point we remark that Qj = interior of Qj is a 8(8) quasicircle ifC\K = (This is. e.g., the case for the Julia set corresponding to Z2 + i.) A most unfortunate complication is that this statement is easily seen to be false if C\K is

for otherwise the John condition would be violated for the corresponding geodesics in n.

n.

(

II·

_-'C''',"_--',_'

---~'"-

258

_">'--

-

_. __ •• _.

, _

__'.~

"."_---­

CHAPTER 12

259

ON REMOVABLE SETS FOR SOBOLEV SPACES 1N THE PLANE

allowed to have bounded components. This necessitates the technical construction of our next section. The reader interested only in the case where n = C\K may skip to Section 7, noticing that Proposition 6.1 has already been proven for quasicircles.

lies in Un and has endpoint on In whenever

10 -

f/JI ~ co 2 •

The proof of the lemma is an exercise in elementary geometry. Now let

5 SOME ADDITIONAL CURVES We now add some additional curves to K. Let OJ be a bounded component of

C\K. Then by the defiqition of the domains nk, each an k intersects aO j in either a connected set or the empty set. Let us for the moment reorder the n k so that ani, anN are exactly those domains such that ann n aO j consists of more than one point (and hence an arc). Let 0 > 0 be a small constant to be fixed later and fix a Riemann mapping h: D --+ OJ so that II, ... , IN are intervals with h(ln) = ann n aoj . By selecting h(O) to lie very close to ani n aO j we may assume £(/1) ~ 2;rr. Let TI be the tent-shaped region bounded by TVI and two straight lines in D which intersect TVI at angle O. The TI is a "thin sliver." Define U I = UI = D\TI so that aUI n T = II. For n ~ 2 let L~, L~ be the two lines which start at the endpoints of In, go into D, and make angle = 0 with TVn' Let ifi In = {(l - o-I£(/n»e : 0 ~ 0 ~ 2;rr}, and let Un be the domain bounded by the four arcs In, L~, L~, In. Then Un almost fills up a rectangle with length (along T)= 0-2£(ln) and width (in the direction orthogonal to T) = o-I£(/n). Then by elementary estimates on the Poisson kernel, (5.1)

Lemma 5.2. ank is an 7/(e, 0) quasicircle.

= OJ n aOj. Then ank = an k U U Ft yf. We first :Iaim nk is an 7/(e, 0) John domain. It is only necessary to find for every Zo E an k an arc y C n k which has endpoints zo and Zk such that

Proo! Let yf

distance(z, and ~ 7/lz - zol,

D n U;I(ZO) whenever f/J-I

~

> O} C U j ,

f/JI, and such that

== £U/Uj-I(zo) + rei'Pm : r

> OJ) ~ Cd(zo)·

!

(5.2)

~

cd(zo) ,

£::j:. m.

Now let Om be the endpoint of r m on an k and let y_I (resp. yd be the John geodesic from {-I (resp. {d in n k to Zk (the John center of nd. Then the curve y = r -I uri U Y_I U YI surrounds {o and by the John condition on nk,

~ 0 ~ 2;rr}

(5.3)

for all

distance(z, r m)

•

where ~ reO) ~ 1

0

+ re ifi , r

Furthermore, Lemma 5.1 allows us to assume that f'm is a Jordan arc and if z E r f and Iz - zol ~ d(zo), A

+ M)-I

~

£(f'm)

£(h) ~ £(h) ~ ... ~ £(/N). AnN N Define Un = Un \ Uk=1 Uk SO that Un=1 Un = Un=1 Un. Recall that a domam U is called an M Lipschitz domain if there is Zo E U and R > 0 such that

(l

E y.

"J

By reordering we may now assume that

= {zo + Rr(O)e ifi : 0

z

If Zo E ank this is clear by Lemma 3.3. We therefore assume Zo E yf for some j. By Lemmata 2.2, 2.3, and 5.1 there are angles f/J-I < f/Jo < f/JI such that Jf/Jf - f/Jm I 02, £ ::j:. m, such that

{w(z, I", D) ~ CIO} C Un C (W(z, In, D) > C2 0}.

au

OJ =

/j(Un). If we consider any nk, we have for each OJ, such that aO j n an k is an arc, obtained a domain oj C OJ (sometimes oj = OJ) with the property that aOj n aO j C ank. Leth = {OJ: aOj n an k is an arc} andledi k = interior of closure of nk U UFt OJ.

distance({o, y)

~

cd(zo).

(Notice here that we are implicitly using the fact that z E oj implies d(z) ~ C. This, in tum, follows from (5.l) and either Lemma 2.2 or 2.3.) Notice also that the interior of Y must lie entirely in nk. Let Yo be the John geodesic in nk from {o to Zk. We claim that the John condition for n k holds on ro U yo. First suppose that ZEro and Iz - zol ~ d(zo). Then by the distortion theorem for h,

0

and

Ir(O) - r(O')j ~ MJO - Oil.

!

Lemma 5.1. Un is a M(o) Lipschitz domain, I ~ n ~ N. Furthermore, if { E DnaUn,thereisf/J E [0, 2;rr]suchthatthelinesegmentDn{{+re ifi : r ~ O}

distance(z, and ~ distance(z, y) ~ clz - zol.

I

260

CHAPTER 12

Now by inequality (5.2), d(z, aQk) ~ d(z, y) ~ elz - zol

~

1

whenever z E f o and Iz - zol ~ d(zo). (Here we have used the John property on nk to obtain distance (z. Y-l U Yl) ~ elz - zol·) We must finally check the John condition on Yo. Ifz E YO and lz - {ol s: ed(zo), the inequality on distance (z, aO k ) follows from (5.3) and the fact that Izo - {ol :::: Cd(zo). If z E Yo and Iz - {ol > ed(zo), the inequality on distance(z, aQk) follows from the John condition distance(z, ank) ~ elz - {ol. We have thus established that Ok is a- John domain. We now claim that Gk = C\(Od is a John domain. We note that by the definition of Qb aOk = aGk. Now fix a point zo E Gk.

ON REMOVABLE SETS FOR SOBOLEV SPACES IN THE PLANE

n

261

capacity. While j is a quasidisk, Qj n K is not necessarily connected. This means we cannot simply apply Lemma 2.3. We state our result as a proposition; its proof will be broken into several steps. The result we state is far from optimal, but it is all we need. Let Qk be the domain obtained by adding to Qk the set C

U{z

E

OJ: p(z, aOj,k) < I},

j

where p is the hyperbolic metric on OJ. The domains

L~. s:

(.

Qk

then satisfy

C.

~roposition 6.1. Suppose H is continuous on the closure of Qk and harmonic on

Qk \K. Then if

r~

Case A. Zo E (Q j) for some j. (Then j I- k.) First draw the John geodesic in n j from Zo to Zj. We then draw the geodesic (in the Poincare metric ofn) from Zj to 00. By the construction of the domains n j (Lemma 3.2) this is a John geodesic. The union of these two geodesics provides the arc joining Zo to 00.

J.!?ik\K we have the estimate

Cap({z

Case B. Zo E n\ Uj Qj. Let Y be the Poincare geodesic in n from Zo to Then by Lemma 3.2, y is a John geodesic in G k • Case C. Zo ¢. n

00.

as A

t :y3

u Uj fl. j . Then Zo

E

Ojo for some jo. Let

A = sup diameter n j, j

so that A ~ I. If d (zo) ~ 2A there is a half line y (to 00 from zo) which is a John geodesic in Gk. If d(zo) < 2A there is a hyperbolic geodesic (which is also a John arc in Gk) Yl from ZO to Zl E Ojo,f where £ I- k, d(z) ~ I on Yl, and £(y) s: C. (This follows from the definition of the domains OJ,k') By Case A there is a John geodesic Y2 from Zl to 00 in G k. The curve Y = Yl U Y2 is the required JolIn arc. The proof is now completed by first observing that a simply connected domain D with aD locally connected, D = Interior(V), and C\V connected is bounded by a Jordan curve, and then invoking the following fact (see [9]): A Jordan domain D is bounded by a quasicircle if and only if D and C\T> are John domains. •

~::~

~K

I i,!I,

It I

f

I I,t ~"

6 AN ESTIMATE ON CAPACITY Ie ~

E

aQk: IH(z) - H(Zk)1 > A}) = 0(1)

~ 00.

aQk n ank: IH(z) - H(Zk)1 > A} and let E2 = {z E aQk\ank: IH(z)-H(Zk)1 > ),,). ThenbyLemmata2.3and2.5,Cap(E I ) = 0(1) as )" ~ 00, so it is sufficient to show Cap(E2) = 0(1) as)" ~ 00.

Proof. Let EI = {z

E

Step 1. Construction of Some Special Points. Let {Zn) be a collection of points in aOk \an k satisfying IZn - zml ~

1

4: d(zn),

V n, m

and

i~f lz - znl

s: "21 d(z),

­

V Z E ank \ank.

We will now form for each Zn a point z~ E nk. For an arbitrary point z E

aoj,k \ank we let K z = {{ E ank: Iz - {I S 2d(z)) so that Cap(Kz ) ~ cd(z).

By the John condition and Lemma 2.5, there is a point z* E nk such that d(z) ~

d(z*) ~ Iz - z*1 and there is a set Kz C K z such that (6.1)

We now seek to imitate the proof given in Section 3. What is required is an estimate implying that IF(z) - F(zj)! is not too large on aQj, except for a set of small

2

IVHI dxdy = 1.

Cap(K z , z*, n), Cap(Kz , z, OJ) ~

C.

Denote by f a Riemann mapping from nk to D with f (Zk) = 0, where Zk is the "center" of nk' We can move z· slightly so that f(z*) has the form (6.2)

f(z*) = ( l - T f )exp{im2- f n}

262

CHAPTER 12

for some.e, mEN. By this method we produce from our collection {Zn} a new even if n =1= m, but then collection {z~}. Notice that it is possible that z~ = d(zn) '" d(zm) '" IZn - zml.

Z;,

Step 2. Another Geometric Construction. Let {Zn} be the collection of points in Step l. Let {In} be a collection of subarcs of an k\an k such that Un In = ank \ank, In n 1m = ¢ when n =1= m, diameter(In) '" d(zn), and Iz - Zn I .::: d(Zn) for zEIn. We also define I; to be the arc

SO

that by (6.3).

E= UE

+ t)rfJr}:

O::s t .::: I}).

Notice that the sets En are pairwise disjoint. Now define a probability measure J.L * by setting J.L * to have uniform distribution on the center half! I; of I;, J.L*(l;\ 4I;) = 0, and

J.L*(l;) = J.L(E n)· By the construction of I; and

See (6.2) for notation. Then I; has diameter'" d(zn) and if Z E I: the hyperbolic distance from Z to z~ (in nk ) is bounded by C. With the notation of (6.1) we also denote by I n the subarc of T I n = {e ill : m2- f Jr < () ::s (m

+ 1)rfJr}

and we denote by Qn the "square" Qn = {re

iO

: ( l - r ).::: r'::: l , i ll E I n}. i

IZf -

z~ I

Let Z E E and z' E

f

log

4In

Iz' -I ~ I

E* =

=1=

dJ.L

*! (0

=

(Iz'-{I~Ad(z)l

::s c

1 1

+

+c +

+

U1n d

m.

1

(Iz'-{I>Ad(z))

{lz'-{I~Ad(z)}

1 log - - dJ.L(O

Iz -

log

(lz'-{I>Ad(z)}

.::: 2c

,

~I

dJ.L(O

+ y,

•

and Lemma 6.2 is established.

U I;

~I

I

Iz -

In we denote by E* the

Step 4. Proofofthe Proposition. By Lemmata 2.3 and 2.5 and by estimate (6.1),

J. EJ'"

where

IH(z) - H(z*)1 ::s 1.

F = {In: In

E

j: and I n is maximal}.

Step 3. A Capacitary Estimate. Let E and E* be sets as in the previous paragraph.

Now let 'D = nk \ UF Qn, where the Qn are as defined in Step 2 and Qn f- I (Qn)' Then by Lemma4.1, 'D is an (£') John domain. We define our collection F to be {In : 3z E In, IH(z) - H (zk)1 2: A}. Then ifIn E -G IH(O - H (zk)1 2: A - c for all ~ E In. (This is why we slightly enlarge k to Qk .) By our previous estimate,

n

Lemma 6.2. Cap(E*) 2: c Cap(E).

Proof Let J.L be a probability measure on E satisfying

f

log _1-dJ.L(O .::: y,

Iz -

~I

Z

IH(z) - H(Zk)1 2: A ­

E

C.

for any In

E

= {z E

E:

Iz - znl .::: Cd(zn) and Iz - zml

> Cd(zm), m < n},

2c

on

F. Setting as before

We relabel the intervals I n E F so that F = {II. J2, ...} and d(ZI) 2: d(Z2) 2: ... 2: d(zn) 2: d(Zn+l 2: .. '. Define

En

Vn

where n satisfies z* E In. Then

::s Cd(zm).

Finally, if:f is a subcollection of {In} and E = set

4I;,

1 I dist( 2" I;, 2" I~) 2: cd(zn),

We now use the standard terminology that an arc Jm is maximal in a subcollection F of {In} if Jm E F and It E F, i =1= m, imply either Ji n Jm = ¢ or Jf C Jm. Notice (by the John condition) that if Ji C Jm ,

(6.3)

n.

n

4

f-1({(l- rf)exp{i(m

263

ON REMOVABLE SETS FOR SOBOLEV SPACES IN THE PLANE

E =

Ul

n

F

and

E* =

U I*

m'

I;,

iiiijjj,"C'jiii>

-

--

-

'I

'I:lIlW

,..

264

~,._-

-

'~-

CHAPTER 12

we have Cap(E*)

~

c Cap(E). Now since

f1

sequence F n

2

IVHI dxdy

~

E

C

OO

(R 2)

ff

1,

as

A -+

00.

fL2

•

distance(Q j, Qk) ~ 1.

It is an exercise to find domains Qjl' .. " Q jN where h = i, iN = k, and oQj~ n aQj~+l is an arc of diameter ~ c, and N ~ C. (Use the fact that Q is a John domain and each Q j is an TJ quasicircle, i.e., Lemma 5.2.) Then by Proposition 6.1,

2

C\ UZkEFo

) 1/2

c'

2

(flzEKC: Iz-zjl.:5C)

IV FI dxdy

)

We also notice by (6.1) that if Z E Qk and d(z) ~ 1,

IF(Z) - F(zdl

~ C ( fkk\K IVFI

dxdy

Putting our last two estimates together we see there is F(z) = F(z) when d(z) ~ 1 and

rr

JJlzEKc: Id(z)I.:5l1

2

IV FI dxdy

~

C

rr

2

IV Fl dxdy.

F

JJIZEKc : Id(z)I.:5Cj

¢. Vo then

Qj is in the unbounded component of

Qk.

inf p(Zj, zd ZkEF.

IP

2

= fic

It is now an exercise (with induction) to construct connected graphs G n with

the following properties: (8.2) Every edge in G n is ofthe form [Zj, zd for some Zj, Zk E T n +1 U Vo where oQj n OQk =f. ¢. (8.3) Every Zj E Tn is in Vn, the vertices of G n. (8.4) If p(Zj, zd is the graph distance on G n,

1/2

<

IV FI dxdy

(8.1) If d(Zj) ~ 1 and Zj

~ C ~ fkj~\K IVFI dxdy -

W1.2 (R 2 )and

We also build Go so that

F(zj~+)1

(

Fl 2dxdy.

To C Vo C {Zj: 1 ~ d(zj) ~ A 2 }.

m=1

N

rr IV JlrzEKc: d(z).:5 ; }

a

N-I

-

C

Let Q be an (e) John domain with compact boundary K of diameter one, let {Q j} denote the Whitney decomposition of Q into dyadic squares [11], and let Z j be the center of Qj. Let A = A(e) be a large constant and define Tn = {Zj: A-n ~ d(zj) ~ A}. It is an exercise with the John condition to construct a connected graph Go such that every edge in Go is of the form [z j, zd where Q j n 0 Qk =f. ¢, and where the vertices Vo of Go satisfy

Let Q j and Qk be two domains satisfying

L IF(Zj~)

~

8 PROOF OF THEOREM 2

7 PROOF OF THEOREM 1

~

2

E

F(z) when d(z) ~ ~ and

=

IV F n l 2dxdy

SincelKI = O,itfollowsthat F

This completes the proof of Proposition 6.1.

IF(Zj) - F(Zk)1

with Fn(z)

JJIZEKc: d(z).:5 ~ }

it follows from Lemmata 2.3 and 2.5 that Cap(E*) = 0(1)

265

ON REMOVABLE SETS FOR SOBOLEV SPACES IN THE PLANE

~

C,

Zk E G n .

(8.5) G n C Vn + l •

) E

C OO (R 2 ) such that

~. 'J

2

IV Fl dxdy.

Here we are using the fact that, by the construction of the Qt, {z E K C : d(z) ~ I} C Uk Qk. Since the John condition is dilation invariant, we may now build a

Notice that we have chosen Go to be connected. Let ZO E Vo be an extreme point of the (planar set) convex hull (Go). We may assume by induction that each G n is actually a directed graph in the following sense. Each edge [Zj, zd is directed in the sense that (perhaps switching i and k) (8.6) p(Zj, zo) = p(zt, zo) + 1. Such an edge is an outgoing edge from Zj. It is not hard to see that we may choose the G n so that (8.7) Each Zj =f. Zo has exactly one outgoing edge.

__.

..-"'-"'-"'-"'.:"'....,,...,...,,.,,"""""'"

.

.

266

CHAPTER 12

Lemma 8.1. The graph G n is simply connected, i.e., it contains no loops. Proof Suppose, to the contrary, that there is a loop in G n . Let Zj be a vertex in the loop maximizing p(Zj. zo). Then Zj has two outgoing edges (by (8.6» and this contradicts (8.7). Let G = limn G n be the limiting graph, so that G is simply connected. It is clear that K U G is connected. Notice by (8.3) that (8.8) For every Zj E G there is an arc y C G from Zj to zo which satisfies the s' John condition in Q. In other words, G is a John graph. For a Whitney square Qj with Zj E G let {.c~} denote all the edges of G with one endpoint being Z j' Define

11 = {z E aQj : distance(z, .c~) <

8 diam(Qj)},

ON REMOVABLE SETS FOR SOBOLEV SPACES IN THE PLANE

Case 1. Z E Q+ n Qj. There is arc y from Zj to some Zk if. G such that length(y) :::: C, d(z) ::: 1 on y, and Zk is not in the convex hull of aQ. By selecting a suitable ray R from Zk to 00 we then see that [Z,Zj]UyUR

is the required John arc.

Case 2. Z E Q_ U If. Let y be a John arc from Ze (the center of the special Whitney square Qe adjacent to zo) to 00. Then if Z E Qj and y' eGis the John arc from Zj to Zo guaranteed by condition (8.7), we see that [z, Zj] U y' U [zo,

zd U y

•

is the required John arc.

where 8 is a small constant, and put Sj = aQj\

U 11,

j

f:.

267

o.

Yale University

k

For the special point Zo E G we select a Whitney square Qe such that Ze if. G, aQo n aQe f:. ¢, and we put So = aQo\(lf U

U If), k

Q = Q\

U Sj'

•

ZjEG

Lemma 8.2. Q is simply connected. Proof Let Q+ = U{Q

n

Qj : Zj ~

if. G}, Q_ = U{Q n ~

~

Q = Q+ U Q_ U

Qj : Zj E G} so that

0

Ie .

By condition (8.1), Q+ is simply connected (in C), so it is only necessary to check that ~'L is simply connected. We first verify that Q_ is connected. Let Z E Qj n Q_ and let y be an arc in G connecting Zj to Zoo Then y' = [z, Zj] U y is an arc in Q_ which connects Z to Zoo

Now suppose that y is a loop in Q_ that is not homologous to zero. It is then an elementary exercise to homotopy y to y', a loop in G that is not homologous to zero. This contradicts Lemma 8.1. It is clear from the construction of Q that aQ c aQ. To verify that Q is a John domain we must look at two cases.

REFERENCES [l] L. V. Ahlfors. Lectures on Quasiconformal Mappings. Wadsworth, 1966; 1987. [2] . Conformal Invariants. McGraw-Hill, 1973. [3] L. Carleson and P. W. Jones. "On coefficient problems for univalent functions and conformal dimension." Duke Math J. 66 (1992), 169-206. [4] A. Douady and J. H. Hubbard. "Etude dynarnique des polynomes complexes, I, II." Publ. Math. d'Orsay, 84-102. [5] F. W. Gehring and K. Hag. "Quasi-hyperbolic geodesics in John domains." Math. Scand. 65 (1989), 75-92. [6] P. W. Jones. "Quasiconformal mappings and extendibility of functions in Sobolev spaces." Acta Math. 147 (1981), 71-88. [7] R. Kaufman. "Fourier-Stieltjes coefficients and continuation of functions." Ann. Acad. Sci. Fenn. 9, 27-31. [8] O. Lehto and K. 1. Virtanen. Quasiconformal Mappings in the Plane. Springer Verlag,

1973. [9] R. Nlikki and J. Viiisiilii. "John Disks." Exp. Math. 9 (1991),3-43. [10] Ch. Pornmerenke. Univalent Functions. Gottingen, 1975. [11] E. M. Stein. Singular Integrals and Differentiability Properties ofFunctions. Princeton University Press, 1970.

=====-----

---~-~~

OSCILLATORY INTEGRALS AND NON·LlNEAR DISPERSIVE EQUATIONS

269

is continuous. If this is true when the time interval (-00, +(0) is replaced by (-T, T), T = T(uo) > 0, we say that the problem is locally well-posed. From now on, X = HS(JRn) = (f E L 2(JRn): (-,-~)'/2 f E L 2(JRn)}, or X = HS(JRn) = {f E Lfoc(JRn): (_M s /2 f E L 2 (JRn)}. To illustrate these concepts, we present two families of well studied examples.

13

Example 1 (Generalized Burger's equation).

Oscillatory Integrals and

Non-Linear Dispersive Equations

Carlos E. Kenig*

au at

-

(1)

{

k

au ax

+u -

=0

k~

ult=o = uo·

Then (see [58]), (1) is locally well posed in HS(JR), s > 3/2, and is not locally well posed in HS (JR), S :::: 3/2, for any k ~ 1.

Example 2 (Semilinear Schrodinger equation in JR x JR). In this chapter I will describe a collection of results, obtained jointly with G. Ponce and L. Vega, on well posedness and non-linear scattering, with data in Sobolev spaces, for solutions to a variety of non-linear dispersive equations. I hope that I will be able to convey, in describing these results, and the meth­ ods used in their proofs, how many of the ideas pioneered and developed by E. M. Stein, such as complex interpolation, the application of oscillatory integrals to the study of restriction theorems for the Fourier transform, and the application of the Kolmogorov-Seliverstov-Plessner method to study maximal operators in which curvature is present, can be used in a natural way in the study of non-linear dispersive partial differential equations. The results thus obtained are sharp in many instances, and do not seem to be attainable by any other approach. When considering an initial value problem (LV.P.) of the form

au + A(u) = at

-

{

0

ult=o = Uo

where A is a linear or non-linear operator and Uo E X, a Banach space, we say that the problem is globally well posed in X if given Uo EX, there exists a unique u E C«-oo, +(0); X) n V'O«-oo, +(0); X) satisfying ~~ + A(u) = 0, with u (0) = Uo, and such that the mapping

Uo

t-+

u

E

C« -00, +(0); X)

n L 00« -00, +(0); X)

"This research was supported, in part. by the National Science Foundation.

au

(2)

{

a2u

-at + i -ax2 + ilulku = 0,

k > 0

ult=o = uO·

Then (see [58], [21] and [62]), for 0 < k < 4, (2) is globally well posed in L 2 (JR), while for 4 :::: k, (2) is locally well posed in H S (JR), S ~ Sk = k;4, and globally well posed for small data in HSk (JR). Analogous results hold in JRn; there 4 is replaced by 4/n, and k;4 by k~k4 . Example 1 is treated by using energy estimates. Example 2 is treated using the extensions due to R. Strichartz ([57]) of the Stein-Tomas ([61]) restriction theorem for the Fourier transform. In our work, the general strategy when studying a non-linear dispersive problem is to use oscillatory integral estimates to obtain sharp inequalities for the homo­ geneous and inhomogeneous associated linear problems. Many times, these are estimates that involve fractional derivatives and are measured in mixed norms, where the time variable norms are taken first. In order to apply those estimates to our non-linear problems, we develop mixed norm (and weighted norm) Leibniz and chain rules for fractional derivatives. Using our linear estimates, we then construct suitable space-time Banach spaces, in which we solve our non-linear problem by using the contraction mapping principle. In the remainder of this chapter, I will describe the most salient results thus obtained. I will then illustrate the proofs by sketching the arguments used in one

CHAPTER 13

270

particular case. This case will clearly show the impact of the ideas of E. M. Stein on the subject at hand. We start out with our results on the generalized Korteweg-de Vries equation:

OSCILLATORY INTEGRALS AND NON-LINEAR DISPERS1VE EQUATIONS

Remarks on Theorems 4 and 5. (iv) in Theorem 4 follows from (i), (ii), (iii) because of the conservation laws +00

h(u) = au a3 u k au -+-+u-=O at ax 3 ax '

(3)

{

k>l ­

ult=o = Uo

For k = 1, this equatiop. was derived by Korteweg-de Vries as a model for long waves propagating in a channel. Subsequently, the cases k = 1, 2 have been found to be relevant in a number of different physical systems. Moreover, these equations have been studied because of their relation to inverse scattering theory and to algebraic geometry.

Theorem 4. (i) For k = 1, (3) is locally well posed in HS(JR), s > 3/4 , with time interval of length T = T(lIuoIIH')' (ii) Fork = 2,(3)islocallywellposedinW(JR),s::: 1/4,T = T(ll u oIIH'/4). (iii) For k = 3, (3) is locally well posed in HS(JR), s ::: 1/12, T = T(lI u oIlH'/12). (iv) For k = 1,2,3, if s ::: 1, then we have well posedness On any time interval [-T, T], and the solution belongs to V·O(JR; H[s](JR». (v) For k ::: 4, (3) is locally well posed in W (JR), s ::: Sk = (k - 4) 12k. (vi) For k ::: 4, (3) is globally well posed for small data in JiSk (JR). It is also globally well posed in HS(JR), s ::: sk,for data small in JiSk(JR). (i) was first obtained in [33], (iv) for k = 2, 3 was obtained in [31], while for k = 1, it was obtained in [33]. The remaining results are in [36].

We have also obtained a sharp non-linear scattering result.

Theorem 5. Let k ::: 4, and assume that lIuollli'k(IR) ~ Sk, Sk the "smallness" prescribed by Theorem 4(vi). Then, there exist unique wt E JiSt(JR) such that (6)

(8)

{

wlt==o

= Wo

h(u)

=

+00 ( / -00

[ l

R

au - Ckuk+2 ) (x, t) dx. (_)2 ax

-R

a2u

1-2 u(x, t)1 2 dxdt ~ C(R, lIuollHI) ax

and that, (for 1 ~ k ~ 3) if Uo (9)

{ LRr I

J1xl
E

L 2 (JR) , there exists a weak solution such that

2 au (x, t)1 dxdt ax

~ C(R, lIuollu).

In [23], it was shown that, for k ::: 2, the weak solution with HI data is unique. Finally, in [22], and using the methods of [32], [33], and [35], results for k ::: 4, weaker than those in Theorem 4(v), are announced. Note that the results in Theorem 4 are very similar to the ones in example 2. The novelty here is that the non-linear term includes derivatives, complicating the analysis. The results in Theorem 4 are sharp (in a suitable sense) for k ::: 4. To explain this, and to clarify Theorem 5, let us fix our attention on the case k = 4. A perfect balance between the dispersive effect and the non-linearity in (3) is represented by the existence of solitary wave solutions

Uc,k(X, t) = ¢c,k(X - ct),

c > 0

where (10)

(7)

2 u (x, t) dx;

Ixl
lim Ilu(t) - W(t)wtIIH"k = 0,

aw a3 w -+-= 0 at ax 3

1

-00

Previously, ([4], [5], [50], and [27]) the energy method showed local well posed­ ness in H S (JR), s > 3/2, just as in example 2. Moreover, using this result and energy estimates for the second derivative, it was shown in [5] and [50] that there is global well posedness in H S (JR), s ::: 2 (with small HI norm for k ::: 4). In [59] and [28], using the conservation laws /z and h global weak solutions with HI data were constructed (for small HI data if k ::: 4). Moreover, in [28], taking advantage of the term ~:~ in (3), a "local smoothing effect" was discovered (which is not true for solutions of (1». Thus, it was proven that there exists a weak solution u, with data Uo E HI (JR) (small data if k ::: 4) such that

t---+±oo

where u(t) is the solution to (3) given in Theorem 4(vi), and W(t)wo = w(t) is the solution to the associated linear problem

271

¢c,k(X) = {

(k

+2 2)c

sech

2

( k )} 1/4 2.,JCx

A simple computation shows that, for k ::: 4, and Sk = k;4,

lI¢c,kllli'k = at.

l­

CHAPTER 13

272 where ak > 0 is independent of c, and if s as

lI¢c,k II !i' ~ 0

c

i=

Sk.

~

I 6 / U(X,

either 0 or +00,

__

5liJ&!lS2!5

I_4!@.~~

2&~

~ __•

OSCILLATORY INTEGRALS AND NON-LINEAR DISPERSIVE EQUATIONS

£0

2

t) 1 dxdt <

_~

__

_

273

Remarks on Theorem 13. (11) generalizes the KdV hierarchy introduced in [41]. In [18], it was shown that the eigenvalues of the time independent SchrOdinger operator

then

Returning to the case k = 4, the solution that we construct to (3) for lI u oliu ~ has the property that

+00 1+00 1-00 -00 ID

_~!§\~~~

00. 2

This result fails for powers k i= 4 as can be seen from Uc,k. and also forlarge L (lR) nonn as can be seen from u 1,4- Similarly, the scattering result of Theorem 5 cannot hold in the L 2 nonn for k i= 4, or for the L 2 nonn k = 4 and large data.

L(g)

=

d 2

dx 2 - q(x)

remains unchanged when the potential q(.) = u(·, t) evolves according to the KdV equation ~ + ~ + u ~; = O. This discovery was the starting point of the inverse scattering method. In [41], it was shown that the same principle holds for the sequence (KdV hierarchy)

au at

(14)

-

+

[B j ; L(u)]

= 0,

(with [A; M] = AM - MA), and B j denotes the skew-symmetric operator Next, we tum our attention to the results that we have obtained for the hierarchy of the generalized KdV equations. Thus, we consider

I

au

(11)

at

+

2 1 a j+ u ax 2j+l

+

(au Q u, ax"'"

2j

a u)

ax 2j = 0

ult=o = Uo

where j is a positive integer, x, t

E JR

Q:

and

JR2j+ 1 -----+ JR

is a polynomial having no constant or linear tenns, i.e., p

(12)

Q(z) = Q(zJ, Z2, ... , Z2j+l) =

L

aa za ,

with

k::: 2.

lal?k

Theorem 13 ([37]). (i) There exist m, So E N, and 8 > 0 such that, for any s ::: so, and any Uo E HS(JR) n L 2(lxl m dx), of norm less than 8 in that space, (11) is well

posed on a time interval (-T, T), T = T(Q; lI uoII w onL2(1xl'" dx»)' (ii) If k in (12) ::: 3, we can take m = 0 in (i). (iii) If Q does not depend on Z2j+l, the results in (i) and (ii) hold without any smallness assumption on uo. (iv) The results in (ii) are global in time if one of the following two hypotheses is verified: (a) k ::: 4j + 3 in (12), or (b) k ::: 5 and Q(z) = Q(Zi, ... , z2j+d with l ::: (2j - 1)/4.

d 2j +l B j = etj dx 2j+1

j-l (

+

L i=O

dU+l b ji dX 2f+l

d U+1

)

+ dx U+1 b ji ,

and the coefficients b ji = bji(u) are chosen so that the differential operator [B j ; L(u)] has order zero. Equations of the type considered in (13) also appear as higher order models in water waves, in elastic media with microstructure ([42]), and in recent developments in physics ([67]). It is interesting to note that the classical approaches-such as energy estimates, abstract semi-group theory, or space-time estimates-used to study other evolution equations cannot be applied to (13) except for very particular fonns of the poly­ nomial Q. Previously, existence results for the KdV hierarchy (14) were obtained in [48], and uniqueness for (14) was shown in [47]. Both sets of results depended heavily on the Hamiltonian fonn of (14). In [45], for some fifth order equations of the type appearing in (11) both in Hamiltonian and non-Hamiltonian fonn, well posedness results were obtained. Next, we tum to higher dimensional results. We have been able to establish local well posedness results for non-linear Schrodinger equations, with first order derivatives in the non-linearity. Thus, we consider

au

(15)

I

- = if),u + P(u, Vxu, U, Vxu) at

t

ult=o = Uo

where u

= u(x, t) is a complex valued function, and P: C 2n +2 -----+ C

E

JR, x

E

JRn

_.

274

CHAPTER 13

is a polynomial, PCl.) = P(ZI, ... ,Z2n+2) = Ld::S:lal::S:p aaza, and d ::: always assume that there exists aao' laao I =I 0, laol = d.)

2.

(ytfe

Theorem 16 ([35]). (i) Assume that d = 2. Then, there exists 0 = o(P) > 0 such that,jor any Uo E HS(IRn) n H 2n +3 (IRn: Ixl 2n +2 dx), with s ::: So = 3n + 4 + 1/2. and IluoIlH'onH2n+3<1xlz.+zdx) = 00::: o,then(15)islocallywellposedon(-T, T), T = T(oo). (ii) Assume that d :::- 3. Then there exists 0 = o(P) > 0 such that,for any Uo E W(IRn), with s ::: So = n + 2 + 1/2 and lIuollH'o ::: 0, then (15) is

locally well posed on (- T, T), T = T(lIuoll H'o), Remarks on Theorem 16. As mentioned in the discussion of Example 2, the semilinear case P = j(lul)u (with j a real valued function) has been studied extensively. The general initial value problem (15) had been treated before mainly when one could either use energy estimates or deal with analytic data and analytic solutions. Energy estimates for (15) can be established when, using integration by parts, one can show that

L JR.[ (~)ap(U' V'xu,"ii, V'xU) (~)a u dxl ::: Cs(I + lIull~,)lIull~" ax ax

lal<s

for any u E W(JR ), with s > 1 + 1, p = pep) E Z+. The above estimate can be guaranteed only if P exhibits an appropriate symmetry. For example, n = I, P = aax (Iulku), k E Z+ (see [63], [64]), n ::: 1 and a.E!!. P, a au P, j = n

aXj

aXj

I, ... , n are real valued functions. In this case the local well posedness in H S (JRn), s > + 1 follows from the argument used for quasilinear symmetric hyperbolic systems ([26]). Indeed, for those P's the same proof works if one removes the Laplacian term from the equation in (IS). Also, when n = 1, P = aax (luI 2 u), in [30] the inverse scattering method is applied.

1

The approach that deals with analytic data uses analytic function techniques to overcome the loss of derivatives introduced by the non-linearity ([25]). Our method works without any special assumptions on P. It works for real or complex valued functions, or for systems of equations. The same method applies to non-linearities given by smooth functions F (u, V'xu, "ii, V'xU) with Taylor expansion at the origin having no constant or linear terms. We have not attempted to obtain the best results provided by our method, since, in any case, it is not clear that they would be optimal.

OSCILLATORY INTEGRALS AND NON-LINEAR DISPERSIVE EQUATIONS

275

Nevertheless, in cases where both the energy estimate and our method apply, our method seems to give better results. An instance of this is our study of the initial value problem for the generalized Benjamin-Ooo equations

(17)

au at

k

au ax

a

- +u­ - ax Dxu = 0, t, x E JR, k E z+

{

ult=o = Uo

where D x = (- aa;2 ) 1/2. In the case k = I this equation was deduced in [3] and [43] as a model in internal wave theory. The generalized Benjamin-Ono equation presents the interesting fact that the dispersive effect is described by a non-local operator and is weaker than the one exhibited by the Korteweg-de Vries equations. The energy method proves local well posedness in H S ,s > 3/2. Our result here is Theorem 18 ([34]). (i) Let k::: 2 and s be such that S>1

k=2

s > 5/6

k

s ::: 3/4

k ::: 4.

I

=3

There exists 0 = o(k) > Osuchthat,joranyuo E HS(IR),lIuolIs::: 0,(17) is locally well posed on (-T, T), T = T(lIuolls, k). (ii) For k ::: 4, s ::: 1 there is global well posedness,jor small initial data. (iii) For k ::: 4, Uo E HI (JR), II Uo 111 ::: 0, if u is the solution oj (17) there exist unique Wo± E HI (JR) such that lim Ilu(t) - V(t)WO±IIHl = 0, t-->±oo

where V(t)wo is the solution oj

av at

I

a ax x = Wo

---Dv=O vlt=o

Remarks on Theorem 18. The previou~ly known results for (17) were: For k = 1 or 2, Uo E HS(JR), s = 0 or 1/2, there exists a weak solution u with u E Lfoe (JR; ~': 1/2) (local smoothing effect) ([20]). For k = I, the same is true for 1 Uo E H (JR) ([20], [60]). For k I, (17) is globally well posed in H 3/ 2(JR)

=

CHAPTER 13

276

OSCILLATORY INTEGRALS AND NON-LINEAR DISPERSIVE EQUATIONS

([44]). For k > 1, (17) is locally well posed in HS(lR), S > 3/2 ([1], [24]). Note that none of these results requires a restriction on the size of the initial data. The results of Theorem 18 extend also to the initial value problem

{ where Pj

:

C2

--+

1,2 are polynomials, with

C, j

Pj(ZI, Z2)

=

Next, in order to illustrate our method, we will sketch a proof of Theorem 4(v),

= 8. Thus, we will prove

Theorem 20. There exists 8 > 0 such that for any Uo IID /4 uoIlU(lR) < 8, there exists a unique strong solution of ' au at

a u ax

8

E

H

14 / (R)

with

au ax

ult",o

= Uo

satisfying

u E C(R;

lIullL I2 L OO ,

Moreover, the map Uo B

=

{f

E

au

liD 1/4 -11L''''L2 ax x t

H1/ 4 (R» n L OO(R; H1/ 4 (R»; <

x

t-+

yE]R

u(t)from {uo

C«-oo, +00); H .

sup II Diyu II L4 L OO <

+00;

max["fIlLoo(]Roli L/4 (]R));

14 / (JR»:

E

x

H1/ 4 (1R):

t

<

+00;

+00.

t.J

=

W(t)uo (Le., fo

=

IIfoliB S CII uoII li1/4.

u(t)

H1/ 4 (IR), v

E Ba

=

we define

av W(t - t')(v 8 - )(t') dt' o ax a(8) such that, if lI uo II lill' < 8, then 4>(B) C B,

= 4> uo(v)(t) = W(t)uo

i

-

t

Notice that once 23 is established, Theorem 20 follows from the contraction mapping principle and Duhamel's formula. We start out with a verification of (22). These are inequalities for oscillatory integrals, which are versions of the "local smoothing effect" of Kato ([28]), the maximal function estimates of L. Carleson and others ([7], [38]), and the L 2 restriction theorem of Tomas-Stein ([61]) and Strichartz ([57]). Note first that the inequality

is Lipschitz.

In order to prove Theorem 20, we introduce the quantities 4

IID 1/ f11u

aw (t)woll

.J

<

sup

B: II fliB S a}. We have the following two

Lemma 24. Let W(t)wo be defined as in (7). Then

I

-00<1<+00

E

is an immediate consequence of the fact that W (t) is a unitary group on L 2 (JR).

IID ' /4 af II Diy fIIL 4 LtOO ] ax IIL oo L2 ; IIfIIL 12 Loo ; sup Y

f31(f)=

{f

f31 (fo) s lI uoII li l /4

lIuolilil/4 < 8} into

IIf11B =

A:

=

(22)

then there exists 8 > 0, a and 4> is a contraction.

-+-+u-=O 3 {

2

t

Claim 21. If Uo E H1/ 4 (R), and we let fo r~:: eiW+X~)uo(~) d~ is the solution of (7», then

Claim 23. If for Uo E

3

= IIf11L:2L~

and for a > 0, we let Ba claims:

'" afJ­ L..a+fJ~2 ·ajafJzl Z2'

case k

f33(f)

x

Y

=0

= Uo

u!t"'O

af oo = liD 1/4 -II ax L L

f34(f) = sup II Diy fIIL 4, LIoo

au au au a2 u - + i -2 + P1(u, u) - + P2(U, Ii) at ax ax ax

(19)

f32(f)

277

+oo}

11

ax

LOOL2 x

s

Cllwollu.

I

Proof -Ix W(t)wo = J ei(t~3+X~)(i~)wo(~) d~. Now, fix x, and perform the change of variables 1) = ~3. Plancherel's theorem in t and a new change of variables shows that, for each fixed x,

1

+00

-00

I-axa W(t)wo

The lemma follows from this.

2

1

dt

= cllwolli2'

•

278

CHAPTER 13 OSCILLATORY INTEGRALS AND NON-LINEAR DISPERSIVE EQUATIONS

Note that this is, for the solution of (7), a refinement of the estimate (9) obtained by Kato for solutions of the Korteweg-de Vries equation. Local smoothing effects for linear dispersive equations have attracted a lot of attention recently ([13], [14], [66], [51]). This sharp form of it was first observed in [65].

(32)

Corollary 25. fj2(W(t)Uo) :::: clluoll/i'/4.

In tum, the left-hand side of (32) squared equals

Lemma 26. For Uo

This is because, by duality, (30) is equivalent to

U.:OO D-'/ W(t)0(-, t)dt 4

E ifl/4 we have the estimate

IIW(t)uoIlL rI2 Lxoo

::::

f (f

D-

=

ff

CII uoll/iI/4.

14 / W(t)g(-,

g(x, t)

(f

t) dt)

D-

L"

Clio II L:"" L) .

(f n-

12 / W(t

Fractional integration in t now finishes the proof.

Lemma 27. Let I!(x) = f+oo ei(le+X~).!!:1. Then , a -00 1~18 •

Corollary 33. fj3(W(t)Uo) :::: CII u oll/il/4.

J

(28)

I Cs IIs(x)1 :::: Ixl H

(29)

Cs IIs(x)1 :::: Itl(l-S)/3

for

I

1 - < (l < 1

2 ­

for

We claim that (30) is in tum equivalent to 00

/+ 11

-00

00

111:

1:

00

D-'I'W(t - t')o(-, t')dtl" "

c

It _

:'1'/6110<-' t')IIL)dt'.

Lemma 34. For Uo E if 1/4 we have the estimate IIW(t)uoIIL 4x LIoo

::::

CII uoll/il/4.

2 ­

IID-I/4W(t)uoIlL:2L~ :::: CIIuo1lL2.

(31)

Hence, (32) and (31) are equivalent. In order to prove (31), we apply Lemma 27, estimate (29), (l = ~, to obtain

1 --<(l
The proof of Lemma 27 is an application of the version of Van der Corput's lemma found in E. M. Stein's article ([54]). See [31] and [36] for the details. Once we have Lemma 27, the proof of Lemma 26 follows the lines of the proof of the Tomas-Stein ([61]) L 2 restriction theorem (see [36]). In fact, 26 is equivalent to the estimate (30)

I /4W(t')g(-, t') dt')dX

- t')g(-, t')dt') dxdt.

This is a version of the Strichartz type inequalities ([57]) for the Airy equation (7). Its connection with the restriction theorem for the curve r = (;, ;3) is that, eiW+x~luo(;)d;, the solution operator to (7) is the dual of since W(t)uo = the operator restriction of the Fourier transform to r (i.e., the extension operator) applied to the density uo(nd; viewed as a measure on r. Lemma 26 says that if 1;11/2Iuo(;)1 2d; < +00, the extension operator yields a function in LJ2LC:. By duality this is equivalent to a restriction estimate. The main tool in the proof of Lemma 26 is an estimate for oscillatory integrals.

J

279

D-I/2W(t - t')g(-, t')dt'll

:::: L12L~

CIIgIlL~2flIL~'

Lemma 34 is a maximal function inequality for the solution of (7). It, of course, implies that limq,o W(t)uo = uo a.e., for Uo E ifl/4. This kind of a.e. conver­ gence problem was first studied by L. Carleson ([7]), who showed that, for the linear Schrodinger equation on JR, Uo E ifl/4 suffices for a.e. convergence. Later on, in [15] it was observed that the exponent 1/4 is sharp, and in [38] the maximal inequality in Lemma 34, for the case of the Schrodinger operator is proved. It was E. M. Stein who first noticed that inequalities as those in 34 are simple con­ sequences of the Kolmogorov-Seliverstov-Plessner method. Lemma 34 was first proven in [65]. To prove 34, note that, arguing as in the proof of Lemma 26, it suffices to show that (35)

II

1 2 D- /

/+

OO

-00

W(t - t')g(-, t')dt'll

:::: L~L~

CIIgIlL~/3L:'

t--~:--~

I

_.7_ ---

-

-

~',-. .-:-...... ,---;-~~

.•

,::'..,=---'_..,.

._~-- ;.~~~!:!.._".--,-

-.~~'--'-

--

280

--

.--.---- - _.::-

....~:

._._--=--_..

CHAPTER 13

To show (35), we use Lemma 27, estimate (28), with 0

1:

00

II D-'{2

=

1/2, to obtain

"ldl'L "c Ix :'/2 * Ilg(-, I) lit:,

W (I - I')g( -,

~",~_."":""",,:._-~---,.,,,~~~,",,"""~~;'~~~~~' .. ,_._.~-_.':""~ .. _ ,_., ..•..__ _.•.• ._ _ "_..• _0'.'_ .,...... •• " ·•. m •• " ••••• , . ,•• _. __ • . •" , . , .

and (35) follows by fractional integration in x.

CIIuoIlIi'/4.

E

*

)IL

dt

~ Ca

9

liD" F(f)IIL xPLqt ~ CIIF' (f)IILP' LIql liD" fII LxP2 Ltq2, x

if r

+

1

+00

-00

~ Co

lIuliB

+C

+00

1

av 4 8 IID I / (v - )lIudt' ax

(i)

~ Co

I

+00 D 1/4 ( VI8 -aVI ax

where

Vi E

-

E

[0, a] with a

=

..!. PI

+

= al + a2. Let ..!., 1 = .1 + .1 P2

q

q,

q2

=

=

(ii) IID"(fg) - fD"g - gD" flip

:::

CIIglloolID" flip.

Let us now explain why (38) should hold. By the Leibniz rule, DI/4(v8 av) '=' v 8 D I / 4 av ax ax

+ D I / 4(v 8 ) av . ax

For the first term, note that

+ Ca 9 •

av

av

IIv 8 D I / 4 -IIL2 ~ IIvll~oo IIv 2D I /4 -IIL2 ax x x ax x v28 -aV2 ) ax

II L;

dt ~

ClIvI -

v211Ba 8 ,

and so

~ loo

+00

Ea , we see that

IIUI -

(0, 1), aI, a2

E

IID"(jg) - fD"g - gD" fIIL"L q ~ CIID"'fIILP'L q , II D"2g II LP2 Lq2. MoreoverJor al 0, the valu; ~I 00 is allow~i x r

Using the following variant of (38)

-00

L~:'(lR), then

liD" F(f)hll p ~ CIIF'(j)lIooIID"(j)M(hrp)l/rPllp,

(42)

by (22). Using (38) we see that

1

E

P. PI, P2, q, ql, q2 E (1,00) be such that'! P Then

av 8 IIW(t - t')(v - ) II Bdt' ax

-00

(39)

> 1, h

Lemma 43. Let a

•

Taking (38) for granted, in order to establish 23, we see that, by Minkowski's inequality lIu(t)IIB ~ IIfoliB

t

where M denotes the Hardy-Littlewood maximal operator.

Ea , then

l:oo IID I / 4 (V 8 : :

(38)

281

the HP theory as developed by C. Fefferman and E. Stein ([17]), as well as the "tent spaces" of Coifman-Meyer-Stein ([12]). The results needed here are:

Also,

If v

".

OSCILLATORY INTEGRALS AND NON·LlNEAR DISPERSIVE EQUATIONS

(41)

This finishes the proof of (22). In order to prove (23) we need a non-linear estimate, which is contained in the following:

Lemma 37.

_~

Lemma 40. Let F be of class C I , F(O) = 0. Let a E (0, n, p, q, PI, P2, q2 E O,oo),ql E O,oo]besuchthat ~ = + -};. ~ = +~. Then

I

Corollary 36. ,84(W(t)Uo) ~

.~~%:~

u211B ~

ClIvI -

8 v211Ba ,

and hence 23 follows. We will now indicate the proof of (38). What is needed here are vector valued versions of the Leibniz and chain rules for fractional derivatives. Previous results in this direction are due to [56], [29], [9], and [58]. Our results are a consequence of the ideas of Coifman-Meyer ([10]) and Bony ([6]), and the vector valued inequali­ ties of C. Fefferman-Stein ([16]), Benedek-Calderon-Panzone ([2]), and Rubio de Francia, Ruiz, and Torrea ([46]). To deal with the endpoint results, we also need

av 8 I 4 IIv D / ax ilL;

+00

~ ( loo IIVII1~ )

1/2

(ff Iv

2

av I 4 D / ax

2 1

1/2 dXdt)

~ ,83(v)6 ,84 (v)2 ,82(V).

For the second term, by the chain rule, it "equals" v7 D I / 4 (V) interpolation theorem ([52]) now shows that IIDI/4(v)IILaL~ ~ C,82(vl,84(V)HI x

II

I

av ax IIL~L~ ~ C,82(V)~,84(V)I-~,

i;. Stein's complex

-_. -

=-""'"-i~-'-'---'';'~~;;:'

~~_.-

<-- ...:

"._.~;:.

~._- ,_:.._;=:.••; •. --;.:.=-~r.::~~;(~T~"V~".:

. . . .""''''-'''''' .•''''.

282

CHAPTER /3

[II]

where

1

OSCILLATORY INTEGRALS AND NON-LINEAR DISPERSIVE EQUATIONS

(1 - (J)

1

=00- + -4 - :a=2+ a

(I-B)

1

(1 - '7)

(J

'7

(1-'7)

=00- + -4 /-L

(J

1

TJ

-v = -2 +

00

1

'

00'

1

-4 = B(1 + -) + (1 4 1

1

= '7(1 + -) + (1 4

- B)O

- '7)0.

(It is in this step that Diy f in the definition of fJ4 is needed.) Then,

IIv?D

1 4 / (v)

av

-IIL 2

ax'

av

:::

CIIvll~oollvD1/4(v)-IIL2­

,

ax '

. One then proceeds as before, and then has to estimate

(ff

!vD 1/ 4 (V)

::

f dXdt) 1/2

This is accomplished by means of Holder's inequality and the above estimates.

Finally, the error terms are handled by using (40) and (43) appropriately. The

details are given in [36]. (39) is proven in a similar manner, and this concludes

our proof.

University of Chicago

REFERENCES [1] L. Abdelouha~, J. L. Bona, M. Felland, and J. C. Saut. "Nonlocal models fornonlinear

dispersive waves." Physica D 40 (1989), 360-392.

[2] A. Benedek, A. P. Calderon, and R. Panzone. "Convolution operators on Banach space

valued functions." Proc. Nat. Acad. Sci. 48 (1962),356-365.

[3] T. B. Benjamin. "Internal waves of permanent form in fluids of great depth." J. Fluid

Meeh. 29 (1967),559-592.

[4] J. L. Bona and R. Scott. "Solutions of the Korteweg-de Vries equation in fractional

order Sobolev spaces." Duke Math. J. 43 (1976), 87-99.

[5] J. L. Bona and R. Smith. "The initial value problem for the Korteweg-de Vries

equation." Roy. Soc. London, Ser A 278 (1978), 555--601.

[6] J. M. Bony. "Calcul symbolique et propagation des singularires pour les equations aux

derivees partielles non lineaires," Ann. Sci. E. N. S. 114 (1981), 209-246.

[7] L. Carleson. "Some analytical problems related to statistical mechanics, Euclidean

harmonic analysis," Lecture Notes in Mathematics, no. 779. Springer Verlag, 1979.

[8] T. Cazenave and F. B. Weissler. "The Cauchy problem for the critical nonlinear

SchrOdinger equation in HS," Nonlinear Anal. TMA 14 (1990), 807-836.

[9] F. M. Christ and M. I. Weinstein. "Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation." To appear in J. Funct. Anal. [10] R. R. Coifman and Y. Meyer. "Au dela des operateurs pseudodifferentielles." Asterisque 57 (1973).

283

. "Nonlinear harmonic analysis, operator theory and P.D.E," In Beijing Lec­ tures in Harmonic Analysis, edited by E. M. Stein. Princeton University Press, 1986. [12] R. R. Coifman, Y. Meyer, and E. M. Stein. "Some new function spaces and their applications to harmonic analysis," J. Funet. Anal. 62 (1985), 304-335. [13] P. Constantin and J. C. Saut. "Local smoothing properties of dispersive equations," J. Arner. Math. Soc. 1 (1988),413-446. [14] W. Craig, T. Kappeler, and W. A. Strauss. "Gain of regularity for equations of KdV type," Preprint. [15] B. Dahlberg and C. E. Kenig. "A note on the almost everywhere behavior of solutions to the SchrOdinger equation, Harmonic Analysis," Lecture Notes in Mathematics, no. 908. Springer Verlag, 1982. [16] C. Fefferman, and E. M. Stein. "Some maximal inequalities." Amer. J. Math. 13 (1971), 107-115. [17] . "HP spaces of several variables," Acta Math. 129 (1972),137-193. [18] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura. "A method for solving the Korteweg-de Vries equation," Phys. Rev. Letters 19 (1967), 1095-1097. [19] . "The Korteweg-de Vries equation and generalizations. VI. Method for exact solutions," Comm. Pure Appl. Math. 27 (1974), 97-133. [20] J. Ginibre, and G. Velo. "Smoothing properties and existence of solutions for the generalized Benjamin-Ono equations." Preprint. [21] . "Scattering theory in the energy space for a class of nonlinear SchrOdinger equations," J. Math. Pure Appl. 64 (1985), 363-401. [22] . "Smoothing properties and retarded estimates for some dispersive evolution equations," Preprint. [23] J. Ginibre and Y. Tsutsumi. "Uniqueness for the generalized Korteweg-de Vries equations," SIAM J. Math. Anal. 20 (1989),1388-1425. [24] N. Hayashi. "Global existence of small analytic solutions to nonlinear Schrodinger equations," Duke Math. J. 62 (1991), 575-592. [25] R. J. Iorio. "On the Cauchy problem for the Benjamin-Ono equation." Comm. Part. Diff. Eq. 11 (1986), 1031-1081. [26] T. Kato. "Quasilinear equations of evolution, with applications to partial differential equations," Lecture Notes in Mathematics, no. 448. Springer Verlag, 1975. [27] . "On the Korteweg-de Vries equation," Manuscripta Math. 19 (1979), 89-99. [28] ~. "On the Cauchy problem for the (generalized) Korteweg-de Vries equation." Adv. Math. Suppl. Stud. Appl. Math. 8 (1983), 93-128. [29] T. Kato and G. Ponce. "Commutator estimates and the Euler and Navier-Stokes equations," Comm. Pure Appl. Math. 41 (1988), 891-907. [30] D. J. Kaup and A. C. Newell. "An exact solution for a derivative non-linear Schrodinger equation," J. Math. Phys. 19 (1978), 798-801. [31] C. E. Kenig, G. Ponce, and L. Vega. "On the (generalized) Korteweg-de Vries equation," Duke Math. 1. 59 (1989), 585-610. [32] . "Oscillatory integrals and regularity of dispersive equations," Indiana Univ. Math. J. 491991,33-69. [33] . "Well-posedness of the initial value problem for the Korteweg-de Vries," J. Amer. Math. Soc. 4 (1991), 323-347.

~'k

<

:.._

__

.:::

~_:~":"

__

_

.m_,:::'~:~;._

~;._

. .:=~~-"",.~",,~

.,.,.

,~~£~~=="="",,,~~~~~~_~~,,;'e:.,,"g:~~~;;'~~

284

CHAPTER 13

[34]

. "On the generalized Benjamin-Ono equation." To appear, Trans. Amer. Math. Soc.

[35]

. "Small solutions to nonlinear SchrOdinger equations." To appear, Ann. Inst. Poincare, Analyse Non-Lineaire. [36] . "Well posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle." To appear, Comm. Pure Appl. Math. [37] . "On the hierarchy of the generalized KdV equations." To appear, Proceed­ ings. Nato Lyon Workshop on Singular Limits of Dispersive Waves. [38] C. E. Kenig and A. Ruiz. "A strong type (2, 2) estimate for the maximal function associated to the SchrOdinger equation." Trans. Amer. Math. Soc. 230 (1983), 239-246. [39] S. Kichenassamy and P. 1. Olver. "Existence and non-existence of solitary waves solutions to higher order model evolution equations." Preprint. [40] D. J. Korteweg and G. de Vries. "On the change offonn oflong waves advancing in a rectangular canal, and on a new type oflong stationary waves." Philos. Mag. S (1895), 422-443. [41] P. D. Lax. "Integrals of nonlinear equations of evolution and solitary waves." Comm. Pure Appl. Math. 21 (1968),467-490. [42] P. L. Olver. "Hamiltonian and non-Hamiltonian models for water waves." Lecture Notes in Physics, no. 195. Springer Verlag, 1984. [43] H. Ono. "Algebraic solitary waves in stratified fluids." J. Phys. Soc. Japan 39 (1975), 1082-1091. [44] G. Ponce. "On the global well posedness of the Benjamin-Ono equation." Diff. and Int. Eqns. 4 (1991), 527-542. [45] . "Lax pairs and higher order models for water waves." To appear J. Diff. Eqns. [46] J. L. Rubio de Francia, F. J. Ruiz, and J. L. Torrea. "Calder6n-Zygmund theory for operator-valued kernels." Adv. Math. 62 (1988), 7-48. [47] J. C. Saut. "Sur quelques generalisations de l'equations de Korteweg-de Vries, II." J. Diff. Eqs. 33 (1974), 320-335. [48] . "Sur quelques generalisations de l'equations de Korteweg-de Vries." J. Math. Pures Appl. S8 (1979), 21~1. [49] J.-C. Saut and R. Temam. "Remarks on the Korteweg-de Vries equation." Israel J. Math. 24 (1976), 78-87. [50] M. Schwarz, Jr. "The initial value problem for the sequence of generalized Korteweg­ de Vries equation." Adv. Math. S4 (1984),22-56. [51] P. Sj61in. "Regularity of solutions to the SchOdinger equations." Duke Math. J. 5S (1987),699-715. [52] E. M. Stein. "Interpolation of linear operators." Trans. Amer. Math. Soc. 83 (1956), 482-492. [53] . Singular integrals and differentiability properties of functions. Princeton University Press, 1970. [54] . "Oscillatory integrals in Fourier analysis." In Beijing Lectures in Harmonic Analysis, edited by E. M. Stein. Princeton University Press, 1986. [55] E. M. Stein and G. Weiss. Introduction to Fourier Analysis in Euclidean Spaces. Princeton University Press, 1971. [56] R. S. Strichartz. "Multipliers in fractional Sobolev spaces." J. Math. Mech. 16 (1967), 1031-1060.

OSCILLATORY INTEGRALS AND NON-LINEAR DISPERSIVE EQUATIONS

[57]

28S

. "Restriction of Fourier transforms to quadratic surface and decay of solutions of wave equations." Duke Math. J. 44 (1977), 705-714. [58] M. E. Taylor. "Pseudo-differential operators and nonlinear P.D.E." Preprint. [59] R. Temam. "Sur un probleme non lineaire." J. Math. Pures Appl. 48 (1969),159-172. [60] M. M. Tom. "Smoothing properties of some weak solutions of the Benjamin-Ono equation." Diff. and Int. Eqns. 3 (1990), 683~94. [61] P. Tomas. "A restriction theorem for the Fourier transform." Bull. Amer. Math. Soc. 81 (1975),477-478. [62] Y. Tsutsumi. .. L 2- s01utions for nonlinear SchrOdinger equations and nonlinear group." Funkcialaj Ekvacioj 30 (1987), 115-125. [63] M. Tsutsumi, and I. Fukuda. "On solutiQns of the derivative nonlinear SchrOdinger equation. Existence and Uniqueness Theorem." Funkcialaj Ekvacioj 23 (1980), 259­ 277. [64] . "On solutions of the derivative nonlinear SchrOdinger equation, II." Funkcialaj Ekvacioj 24 (1981),85-94. [65] L. Vega. Doctoral Thesis, Universidad Autonoma de Madrid, 1987. [66] . "The SchrOdinger equation: pointwise convergence to the initial data." Proc. Amer. Math. Soc. 102 (1988),874-878. [67] E. Witten. "Two dimensional gravity and intersection theory on moduli space." Preprint.

SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS

14

Singular Integrals and

Fourier Integral Operators

D. H. Phong*

I INTRODUCTION The theory of singular integrals and pseudo-differential and Fourier integral oper­ ators [17], [39], [43], [45] constitutes one of the most impressive achievements in classical analysis of the last few decades, both in its wide ramifications and in the coherence and unity of its structure. The operators falling outside of the scope of this classical theory have usually surfaced in widely disparate areas of analysis and geometry, often to the extent that their mutual existence was hardly known. They required methods devised individually, and with more limited success in many cases. However, they have been appearing increasingly frequently in recent years, and remarkably, many common threads have begun to emerge. In this chapter we shall focus on two themes which seem to unify many of the very disparate operators not covered by the classical theory of Fourier integral operators. In the first theme the Lagrangian manifold may fail to be locally the graph of a canonical transformation. This was the case for diffraction theory [19], [20], [41], [42], where the Lagrangian is a Whitney fold. More systematically, it is generically the case whenever we deal with Radon transforms along submanifolds of codimension higher than 1. Thus Radon transforms and maximal functions along families of curves [34], [25], [40] and X-ray transforms [10], [12] are prime examples, as are analysis and deformation theory on manifolds of geodesics [10], [14], [44]. In the second theme, the densities of the Fourier integral operators 'This research was supported, in part, by the National Science Foundation, under grant DMS-90­ 04062.

287

are allowed singularities of an either fractional or Calder6n-Zygmund type. This again seems dictated by rather general situations. A phenomenon probably typical of subelliptic problems is the following. The main term of the Green's function for the a-Neumann on the Siegel upper half-space [29], [32], [3] is smooth outside of the diagonal of the Heisenberg group. Yet it satisfies no better estimates than a ker­ nel supported on the distribution of maximal complex hyperplanes with additional Calder6n-Zygmund singularities, and must be treated as a singular Radon trans­ form [31], [32]. Another case of a singular density is the fundamental solution of a hyperbolic partial differential equation [21], [15]. Its wave front set is contained in the union of the cotangent space at the origin with the flow out of its intersection with the characteristic variety of the equation. This occurrence of two intersecting Lagrangians is shared by singular Radon transforms, with the Lagrangians given in this case by the diagonal and the normal bundle of the distribution of hyper­ planes. Recognition of this common feature has already borne fruit, as it has led to a microlocal proof of the boundedness of singular Radon transforms [11]. It is very intriguing that the two themes we have just described seem to be themselves intimately related. In fact, composition of the Fourier integral operators singular in the first sense leads under different guises to Fourier integral operators singular in the second sense. Examples of such occurrences are given in Sections 11.3 and IV below. It can be hoped that a theory with composition calculus and Sobolev and LP - Lq bounds can be constructed which encompasses these degenerate Fourier integral operators and singular Radon transforms. Here we shall discuss encouraging progress in a number of directions [34], [35]: first, in a class of operators modeling Fourier integral operators with arbitrarily high order of degeneracy, and whose bounds can be formulated in terms of the stratification of the Lagrangian; second, in the LP - Lq bounds for Radon transforms with folding canonical relations and their relation with singular Radon transforms; third, in establishing bounds even in some cases when a normal form for the Lagrangian may not be available; and finally, in the understanding of the role of the Monge-Ampere determinant in the localization procedures required for the Sobolev estimates. This paper is organized as follows. In Section II we provide an introduction to several problems in Fourier analysis and integral geometry leading to the operators considered here. In Section III we describe bounds for the model Fourier integral operators. The key ingredients in our approach are a refined method of stationary phase with uniform bounds and sharp estimates for the distances between the roots of an algebraic equation. In Section IV we present some aspects of Radon transforms on curves in two dimensions, with special emphasis on the emergence of the Monge-Ampere determinant in the hard analysis, and of singular densities in the composition.

~~<'t!~ijtlt&""i!lit\\i&t
288

CHAPTER 14

This paper describes joint work with E. M. Stein, some of which stretches back to the days when I was one of his graduate students. It has been an exceptional privilege to carry on a research program with him continuously since that time. On this occasion of his sixtieth birthday, I would like to express my immense gratitude to him, and also my pride and happiness at participating in an enterprise which, if I were to judge from the boundless energy and enthusiasm he brought to it, must have been especially close to his heart.

U BASIC EXAMPLES In this section we provide a brief discussion of several areas of analysis and integral geometry leading generically to singular Fourier integral operators. The singular­ ities emerge under many different guises, either in the symbols or in the projections of the Lagrangians, and it is remarkable that they all seem to be intimately related.

1 Radon Transforms along Curves We consider first the distribution of curves (Mp = P + y(t), P ERn} in Rn given by translates of a fixed curve y(t ), and the corresponding integral transform (1)

(Tf)(P)

==

r+

oo

J-oo

f(P

+

y(t»X(t)dt

==

r

JM

r+ J-oo

C = {(P, Q)

(4)

E

R n x Rn ; Q

m(A) =

ei{A,y(I»)

X(t)dt.

The most natural geometric condition, namely that y(t) have non-zero torsion at t = 0, i.e., that the n vectors y(O), ... , y(n)(o) be linearly independent, is also the one guaranteeing uniform bounds in Afor the phase function in (2)

(3)

I(A, y}(O) I + ...

+ !(A, y(n)(O)}1

~ ciAI.

The van der Corput lemma implies Im(A)1 S CIAI- I/ n, and we obtain the sharp estimate Theorem 1. If y(t) has non-vanishing torsion at 0, then the operator T of(1) is boundedfrom H(s)(Rn) to H(s+l/n)(Rn).

E M p }.

Its wavefront is contained in the Lagrangian manifold N*(C) C T*(R n) x T*(Rn). The calculus of Fourier integral operators applies and gives L 2 estimates when the projections JrL and JrR from N*(C) on the left and right factors

/N'(C)~

(5) T*(R n )

T*(R n )

have invertible differentials. By convention the P variables are on the left and the Q variables a~e on the right. Now an element (P,~; Q, 1/) is in N*(C) when Q = P + y(t), ~ + 1/ = 0, and

(6)

(1/,

An element (8P, (7)

8Q

8~; 8Q,

= 8P + y8u,

y) = O.

81/) is then in T(P.~;Q''1)(N*(C» when (81/, y)

+ (1/,

ji)8u = 0,

for some 8u in R. In particular the kernel of dJrL at exactly at points where

oo

(2)

289

This simple example already falls outside the scope of the classical theory of Fourier integral operators. Indeed it exploits an n-th order derivative condition (torsion) rather than the second order conditions of Fourier integral operators. More precisely the kernel of (1) is the Dirac 8 measure along the defining relation

f

Co

---:&f:i:it:>£i6:,~

SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS

p

cut-off function, == I and supported in a small enough where X is a fixed neighborhood of O. The question is to determine geometric conditions on the curve y(t) which insure the optimal regularity properties for T. As a convolution operator the multiplier of T is given by

"l~~

. ii'iMiil4tiiii;Jtiiiij

(8)

(1/, ji)

(P,~;

8~

+ 81/

= 0

Q, 1/) is non-trivial

= O.

For n > 2 there are at least n - 2 directions 1/ satisfying both (6) and (8), and the Lagrangian N*(C) fails there to be a local graph. What are the optimal estimates when the condition that N*(C) be a canonical graph breaks down? A classic theorem of Hormander [16], [17] says that when the kernels of dJrL,dJrR have dimension at most k, then the corresponding Fourier integral operator satisfies bounds which lose -k/2 derivatives compared to when the Lagrangian is a local graph. In the formalism of Fourier integral operators, the Radon transform along curves (1) is of class I- 1/ 2 (M, M, N*(C». The fact that it is of order -1/2 can be seen by choosing n - 1 defining functions <1>1 (P, Q) = ... = <1>n-l (P, Q) = 0 for (4) and representing the Dirac measure along (4) as a Fourier integral distribution with oscillating variables AI, ... , An-I. phase

290

CHAPTER 14

L~:: Ajj(P, Q), and symbol I

=

8c
f

(12)

exp(i

n-I

I: Ajj(P, Q»

n

j=1

j=l

n-I

dAj.

The order of the operator is then order(symbol) +

I

(2" #oscillating variables)

I . b - - - (#P vana les + #Q variables) = -1/2 4 and would have been the amount of smoothing, had the projections Jr L and Jr R been locally invertible. In this case the kernels of dJrL and dJrR are one-dimensional, and the sharp estimate in the absence of any additional information as given by Hormander's theorem is then boundedness on H(s). Theorem I shows that this can be improved by natural and generic assumptions on the geometry of Jr Land Jr R. The next issue we address is whether the torsion condition, natural as it is for the family of translates of a curve, can be extended to general families of curves. This is the case, as the torsion condition for distributions of translates of a curve arises naturally from the stratification structure of the singular varieties of Jr L and JrR. Let

(9)

!lli' I

1~lnl

:~

(10)

I:

= {(P,~.

Q. 7])

E

N*(C) where JrL. JrR are not locally 1 - I}

be the singular variety of dJrL and dJrR. The key geometric information is the position of the kernels of dJrL, dJr R relative to the tangent space to I:, viewed as subspaces of T(P.~;Q.'1)(N*(C». We discuss first, say, the left projection JrL, and make the generic assumption that the kernel of dJrL is one-dimensional and the determinant of dJrL vanishes only of first order along I:. The variety I: is then smooth, and Kerd(JrLIl;)

= KerdJrL n TI:.

In particular, the transversality of Ker dJrL and T(p,~; Q,'1)(N*(C» is equivalent to the local injectivity of Jr L when restricted to I:. This leads us to define successively I I!I

(11)

I:2 = I:, I:1

= {(P,~;

Q. 7])

E

L

I:i

I

I', I

.

i~1

2 - )

= {(P,~;

Q, 7]); Q

= P + yet),

(7],

y) =

(7], ji)

= ... =

(7]. y(n»)

291

= O}

and the condition that y have torsion is equivalent to I:£-2 be empty. Finally we come to the issue of possible asymmetry between the left and right projections. To define I: there is no need to distinguish between them, because one of them is singular if and only if both are. This is a consequence of the fact that N*(C) is Lagrangian and can be seen as follows. First observe that dJrR(Ker dJr d and I mage(dJrR) are orthogonal complements with respect to the symplectic form on T*(R). In fact 88, 8n are in these spaces respectively when (0,88) and (8'11, 8n) are in T(N*(C» for some 8'11. Since N*(C) is Lagrangian we have 0= w;"(C)(O, 88); (8'11, 8n»

= Wp(Rn)(8?, 8n).

This implies that dJrR(Ker dJrd and Ker dJrR have the same dimension, and thus that Ker dJrL and Ker dJrR have the same dimension since dJrR restricted to Ker dJrL is manifestly one-to-one. For translation invariant distributions of curves, there is symmetry between left and right projections to all orders of stratification. In general, however, we also need to consider the stratification varieties I:~ of JrR, as well as the mixed singular varieties defined as follows. Let W = (WI, ... , wd denote a sequence of k indices Wi which can be either an L or a R index. Then

(13) I:kW1W2"'Wk =- I:k-I W2'''Wk

n {(P,~; Q,

7]); KerdJrw, (P,~; Q. 7]) C T(P,~;Q,'1)I:~~'Wk}'

This leads us to the extension of the notion of torsion for a single curve that we were looking for. In [34] a (not necessarily translation invariant) family of curves was said to have left (respectively, right) torsion if the variety I:£-2 (respectively, I:~-2) is empty. More generally it will be said to have non-vanishing "w-torsion" at (P, Q) if for some k and w of length k

I:~

(14)

n N(p,Qj(C) = 0.

The simplest family of curves in Rn incorporating an asymmetric behavior is given by the parametrization

I:~-I; KerdJrL(P,~; Q. 7]) C T(P,~;Q,'1)I:~-I}.

In the language of singularity theory, I: = 0 means for example that the projection JrL is a (Whitney) fold. For the distribution of translates of y (t) it is readily seen that n

SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS

M(x,f)

= ley, s); s = t + sex. y)}

C = {(x';; y, s); (y, s)

(15)

with x. yin R, and S, t, Sex, y) vectors in R n (16)

..

-+,

-

I.

-+,

E M(x,i)}

The Lagrangian becomes -+

N (C) = {«x. (A, Sx(x, y)}), (t, A); (y, (A, Sy(x. y)}), (s, -A»; A E R

n-l}

.

292

CHAPTER 14

In particular it can be parametrized by (x, y, t, X), and its tangent space by (8x, 8y, 8t, 8A). Linearizing the defining equations for N*(C), we see that the singular variety 1:; and the kernels of drrL, drrR at 1:; are given by 1:; =

KerdrrL = (8x = 8t = 8A = 0, 8y

(17)

KerdrrR = (8x

E

E

= IWJ. ••• wk

U {(a

+

E IWJ.

n

IwlWJ. •••wk

=

IWJ.· ..wk

U {(a, fJ

+

I); (a, fJ) E

IWJ.

if

wk},

WI WI

+ -

I

.j2

T.

Then the a-Neumann problem is the following boundary value problem

= {(x, y, t, A);

(A, a:+la:+IS(x, y»)

= 0,

(a, fJ) E

It is perhaps a surprising fact that Radon transforms arise in Green's functions for subelliptic problems. Indeed they are normally associated rather with propagation of singularities phenomena inherent to hyperbolic problems. More precisely the singular support of the Green's function of a subelliptic problem is contained in the diagonal, whereas the singular support of Radon transforms is usually given by a manifold of higher dimension. x R be the Heisenberg group, with the group operation Let Hn = (z, t)(z', t') = (z + z', t + t' + 2Imz . Z'). The vector fields = -

a

aZj

a -

a T = - i -a -iz·J a at Zj t'

t

p

where fez, t, p) is a given Co function on iI. More precisely U can be identi­ fied with the Siegel upper half space {(z, Zn+I); ImZn+1 > Iz1 2 } by (z, t, p) ~ (z, Zn+1 = t + i(p + IzI 2 The Levi metric is the metric which makes d z, j J. -I ... , n and W n+1 -- - 21/ 2 " n -·d· ' W j -L..,j=1 ZJ ZJ - I'2- 1/ 2d Zn+l mto an orthonormal basis of (I,O)-forms. The corresponding Laplacian on (O,I)-forms Lj~: UjW j is diagonal. The boundary condition on Un+1 is the Dirichlet condi­ tion U n+ II p =o = 0, while the Laplacian and boundary conditions on the remaining components U j, j = I, ... , n are given by (24). The L 2[7], U and Schauder estimates [8], [13] for both problems (22) and (24) are by now well-known. Here we would like to discuss instead explicit formulas for their Green's functions (more accurately, their parametrices) and bounds for the corresponding classes of integral operators. Translation invariance shows that the Green's function for 0:;:1 is a convolution operator T on the Heisenberg group

».

(25)

(T¢)(z, t) =

II

Gb«Z, t)(w, s)-I)¢(w, s)dwds.

On the other hand, the parametrix for the boundary value problem (24) must be of the form

1 II + 1 II 00

(Nf)(z, t, p) =

a

+iz·z· = -aJ a 'J

a ap

(-+i-)ul=o=O

IwtWJ. ...wk}.

2 Green's Functions for Subelliptic Problems and Singular Radon Transforms

en

a at

Ou = f,

(24)

We can finally formulate the first main goal of a theory of Fourier integral op­ erators with degenerate Lagrangian manifolds: to establish optimal estimates, or, more precisely, identify the nature of TT* and T*T, for each possible stratifi­ cation structure of the projections rrL and rrR, as described by the subvarieties 1:;~. In Section III we shall provide a systematic study of a class of models which shows how the L 2 estimates and smoothing properties are indeed dictated by the stratification of the Lagrangian.

z·J

'

a

= L.

(19)

(20)

(n - 2)T

I a2 D=---+Ii.. 2 p2

= R

Then we have

1:;~tWJ."'Wk

-

This problem is closely related to the a-Neumann problem on the domain U = H X R+. Let (z, t, p) denote points in U, and set (23)

(18)

JJ

0+ = li. 1/ 2

(22)

if

wk},

IL: (Z·Z· + Z·Z·) JJ -

and our task is to study the Green's function for the following operator

R)}

R, 8t = 8A = 8y = O)}.

I, fJ); (a, fJ)

1

Ii. = - T 2 - 2 2.

J=I

To each W we associate now a set I w of indices (a, fJ) E N 2 by the following inductive process reflecting that of (13) IwtWJ.···W!

are left invariant and form a basis for the Lie algebra of Hn. The Laplacian is given by (21)

N*(C) n {(A, S;y(x, y)) = O}

293

SlNGULAR lNTEGRALS AND FOUR1ER 1NTEGRAL OPERATORS

Go«z, t)(w, S)-I;

Ip - J-Ll)f(w, s; J-L)dwdsdp

00

(26)

G«z, t)(w, S)-l; p

+ J-L)f(w, s;

J-L)dwdsdp.

294

CHAPTER 14

The first term on the right hand side of (26) is pseudolocal and is the same as for, say, the Dirichlet problem. The second term is the key one insuring the -Neumann condition. Such terms are called Hilbert integral operators. There is a remarkable relation [30J linking directly the Green's function of 0+ to that of (24)

a

0+ 41 = -(N(¢ ® 8(JL))Ip=o. 1

(27)

SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS

• If k + 1 = 2n + 4, k value 0 on the z-sphere (29)

= 2n, and the kernel Ek(Z, t; p)H/(z, t; Izi = 1 for any (t,

1

295 p) has mean

p), i.e.,

Ek(Z, t; p)H/(z, t; p)dz = 0

£
for any E, 8,0 < E < 8, then H is bounded on LP(U n x R+)for 1 < P < 00.

It is now well known [29], [32J, [3J that G(z, t; p) is given by an asymptotic expansion in kernels of the form Ek(Z, t; p)H/(z, t; p), where Ek(Z, t; p) and H/(z, t; p) are homogeneous of order -k and -1 with respect to the two different notions of dilations

The conditions k + 1 < 2n + 2 and k + (1/2) < 2n + 1 of Theorem 2 define a region in the (k, 1) plane, inside of which the kernel is locally integrable. On the edges k + 1 = 2n + 2, k < 2n, away from the intersection point k = 2n, 1 = 2, the kernel behaves as the parabolically homogeneous kernel Ek(Z, O)H/(z, t) which has singularities along the plane z = O. These singularities are, however, H/()..z, )..2 t ; )..2 p ) = ).. -/ H/(z, t; p) (28) Ek()..z, At; )..p) = rkEk(Z, t, p), integrable in view of the condition k < 2n, and the condition that the mean value on the (z, t)-sphere of Ek(z, O)H{(z, t) is 0 leads to LP bounds just as in the case of respectively. Similar notions of homogeneity are introduced for kernels which are functions of (z, t) alone. The sharp estimates for the a-Neumann problem smooth kernels homogeneous of degree 2n + 2. The case of k + (1/2) = 2n + 1, and for 0:;' are given by bounds on ZjZkN(f), Z/ZkN(f), Z/ZkN(f), and 1 < 2 is similar, with the kernel Ek(Z, t)H{(O, t) homogeneous of the (euclidian) Z j O:;I¢, ZjO:;'¢. The theory of singular integrals with homogeneous kernels critical degree 2n + 1 and having integrable singularities on the (euclidian) (z, 1)­ on the Heisenberg group has been developed in [8J. The above expressions lead, sphere. The estimates for ZjO+'¢, ZjO:;! fall in these cases. The case k = 2n, however, to convolutions on the Heisenberg group and Hilbert integral operators 1 = 2 is not needed here, but may be of independent interest. It is likely that it with products of kernels of two different homogeneities. The following theorems can be treated by the method of singular Radon transforms used for Theorem 3, provide bounds for such operators [30J, [31J, [32J: to be described next, but this has not been established. We tum now to Hilbert integral operators. Inside the region k + 1 < 2n + 4, k + (1/2) < 2n + 2 the kernel Edz, t; p)H/(z, t; p) is locally integrable, and Theorem 2. Let Gb(z, t) = Ek(Z, t)H/(z, t) be a kernel of mixed homogeneity, the first statement of Theorem 3 follows at once. On each of the edges k + 1 = and let T be the corresponding convolution operator (25). Then 2n + 4, k < 2n and k + (l/2) = 2n + 2, 1 < 4 the kernel behaves again as a • If k + 1 < 2n + 2, k < 2n or k + (1/2) < 2n + 1,1 < 2, Tis boundedfrom homogeneous kernel with integrable singularities on the sphere, the homogeneities Lfoc(un) into itselffor 1 ::: p ::: 00; being parabolic and euclidian, respectively, in their critical degrees 2n + 4 and • If k + 1 = 2n + 2, k < 2n, and the kernel Ek(Z, O)H/(z, t) has mean value 0 2n + 2. Such kernels lead to operators bounded on LP(U n x R+), by arguments on the parabolic (z, t)-sphere Izl 4 + t 2 = 1, then T is bounded on LP(Un)for similar to the one-dimensional case of K (x, y) = (x + y) -Ion functions on 1 < P < 00; U(R+). Unlike in the case of 0:;', sharp bounds for Nf lead to Hilbert integral • If k + (1/2) = 2n + 1,1 < 2, and the kernel Ek(Z, t)H1(z, 0) has mean value 0 n 2 operators at the doubly critical degree of homogeneity k = 2n, 1 = 4, requiring on theeuclidian (z, t)-sphere Id +t = 1, then T is again bounded on U(U ) singular Radon transforms on the Heisenberg group. for 1 < p < 00. The key geometric ingredient of singular Radon transforms on un is a distribu­ tion of hypersurfaces M p passing through each point P E un. At P = (0, 0). Theorem 3. Let G(z, t; p) = Ek(Z, t; p)HI(z, t; p) be a kernel ofmixed homo­ Mp is the maximal complex space M p = {Q = (w, 0); WEen}. For general geneity, and let H be the corresponding Hilbert integral operator defined by the P = (z, t) it is the image of M(o,o) under the group multiplication by (z, t) second terms on the right hand side of(26). Then • If k + 1 < 2n + 4, k < 2n or k + (1/2) < 2n + 2, 1 < 4, H is bounded from L{oc(un x R+) into itselffor 1 ::: p ::: 00; • If k + 1 2n + 4, k < 2n, or k + (1/2) 2n + 2, 1 < 4, then H is bounded n on LP(U x R+)for 1 < P < 00;

=

=

(30)

M(z.t) = {Q = (z

+ w, t + 2/mz . w)}.

Let Lo(w)dw be a density on the hyperplane M(o,o), which we can transport to a density dJLp(Q) on each M(z.t) again by group multiplication. Typically Lo(w)

296

CHAPTER 14

is a homogeneous kernel on en, and, in particular, singular at w = O. The singular Radon transform associated to the distribution of hyperplanes M p with their densities L p is given by averaging over each M p with respect to the density dlLp(Q) (31)

R¢(P)

= {

lMp

¢(Q)dlLp(Q)

= {

¢(z

len

+ w, t + 2lmz . iiJ)Lo(w)dw.

When Lo(w) is a Calderon-Zygmund kernel on bounded operator on LP (32)

IIR¢IIU(H") :5 CII¢IIU(Hn),

en

the operator R extends as a

Hf(z, t, p)

=

00 1+00

1

o

-00

(Rs,p,IJ.¢S)(Z' t)dsdlL.

Now the densities K (w, s, P+IL) are smooth in w, but their bounds as Calderon­ Zygmund kernels are only O(s2 + p 2 + 1L 2)-1. Thus (32) implies IIRs,p,1J.1I :5 C(s2

+ p 2 + 1L 2)-1.

It follows that

II(Hf)(·", p)llU(Hn) :5 C < C

(34)

00 1+00 ( 2 o -00 s 00 I

1 1 o

+

I 2 P

+

?

IL

II¢sllu(H")ds)dlL

--lIf(-' " 1L)IIU(Hn)dlL· p + IL

The LP boundedness of the one-dimensional Hilbert integral operator gives the desired result, establishing Theorem 3. In [9] the estimate (32) was proved using the group Fourier transform on the Heisenberg group. The natural setup for singular Radon transforms is, however, in integral geometry, and the following general theorem holds [31], [32]:

Theorem 4. Let M p be a smooth distribution of hypersurjaces passing through P in a Coo compact manifold M, and let dlLp(Q) be a density on each M p with a Calder6n-Zygmund singularity at Q = P. Define the corresponding singular Radon transform and maximal operator by (R¢)(P)

= {

lMp

¢(Q)dlLp(Q)

(M¢)(P)

= sup~ , ~/ ~

0"

lB(P,~) (

¢(Q)dlLp(Q)

where we have chosen a metric on M and denoted by B(P, 8) the geodesic ball centered at P and of radius 8. Assume that the distribution of hypersurjaces M p has non-vanishing rotational curvature along the diagonal, i.e., if (P, Q) = 0, 'VP,Q #- 0 is a defining equation for the relation manifold C = {(P, Q); Q E M p }. then

00.

The key observation is that Hilbert integral operators are effectively averages of singular Radon transforms. For fixed s, p, IL we can construct the singular Radon transform Rs,p,1J. with density Lo(w) = K(w, s, P + IL) and the function ¢S,IJ. on the Heisenberg group defined by ¢S,IJ.(z, t) = f(z, t - s, IL). Then (33)

(35)

(36)

1< P <

297

SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS

det(])

==

(P, Q) ( 'VQ(P, Q)

'Vp(P, Q) 'Vp'VQ(P, Q)

)

#- O.

Then the operators R. M are bounded on LP(M)for 1 < p <

00.

It should be pointed out that (36) is exactly the Monge-Ampere determinant ] (
(a()..)1jr)(x)

=

f

ei'-S(x,y) K(x - y)1jr(y)dy.

Since rotational curvature is a second derivative condition, we assume that the phase S(x, y) is a quadratic form. We also setA = 1. The analogue of Theorem 4 is [32]:

Theorem 5.· Let the rank of the quadratic form S (x, y) be k, and let -IL be the degree of homogeneity of the kernel K (x). When IL = n we also assume that K(x) has mean value 0 on the unit sphere. Assume also that k > O. otherwise (37) reduces to a familiar fractional integral. Then the operator a is bounded on LP(an)for 11/2 -11pi < (IL -n +k)/(2k), 1 < p < 00, andfor 11/2 -11 pi :5 ILI(2n) when k = n. More recently progress has been made in several different directions. First it has been shown in [27] that the singular Radon transforms associated to the distribution of hyperplanes (30) in the Heisenberg group and a density Lo(w )dw homogeneous of degree 0 are smoothing of order -n on Sobolev spaces. This is a generalization

m_"

~.

_"C"'-"

.,_~

298

._••_"••

-

•

-

.",n"

__"._-_

•

i:C:-"-_C·"'=-=-=--"'""--""'-"'"="='=""""'·_·.~.'-C--"-.""~

CHAPTER 14

of the similar estimate for smooth densities which would follow from the classical

theory of Fourier integral operators. Although we may expect a similar result to

hold in the more general setup of Theorem 4, this has not been established to date.

Second, Theorem 5 suggests that the condition that the rotational curvature form

be non-degenerate can be weakened considerably if we restrict our attention to

LP bounds for singular Radon transforms with Calder6n-Zygmund singularities

or maximal functions. In the context of nilpotent groups, such improvements can

be found in [4], [23], [36]. This has now been done in full generality by Christ,

Nagel, Stein, and Wa~nger [5], who showed that the LP bounds hold as long as the

rotational curvature form does not vanish to infinite order. At the other extreme,

when the densities are smooth, singular Radon transforms become Fourier integral

operators, and non-vanishing rotational curvature is essential to insure the optimal

bounds since it reduces to the condition that the Lagrangian be a local graph. It

may be interesting to determine exactly how much rotational curvature is needed

for a fractional singular Radon transform.

3 Integral Geometry In a remarkable parallel development, operators sharing many of the features of

the previous examples arise in integral geometry, particularly in the problems

of inverting the X-ray transform and constructing Lorentz metrics, all of whose

light rays are closed. More precisely, they are certain Fourier integral operators

X;: whose Lagrangian exhibits an asymmetric type of singularity as discussed

in Section 1.1, and whose composition X}X;: is a Fourier integral operator with

a singular density, analogous to the singular Radon transforms of Section 11.2.

This important phenomenon was discovered independently by Guillemin [14] and

Greenleaf and Uhlmann [10], [12], and we discuss it briefly here.

The main example is the X-ray transform on the family of light rays in R 3 • Let

a light ray 1 in R3 be any line making an angle of Jr /4 radians with the X3 axis,

and let F be the 3-dimensional family of all light rays. The corresponding X-ray

transform is the operator sending a function I E CO(R 3) to the function X;: I

on F, defined by averaging I on each 1 E F. If we parametrize a light ray 1 by

its intersection with the (Xl, X2) plane, and the angle B that its projection on the

(XI, X2) plane makes with the x-axis, then

(38)

(X;:f)(xJ, x2,B) =

i:

I(x] +sCOSB,X2 +ssinB,s)X(s)ds

where we have restricted the support of I to a fixed compact subset K of R 3 , and

chosen a cut-off function X (s) which is even, and == I in a neighborhood of K.

-...,:""_":-=--="'-=:~-c

SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS

As usual we define C = {(l, Y); Y N*(C)

=

E

299

I}, and the Lagrangian of X;: is

(((XI,X2,B), S); ((YI

= Xl + s cos B, Y2 = X2 + s sin B, Y3 = s), \11)}

S = (-J.LI, -J.L2, Y3(J.Lt sin B - J.L2 cos B)) (39)

\11

= (J.L I, J.L 2, -

(J.L I cos B

+ J.L2 sin B)).

It follows that N* (C) can be parametrized by (XI, X2, B, Y3, J.LI, J.L2) and its tangent space by infinitesimal variations (hI, 8x2, 8B, 8Y3, 8J.LI, 8J.L2). In this instance JrR is the projection on the Y, \11 variable, so that its kernel requires 8J.L] = 8J.L2 = 8Y3 = 0,8xI = Y3sinB8B,8x2 = -Y3cosB8B, and (-J.LIsinB + J.L2cosB)88 = O. Thus Ker dJrR is non-trivial exactly on the variety (40)

b

= N*(C) U {-J.LIsinB + J.L2cosB = O}

and is generated by a non-vanishing 8B. In particular it is one-dimensional and transversal to b, and the Lagrangian is a fold with respect to rr R. On the other hand, it is easy to see that dJrL is singular along b, but its kernel is one-dimensional and generated rather by a non-vanishing 8Y3. This implies that Ker dJrL is tangent to b, which is an asymmetric behavior. In view of later generalizations, it is helpful to note that the description of \11 in (39) is equivalent to its being perpendicular to the line 1. Furthermore, the condition (40) defining b just means that \11 is also orthogonal to the vector e2 = (-s i nB, cosB, 0) which, together with 1, generates the tangent plane to the light cone. Thus b is actually characterized by requiring that \11 be perpendicular to the light cone. The statement about X}X;: can be verified by an explicit formula which we give below. The family of light rays is only a special case of a rather general setup, as shown by Greenleaf and Uhlmann. Let £, be the family of all lines in Rn. We shall see shortly that it is a smooth manifold of dimension 2n - 2. Given a function I E Co(Rn), its X-ray transform is the function on £, defined by (Xf)(l)

=

ii,

1 E £'.

Recapturing I from XI is an overdetermined problem. Gelfand raised the problem

of finding sets of uniqueness for X, i.e., find n-dimensional families F of lines 1

for which

(41)

X;:I =

XII;:

suffices to characterize I. This problem was solved in the case of complex lines in

C n , the sets of uniqueness being the ones satisfying the following cone condition.

For each X let r x be the cone generated by the lines passing through x. Then r x

300

CHAPTER 14

is tangent to r y along the line joining x and y for any y in r x' Greenleaf and Uhlmann addressed the analogous problem for real lines, namely to reconstruct explicitly as much of f as possible from XF f when F satisfies the cone condition. Although not generic, this condition is satisfied by many families of lines, such as the family of light rays discussed above, and the family of all lines passing through a given curve. It turns out to imply that XF is a Fourier integral operator with a fold on one side and a blow-down on the other. Furthermore X;'X F is again essentially a singular Radon transform. We provide now some details in the simplified setting of families of lines in R 3 •

Asymmetric Behavior of the Lagrangian of XF The first task is to obtain an explicit parametrization of the Lagrangian of X F. A line 1 can be described by a reference point P and a unit vector y. Locally we may view then the space of all lines C as a submanifold in T(R 3 ). As a consequence T*(C) can be identified with the restriction to T(C) of T*(T(R 3 )), viewed as linear functionals on T(T(R 3 )). Similarly T*(F) is the restriction to T(F) of T*(T(R 3 », and in particular

SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS

orthogonal to y, and C would be (2n - 2)-dimensional, as asserted earlier. When y) E T(R 3 ), the deformation (44) is expressed as

1 is viewed as a pair (P,

= {«P. Y).

(42)

where

+

Y + X)

X(O).

so that a tangent vector in Tt(C) is identified with the pair

X)

(X(O),

(47)

x Tp(M).

E Tp(M)

Since Tt(F) is a subspace of Tt(C), it can also be described by Jacobi fields as in (45) and (47), and the covectors (Eijk») Yk. Eijk») (Q - p)k) appearing in (43) pair off naturally with vectors in (46). The next task is to introduce a basis 4>" 4>2, 4>3 of I-forms on ~*(F), in which the covectors of (43) take a succinct form. Here we exploit the cone condition satisfied by the family F. Now F is a codimension 1 submanifold of C, so its normal space is a one-dimensional subspace of T* (C). An important ingredient of the geometry of C is that it is a symplectic manifold, with symplectic form 2

2

w.c
2

+ bi)eit L(CjS + dj)ej) = j='

L (bicj - aidj). i,j='

This allows identifying T (C) with T* (C) via VI ~ w~J VI. In particular the normal space to F corresponds to a Jacobi field X 4 (s) along l. The cone condition asserts that X4(S) is proportional to a fixed vector as S varies. If we consider the cone r Q generated by the lines in F passing through Q

l}

Q); Eijk(Q - p)jyk

~ (P

y)

i='

More explicitly the defining relation for XF is E

(P,

(46)

~*(F) = (T;(R 3 ) x T;(R 3 »IT/(F)'

C = {(I, Q); Q

301

= O}

is the antisymmetric symbol. In the above formalism the Lagrangian N*(C) becomes

(48)

(43)

this means that the cones with vertices along 1all share this vector as normal vector. We choose the vector used earlier in the construction of Jacobi fields to be this common normal, and the vector e2 is determined accordingly. We note that e2 is tangent to r Q. It is now not difficult to obtain a basis for Tt(F). It consists of

Eijk

N*(C)

=

. k {«P, y). (-EijkA,1 y ). ~

.

k -EijkA,1 (Q - P) )); (Q.

. k EijkA,1 Y »I1i(F)}'

At this point we need a more concrete characterization of Tt(C) and Tt(F). Let s be the parametrization of 1 by arc length, starting from P, and let eo == y, e,. e2 be an orthonormal basis of vectors in R 3 • The line 1 can be deformed to another line by 2

(44)

P

+ sy ~

(P

+ sy) +

L(aiS

+ bi)ei

i=' where ai, bi are any constants. This means that the vector fields along 1 given by (45)

X(S)

= ei,

or

sei.

i = 1,2

can be viewed as a basis for Tt(C). They are called Jacobi fields. We note that this same discussion is valid in Rn, in which case there would be n - 1 vectors ei

r Q ==

UQEl,lEFl

e,

(49)

X,(s) = (as

+ b)e"

X2(S)

= e2, X3(S) = se2.

The constant coefficients a, b are specified up to a constant multiple by the re­ quirement that w.c<X" X 4) = O. We can now define 4>i, i = 1,2,3 to be the dual basis to Xi, i = 1,2.3. Returning to the expression (43) for N*(C). we let Q = P + sy and write (50)

( EijkA j

y-k ,EijkA j ( Q

-

P) k)

==

-

(A,e,

~ ~ ~ » + e2. s(A,e, + A2e2 •

In terms of the (P, y) parametrization, the Jacobi fields X" X2, X3 of (49) become (be,. ae,), (e2, 0), and (0. e2). respectively. Evaluating their pairings with (50)

._..,,""==-----'----==,.,-'-"""=_""""~~"_=,

302

__._._.

._=~"""'_=",.,.~,=__"=~==.",..__","_.,,~,_~

CHAPTER 14

gives N*(C)

(51)

=

==

{«(P, y), 8

-[(2as

+ b)AI1 + Azz + sA z3)];

+ Azez))}.

(Q, Aiel

Finally we can evaluate drr L and drr R. First we note that N* (C) can be parametrized by (P, y) E F, s, and AI, AZ. Let the variations of the parameters be 3

(52)

8cxiXi. 8s. HI, Hz.

L

.....o...;."'_.~ ..,---'"

"";;";';~""'':'';_:::'',"--~~"'~'';-'"'_~.','_.';"_

.".""".",.__;,;,=",;""""",~~~~,,",,.,...;..-~.,--;

.•._

."'o_ •. ~. ",,,,,,..,_ ''"''''''''''-'''"'''''-''''''''~''"""-;-,,,-,,,,,,",,_.",_,,_-=",,-''C"=_~.'''~~-'-~=_~''''''--''O''''""""""=

,:~~~~,·:,7;;:'::;;;;;;"':;C':;;."':'

303

SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS

the curve of intersection between r Q and the unit sphere. If we assume that the cones r Q have non-vanishing geodesic curvature, it follows that the elements (52) corresponding to Ker(drrR) must have 8Az i- O. This means that the kernel of drrR is transversal to the singular variety ~. This is the asymmetric behavior of the Lagrangian N*(C) that we wanted to establish, X:FXF and Singular Radon Transforms Formally the operator X:FXF is given by

i=l

The corresponding infinitesimal variations drrLCT(N*(C))) are given by 3

8P

(57)

8y = L8aiX

(X}g)(YI, Yz, Y3) = X(Y3)

J

g(YI - Y3COS(}, YI - Y3sin(}. (})d(}

o

i=1

= 8S(2aA I 1 + Az3) + 8AI(2as + b) I + 8Az(z + s3)' For (50) to give an element in Ker(drrd we need that 8a; = 0, i = I, 2, 3.

and the operator X:FXF can be written explicitly as

88

Furthermore for Q generic, 2as + b is different from 0, and Ker(drrd is non­ trivial (in fact, one-dimensional) at exactly the subvariety of N*(C) defined by AZ = 0 ~ = {(((P, y), 8 = (2as

+ b)Al <1>1);

(Q. \II = AIel))}.

Since T(~) = T(N*(C» n {8Az = OJ. and Ker(drrd is generated by (52) with 8ai = Hz = 0, it follows that Ker(drrd is tangent to ~. On the other hand we show now that if the cones r Q satisfy a generic curvature condition, the kernel of drrR is transversal to~. In fact. the infinitesimal variations drrLCN*(C» under (52) are 3

8Q = L

8aiXi(S)

+ 8seo

i=1

z (55)

!(P)d/LQ(P) Q

{Zrr

3

(54)

Jr

where d/LQ is a density on the cone r Q (48), which is singular at Q. For example, when F is the family of light rays in R3 we have

= L8a;X;(0) ;=1

(53)

(X:FXF!)(Q) = {

(56)

8\11

=

L i,j=l

z Aj 8ei

+

L

8Ajej.

j=1

For (55) to vanish, we must have 8s = 8al = 0 and 8az + s8a3 = O. This means that we need evaluate 8ei only when the line I is deformed infinitesimally along the tangent plane to r Q. keeping the point Q fixed. At ~, AZ = 0 and we need only evaluate 8e\. Now the variation of el as we vary I along ez is proportional to ez, with the coefficient of proportionality given by the geodesic curvature of

(58) (X:FXF)(f)(Xl, Xz, X3)

= {{ !(XI -

JJ

U,

Xz - V. X3

±

Ju z + VZ)X(z

±

Ju z + vZ)

dud v

..Ju z + vZ

.

Thus X:FXF is ofthe form (35), with the difference that the submanifolds passing through each Q are cones with vertices at Q and thus singular at Q, and the densities d/LQ are of fractional degree rather than critical degree. Also as the dimension n of the ambient space increases, r Q remains of dimension 2 instead of staying a hypersurface. From the microlocal viewpoint WF'(X:FXF) C ~T'(R3) U N*(9)'

(59)

where~T'(R3) C T*(R 3) x T*(R 3) is the diagonal and 9 = {(Q. P); P E r Q} is the defining relation of X:FXF. Operators whose wave fronts are contained in a

pair of intersecting Lagrangians have appeared in a number of other contexts, and have been extensively studied [20], [15], [I]. Nevertheless a calculus powerful enough to yield Sobolev estimates is only available in certain special cases, notably when the Lagrangian N*(9)' is a flow-out [I]. A crucial observation of Greenleaf and Uhlmann is that this turns out to be the case for N* (9)' when the family F satisfies the cone condition. For families of lines in R 3 we can illustrate directly the underlying geometry as follows. For each Q in R n , we parametrize the unit vectors generating the cone r Q by Y(Q. (}). The orthonormal frame eo. eJ, ez introduced earlier is now given by eo

= y,

ez =

ooY/ll ooYIl.

~

el

= ez ~

~

x eo.

••

304

CHAPTER 14

The dual cone to r Q is the cone r Qof all covectors orthogonal to r Q at some line in rQ. It is generated by the vectors el. We set r* = uQr Q. Then N*(Q)

= {«Q, ~), (P = Q -

sy(Q, 0), ~lIel»}

1/i = ~i - saxiyk~k'

(60)

Under the condition that each r Q have geodesic curvature, the closure N*(Q)' extends smoothly to t. P (R3), and (61)

N*(Q)'

n

t. P

(R3)

r

(aoY)(Q - sy(Q, 0), 0) x (aoy)(Q, 0)

= O.

Differentiating the first equation with respect to s shows that

305

along 1/. This identifies the Lagrangian N*(Q) as the flow-out of r*, which is the desired statement. The asymmetric behavior of the Lagrangian of XF can lead to very different operators for X}XF and XF X}. Again returning to the example of light rays in R 3 and the operator (38) we find

(66)

(XFX}g)(l) =

{{ JJ g(x + u, y + v, 0 -1/I)X

(

u

2

2(u cosO

2

++vv sinO) )

lu cos 8

+ v sin 01

where 1/1 is the angle defined by cos 1/1 = 2uv(u 2 + V2)-I, sin 1/1 = (v 2 - u 2 )(u 2 + v2 ) -I. We note that the kernel of the integral opemtor (66) has singularities along a line, but its support is narrow enough to make it integrable. Operators of type XFX} are central to the problem of deforming the Minkowski metric on R 4 to metrics all of whose light rays are closed. We refer to [14] where a detailed microlocal analysis for them is provided. It would be interesting to explore these operators also from the local viewpoint, with an aim towards LP - L q estimates.

y iaxiyJ. = 0

(63)

so that axi yk ~k is orthogonal to y. Differentiating the first equation with respect to sy(Q, 0), 0) is parallel to aoy(Q, 0) for any s i= O. Letting s tend to 0 gives

III MODELS OF DEGENERATE

FOURIER INTEGRAL OPERATORS

o and using the second one shows next that (aOyi)(Q, O)axiY(Q (64)

..e·.

dudv

y(Q - sy(Q, 0), 0) = y(Q, 0) (62)

.

~_.

SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS

= t. r •.

Next we derive two key consequences of the cone condition. The first is that 1/ in (60) is actually parallel to ~, and the second is a simple characterization of the symplectic complement of T (f*). In fact, the cone condition says that the tangent planes to P and Q are the same along the P, Q line, Le.,

r

...

===="=·~""'~='===;O~="=='-=--=:;="='="'='===='="'='_"~"=='m ,.~

=,=~-=-=,.====,

(aoyi)(Q, O)axiy(Q, 8)llaoy(Q, 0).

Since ~ in (60) is parallel to y x aOY, it follows at once that (1/, y) = (1/, aoY) = 0

A full theory of Fourier integral operators encompassing the stratification (11), (13) of the Lagrangian and singular densities is still far ahead at this point. it is important first to construct and study simple models which reflect the high order of degeneracy of the projections ]fL and ]fRo Returning to the family of curves (15) in R n , the corresponding Radon transforms can be expressed as

and 1/ is parallel to~, as claimed. Next T(Q,~)(r*) is generated by infinitesimals (8Q, 81/) ofthe form (65) 81/ = -s(y x a~oY

+ aQy

Rj(x,t)

=

y, while

fJi = ElpqaxiypaOyqyl.

In view of (63), (64) fJ is orthogonal to both y and aOY, and hence must be parallel to y x aOY. We can identify now the bicharacteristic through (Q, 1/). By the curvature condition on each cone, the covector 1/ corresponds to a unique ray y(Q, w). Points (P, ~) are on the bicharacteristic when P flows along y(Q, w), and ~

j

MexJ)

x aoy)8Q - (y x aoy)8s - s(y x a~oY)80.

Thus w «a, fJ), 8 Q, 81/» = 0 implies that a is proportional to

1

(67)

=

!

1] n-I

ei()...i-S)[! ei()",S(x,y») X(y)!(y,

)..)dy]

d)..j

!

where j (y, s) is a Coo function with support in a fixed compact set, (y, )..) is the partial Fourier transform of j with respect to and X (y) is a cutoff function which is identically 1 on the support of j. As in (37) the operator (67) can be viewed as a pseudo-differential operator whose symbol a()..) is the operator from Coo functions of y to Coo functions of x given by the expression between brackets. For each fixed)" this is an oscillatory integral with phase S)..(x, y) = ()../I)..j, Sex, y»).

s,

_ _ _ _ _. . . . . . . . .

~

7"""'7'

~

. .,

~

306

CHAPTER 14

Non-vanishing right torsion at (x, y) means that for any A

'I-

0, or Sex, y) is proportional to either xyz or xZy. In the first case, the operator T is equivalent to a similar operator with phase

alxZy with aoal

n-Z

(68)

L

aZ Sex, y) = «_0 )1/3 x 3al

= 0 ===? a;-Iays).(x, y) 'I- O.

la:ays).(x, y)1

01=1

Similarly the condition that any of the varieties E~-z be empty translates into the non-vanishing of an n - th derivative of S).(x, y) when certain derivatives of order :s n - I vanish. Fixing A we are led then to the following models for degenerate Fourier integral operators (69)

-

(T¢»(x)

=

f

eiAS(x,y)X(y)¢>(y)dy

where Sex, y) is a homogeneous polynomial in (x, y) of degree n n-I

(70)

Sex, y)

=L

ai_Ix i yn-i.

i=1

We assume that at least one of the coefficients ai, i = 0, ... ,n - 1 is not O. Evidently the norm IITII is unaffected by the inclusion of terms proportional to either x n or yn, so we can ignore them. The models (70) generalize oscillatory integrals with quadratic phases which are models for Fourier integral operators associated to canonical graphs [18], as well as the well-known Airy operator (71)

(Ai¢»(x)

+00

=

-00

ei ).(x-y)3 X (y)¢> (y)dy

which is the prototype of Fourier integral operators where both rrL and rrR are Whitney folds [19], [41]. It is remarkable that no systematic study of the operators (69) has been undertaken to date, and we shall address this issue here [35].

1 Low degrees of degeneracy n

Theorem 6. For each n ::; 4 the following bounds hold and are sharp

1IT11 ::;

(74)

IK).(x, y)1

=

·

If

I I

e').(X-y)z 2 x(z)Zdz '"

irr

A(X - y)

I'/Z

which shows that 1IT11 = 0(\AI- ' /4 ), as stated in (73). We tum next to n = 4. When S(x, y) is proportional to either xy3 or x 3y, the kernellK). (x, y)1 of either TT* or T*T can be evaluated to be proportional to I(A)-1/3 (x - y)-I/3I. It follows that II TIl ::; 0(IAI- ' / 6 ), which is the estimate (73) for n = 4. We now show that the better estimate II TIl ::; 0(IAI- ' /4 ) holds for any other phase function which is a homogeneous polynomial of degree 4. If Sex, y) = x Zy 2, the kernel K).(x, y) of TT* is IK).(x, y)1 '" IA- 1/ Z(X - y)-1/2(x + y)-I/zi. Thus TT* is bounded on L 2 (R) with norm O(IA\-I/Z) and IITII = 0(IAI- I/4 ) as was to be shown. Otherwise we may assume that the phase Sex, y) is given by Sex, y)

(75)

is given by (76)K).(x, y)

= x l + alx 2l + a2 x3 y

than 2. In these cases T satisfies the weaker estimate

::; C1AI- 1/(Z(n-l)).

Proof The case n = 2 is trivial, since the operator T reduces to the Fourier transform (acting on X¢» after a rescaling. For n = 3 either Sex, y) = aoxyZ +

=

f

ei ).(S(x,z)-S(y,z»X 2(z)dz

= ei).(x-y)[?7 0I;(X+y)3- t OI\OIz(x+y)(xz+xy+yz)] x

f

ei).(x-y)(z3+[az(xz+xy+i)- ta;(x+y)Z]z)Xz(z

+ ~3 (x + y))dz .

The second integral in (76) is of the form (77)

CIAI- I/ n

except when Sex, y) isproportional either xyn-I or xn-Iy, and n is strictly greater

II TIl

3ao

After a linear change of variables, T reduces to a convolution operator whose multiplier is readily seen to be uniformly bounded by IA\-1/3. In the second case assume, say, that sex, y) = xl. The kernel K).(x, y) of TT* is then given by

= 2, 3, 4

For low degree of degeneracy we have a complete understanding for the L Z bounds for (69)

(73)

Z

a + (_I )1/3 y )3.

with at least one of the coefficients ai, az different from O. The kernel K).(x, y)

/

(72)

307

SINGULAR lNTEGRALS AND FOUR1ER lNTEGRAL OPERATORS

I (A)

+00

=

-00

ei).(Qz- t z3) X (z)dz

/

with A(X - y) replaced by -A/3. An explicit analysis, say, using Airy functions, yields (78)

(79)

II(A)1 ::; C min{IAI- I / 3, (IA\IQI'/z)-,/z}

If A < 0, then II(A)\ ::; C([).IIQi)-1

where C is a constant independent of A and A.

-:"'iiiO'-­

~

308

CHAPTER 14

In particular, we obtain the following bound for the kernel K).. (x, y)

(80)

IK)..(x, y)1 ~ C/A(X - y)I-I/2/ a2

(X 2 + xy

+ l) -

I 3a~(x

+ y)2/-1/4.

The right hand side of (80) is homogeneous of degree -1. Lemma 1 below asserts that the corresponding kernel is bounded on L 2 (R) with norm 0(IAI- / 2) as long ' as it has at most a finite number of singularities 8i on the circle x 2 + y2 = 1, near which it is bounded by

/Arctan(y/x) - 8d- I +Si for some ~i > 0 in general, and ~i > 1/2 if the singularity 8i is on one of the x, yaxes. These conditions can only be violated if the quadratic form in (80) has double roots at either x = 0, y = 0, or x = y. It is readily seen that the first two possibilities cannot occur, while the third one occurs exactly when 9a2 - 4a~ = O. In particular, we have proved the desired estimate for all phase functions of the form (75), except when

S(x, y) = xy3 + 3ax y2 + 4a 2x 3 y 2

for some a

=f.

O. In this case, however, the phase in (76) becomes A(X - Y)[Z3

a 2(x - y)2 Z]. Since a 2(x - y)2 ::: 0 we can apply (79) and obtain

(81)

+

/K)..(x, y)1 ~ Cmin{lA(x - y)I-I/3, IAI-I/x - Yr 3 }.

i:

As a consequence, (82)

IK)..(x, y)/dx

=

r

+

~

IK)..(x, y)ldx

At-YI
r

IK)..(x, y)ldx

1

Ix - yl-I/Jdx

r

Ix - yl-Jdx

+ C/AI- I

~-_.-----

SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS

Theorem 6 suggests that the sharp decrease O(IAI-I/n) for the operator norm II Til holds when at least two varieties I;~-2, I;:;-2 with suitable w and Wi are empty, or when n is even and I;~-2 is empty when w contains as many L as R indices. Strong evidence for these phenomena is provided by the following theorem which extends Theorem 6 to all orders n of degeneracy [35]: Theorem 7. Let S(x, y) be as in (70), T the corresponding operator (69). Then T extendsasa bounded operator on L 2(R) with norm II Til ~ C/AI-I/n when • an-2 =f. O. n is even, and la(n/2)-" + + lao I > 0; • a n -2 =f. 0, n is odd, and la(n-I)/2-d + + laol > 0; • n is even, a(n/2)-1 =f. 0, and ai = 0 either for all i ::: n/2 or for all i ~ (n/2) - 2. Evidently the roles of x and y can be interchanged, and an analogous state­ ment holds starting from the assumption ao =f. O. Perhaps the most encouraging feature of Theorem 7 is that sharp bounds depend only on the non-vanishing of individual coefficients in the phase function S(x, y), instead of, say, the rank and signature of some quadratic form. This is certainly a necessary condition for any eventual generalization to a full-fledged theory of Fourier integral operators based on stratification of the Lagrangian by the singular varieties of 1rL and 1r R. The key estimates required for the proof of Theorem 7 are contained in the following Theorem 8, which can be viewed as a composition theorem for these models of degenerate Fourier integral operators [35]:

the kernel K)..(x, y) of the operator TT* satisfies the following bounds: (a) (83) IK)..(x, y)1 ~ C/A(X _ y)I-'/(n-') (b) There exists a finite number ofpairs (ai, bi ), i ~ N, at

Ix-yl
J\X-YI> 1)..\-1/4

(84)

~ C/Ar l / 2.

Similarly, the L I norm in y of K)..(x, y) is 0(IAr l / 2). Thus, we still have O(lAI- ' /4) in this case, and the proof of Theorem 6 is complete.

Ii I I!I "

1~ !l j.

il~!

309

Theorem 8. Assume that one of the sets ofconditions in Theorem 7 holds. Then

J1X-YI> 1)..\-1/4

C/AI- ' / 3

-.,--- -

IK).. (x , y)1 ~ C/AI- 2/ n- v

+ b; = l, such that

N

L laix + biy\-IHi(!xl + Iyl)-Si- un i=1

IITII

~

General Order of Degeneracy n In the formalism of Fourier integral operators, the non-vanishing of a coefficient ai in S(x, y) in (70) corresponds to the condition that some variety I;~-2 be empty.

for some v ::: 0, and (x, y) outside ofa conic neighborhood ofthe line x = y. Here the exponents ~i are strictly positive, and strictly greater than 1/2 if either aj or bi vanishes; (c) In the conic neighborhood of x == y omitted above. either an estimate of the form (84) holds, i.e., (85)

IK)..(x, y)1 ~ C\AI- 2/ n- ulx _ YI-I+S(lxl

+ Iyl)-S-vn

310

CHAPTER 14

for some /) > 0, or we have an estimate of the form I"i

(86)

.::s CI)..I- v- 2/ n lx _ yl-I-vn

IKJ..(x, y)1

where A is a positive constant. The Sj 's are called "critical points," and are allowed to be complex. If Sj has an imaginary part, then its conjugate also occurs in (89). We set

with v ~ O.

Sj == aj + ib j ,

(90)

I,

Theorem 7 follows from Theorem 8, according to the following lemma: Lemma 1. Let K)..(x, y) be any kernel on R x R satisfying either the conditions

1:111

I

!IIIIII

(84) and (85), or the conditions (83) and (86). Then the corresponding operator is bounded on L 2(R) with norm O(I)..I- 2/n).

When the kernel satisfies (84) and (85), Lemma 1 is an easy generalization of the standard Hilbert integral lemma which establishes the L 2(R+) boundedness of the operator with kernel (x + y) -I. When it satisfies (83) and (86), the desired estimate can be established as in (82) assuming (81).

311

SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS

j = 1, ... , N;

P"(z) changes sign only a finite number M of times.

By a cluster L around a given critical point Sk we designate any subset L of the sj's which contains Sk. U will denote L \ {sd and lUI will denote the number of critical points in U. For each integer m ~ 2 and each k, k = 1, ... , N, we define KJ..(k, m) to be KJ..(k, m)

(91)

==

minu,lUl=m-2(1A)..1

IT lak -

SjD- I / m

j¢L

when Sk is real, and K)..(k, m)

==

IT lak - SjDm minu,lul=m-2(IA)..ll bkl IT lak - SjD, minu.IUI=m_I(IA)..llbkI2 IT lak - SjDI m / ,

min{minU.IUI=m-3(1A)..!

j¢L

We discuss now the main ideas in the proof of Theorem 8. The kernel K)..(x, y) of T T* can be expressed as an oscillatory integral K).. (x, y) = (87)

P(x, y, z)

==

J

ei)..(x-y)P(x.y.z) X (Z)2dz

lb

z».

ei)..P(z) X (z)dz

where P(z) is a C 2 phase function on R, X(z) is a C l function which is taken to be compactly supported if either one or both of the bounds a and b are infinite. Assume that N

(89)

l m / }

j¢L

(x - y)-I (S(x, z) - S(y,

K).. =

/

j¢L

(92)

In principle, the asymptotics in ).. of such integrals are given by the method of stationary phase or the van der Corput lemma, except that here it is crucial to keep track of the dependence on x, y. For example, the estimate I)..QI/21- 1/ 2 in (78) and (79) can be worse than the estimate).. -1/3 if IQI is small enough. What is needed here is a version of the method of stationary phase which keeps track of the dependence on external parameters. Since the coefficient I Q11/ 2 in (78) can be interpreted as twice the distance separating the critical points ± Q1/2, we are led to the following theorem which may be of intrinsic interest. Let K).. be an oscillatory integral of the form (88)

I

IP'(z)j ~ A

IT Iz j=1

Sjl

when Sk = ak

+ ibk has a non-vanishing imaginary part.

Then

Theorem 9. The oscillatory integral (88) can be estimated by (93)

IK)..I .::s

CM.Nmaxl~k~Nminm~2[K)..(k, m)]

where the constant CM.N depends only on M, N, and the C I norm of

x.

In Theorem 9 we can evidently choose to deal only with real critical points, simply by dropping the imaginary part b j of Sj from the bounds (89), (91), and (92). This will lead to slower decays for K)... The proof of Theorem 8 actually requires the full Theorem 9 with complex critical points. To illustrate the estimate (93), we take A = 1, Sj = aj to be real, and choose min (91) to be 2. Then Theorem 9 implies the following bound for K)..:

IK).. I .::s C M ,NI)..r l / 2

N

L IT lak k=1

ajl-I/2.

j#

This is the estimate predicted by the method of stationary phase, with the key improvement that the constants are uniform in all phases with bounded number of critical points and bounded number of intervals of monotonicity for P' (z).

~~

_efts• .,...--

. ~....._

,.iiilliii

.119"",___ CHAPTER 14

312

Returning to the kernel K)..(x, y) of TT* (87), we replace).. by )..(x - y), the phase is given by P(x, y, z), and thus the critical points ~j of (90) are the (possibly complex) roots ~j{x, y), j = 1, ... , n - 2 of the polynomial in z(x ­ y)-l(S~(x, z) - S~(y, z». They are homogeneous functions of (x, y) of order 1, so we may assume that (x, y) is on the circle x 2 + y2 = 1. For each fixed phase Sex, y) there are at most a finite number of points Arctan(y Ix) = Ok. k = I, ... , N on the circle at which two or more of the roots Sj coincide. Theorem 6 shows that estimates for the kernel K)..(x, y) reduce to estimates for l~i(X, y) ­ Sj(x, y)1 as (x, y) i:" close to the line of slope Ok' Exploiting the factthat these roots are algebraic functions, we can show that under the hypotheses of the Theorem not more than (n 12) - I when n is even, and (n - 1) 12 when n is odd can get too close together in a precise sense. This combined with Theorem 9 leads to the bounds in Theorem 8 for K)..(x, y), and Theorem 7 follows.

IV THE MONGE.AMPERE DETERMINANT AND COVARIANCE The models we discussed in Section III correspond to Fourier integral operators with "constant coefficients," For example the symbol a()..) of (67) does not depend on 1. The task ofconstructing a full theory defined on smooth manifolds even when these models are already well understood turns out to be surprisingly difficult. The only reasonably general results are for n = 3, that is, when at least one of the projections JrL or JrR is a fold. There it is known that • Fourier integral operators whose Lagrangians are folds with respect to both JrL and JrR can be microlocally conjugated to the Airy operator [19], [20] and are smoothing of order -1/3; • If JrL is a fold and JrR is a blow-down, then the FlO is smoothing of order -1/4 [11], [12]; • Sobolev and LP - L q estimates are known for Radon transforms along general variable distributions of curves in any 2-dimensional manifold [34]. The methods for the first two classes of results are microlocal in nature. Here we shall discuss the local approach of [34], which makes the role of the Monge­ Ampere equation (36) transparent and also confirms the intimate relation between degenerate Fourier integral operators and singular Radon transforms. Let M be a Coo two-dimensional manifold, and let M p be a curve passing through P for each point Pin M. If dJ-Lp(Q) is a Coo density on M p supported in a small neighborhood of P we set (94)

Tf(P)

= [

iMp

f(Q)dJ-Lp(Q).

~tti\'1RI~ti"=---·-

SfNGULAR fNTEGRALS AND FOURfER INTEGRAL OPERATORS

313

Let it L, itR be the projections from the Lagrangian manifold N* (C) on the left and right factors as in (5). Then [34] Theorem 10. • If either itL or itR is a fold along the singular variety ~, then T is bounded

from H(s)(M) to H(s+1/4) (M); • Ifboth JrL and itR are folds, then T is boundedfrom (a) H(s)(M) to H(s+I/3)(M); (b) LP(M) to U(M)for (lip, l/q) in the intersection of the closed triangle with vertices at (0,0), (1, 0, (2/3, 113) and the closed half-space of the equation (95)

I

1

p

q

I

4 ~ o.

As mentioned above, Part (a) of the second statement is the well-known estimate of Melrose and Taylor [19], [21] in diffraction theory. The estimate in the first statement has been obtained by Greenleaf and Uhlmann [12] for any dimension, under the additional hypothesis that the other projection is completely degenerate, or, more precisely, a blow-down. We discuss in some detail two ingredients of the proof. We focus on the case when both itL and itR are folds since its treatment also contains many of the ingredients required for the case when only one projection is assumed to be a fold. It suffices to prove the theorem for M = R 2 and f is supported in a fixed small 2 neighborhood of the origin. We can choose coordinates (x, t) on R so that locally the curve M(r ,r) is the graph of a function

M(x,r) = ley, s); s = t - sex, t, y)}

(96)

with Sex, t, x)

(97)

== 0,

S(O, t, y)

==

o.

The operator T can be written as

(98)

(Tf)(x, t)

=

fff

ei )..(s-r+s(x,r,Y»1/J(y)f(y, s)dydsd)"

where 1/J(y) is a suitable cutoff function in y. The standard almost-orthogonality argument reduces the Sobolev bounds to showing that the localized operators in )..

(99)

(Tmf)(x, t)

=

fff

ei)..(s-r+S(x,r,y» X(T m)..)1/J(y)f(y, s)dydsd)"

--

-~"-----

------------------"-------------------------_._----------_._------­

----'-'--'-=-,

314

CHAPTER 14

satisfy the unifonn estimates

Lemma 2. In a neighborhood ofthe origin, the phase function S(x, t, y) satisfies

II Tm II

(100)

S

crm / 3 •

Let J (x, t, y) be the Monge-Ampere detenninant (36) with cI> (x, t; y, s) t+S(x,t.y) (101)

J(x. t. y)

= (1

=

s _

- S;)S:y - S~S:~'

x

III

The Sobolev bounds for T are consequences of the following two bounds for Ii,m

IlIi,mll S

S(x, t, y) be a phase function satisfying the condition (l07), together with fold conditions on both sides. Then • In a sufficiently small neighborhood ofthe form It - s I < clx - YI, the equation S~(x, t, z) - S~(y, s, z) = 0

(108)

admits a unique solution Zc (x, t, y, s); • Let y = x(x, t) denote the zeroes of the Monge-Ampere determinant corresponding to S(x, t, y), and set X(X, t, y, s)

(109)

= S(x, t, i(x, t»

The index m is fixed from now on, so we drop it from our notation. Now the fact that both JrL and JrR are folds implies that the support of the kernel of Ii in each variable is of length "-' 2-1 when the other variable is fixed. The first estimate follows at once. We note that we would obtain only the weaker estimate IlIiII S C2- 1/ 2 if only one projection is assumed to be a fold. The key to the second estimate is a further localization (104) (Ii,j/)(x, t)

x

= x (ci(x(x, t) - Xj»

III

e i,\(s-t+S(x,t,Y))1{t(rmA)X(i J(x, t, y»f(y, s)dydsdA

where cis a large constant, and x j are points regularly spaced at a distance (c) -12-1• Since (x, t) ~ (x(x, t), t) is a smooth change of variables, we may define (105)

ci;,j/)(x, t)

== <1i,j/)(x, t).

The second estimate for Ii can be verified to follow from a similar estimate for In tenns of (y, s) and (x, t) the operator is of the same fonn as (104) but with a different phase function

t,m'

(106)

t,m

S(x, t, y)

Then in the region where c2- l lx - yl we have (110)

Iz -

- S(y, s, i(x, t».

1)1 S 2-1 ,

i(X,

It - s - X(x, t, y, s)1 "-' It - s - S(x, t,

It - s - X(x, t, y, s)1 >

z)

+ S(y, s, z)1

while in the region

cr l

IlIi,mll S qrmi)I/2.

(103)

= 0(1).

Lemma 3. Let

(102) (Ii,mf)(x, t)

ei,\(s-t+S(x,t,y))x (2- mA)X (i J(x, t, y»1{t(y)f(y, s)dydsdA.

IS:~I

= 0(1),

IS:I

(107)

The condition that Jr L (respectively Jr R) be a fold along its singular variety translates into the fact that the zero set of J (x. t. y) can be parametrized as the graph of a smooth function of y = x (x, t) (respectively x = y(t, y». In particular the map x ~ (x, t) is invertible for each t if both projections are folds. We introduce a dyadic decomposition Tm = :L;:o Ii,m away from the zero set of J (x, t. y)

=

315

SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS

== S(x(i, t), t, y).

It - s - X(x, t, y, s)1

(111)

s 2- l lx - yl

the critical point zc can be approximated by the zero of the Monge-Ampere determinant Izc(x, t, y, s) - i(x, t)1 s C(lx -

(112)

yl

+ (c)-lrl ).

Lemma 4. Let j (x, t, y) be the Monge-Ampere determinant determined by the phase function
(113)

We sketch the proof of the second statement in Lemma 3, as it is one of the key steps in the argument. First let x(x, t) be any function of (x, t) and define X as in (109). We note that X can be expanded in tenns of (x - y) and t - s - X with smooth coefficients (114)

X

== (x - Y)P(x, t, y, s)

+ (t -

s - X)a(x, t, y, s).

In fact, a Taylor expansion gives X == (x - y)f3o + (t - s)ao, and since IS:I = 0(1). The equation (114) follows with (115)

P = f3o(1

- ao)-I,

a = ao(1

- ao)-l.

laol =

0(1)

"-"-'-'ii_-~-'--

",irliIr-

:iti:~

-

-_~_~

-.·s-...

'-iiiiii"j"'''"iliii'iiiiiiiiii"i"

316

n

'Dif;;--

CHAPTER 14

Next we rewrite the equation for the critical points in more explicit fonn. Setting S(.x, t, z) - S(y, s, z) == (x - y)y + (t - s)8 we find

(116)

azW(x, t, y, s, z)

= [y + P8] +

t-s-X

_

x-y

(8

a

+ ~8) = 0

with up to negligible tenns (118)

Y = S~(x, t, z),

8 = S;(x, t, z),

Finally, we derive explicit fonnulas for ao, (119)

ao

= S;(x, t, x(x, t»,

-

f30

P=­ 1 - ao

Po, up to similar negligible tenns Po = S~(x, t, x(x, t).

Altogether the equation (116) for zc(x, t, y, s) becomes (120)

S;/x, t, zc)(l - S;(x, t, x(x, t)))

+

in the region defined by (11l). Since Ix kernel of tt­

Iz -

S~(x, t, x(x, t))S;~(x, t, zc)

zc(x, t, y, s)\ ""

= o.

Thus for the critical point zc to coincide with x(x, t), we must choose x(x, t) to be the zero of the Monge-Ampere detenninant! This establishes the key estimate (112) in Lemma 3. We return to the proof of L 2 bounds for 1;, j' Forming 1;,j 1;~j' we obtain an operator whose kernel is an oscillatory integral in z and in ).. with phase given by

(Tf)(x, t) =

i

X (ci(x - xj»X(c21(y - xj»X(2 ](i, t, z»X(2 l ](y, s, z».

The main contribution to the oscillatory integral comes from its critical point zc(x, t, y, s). Now Lemma 4 implies in particular that the zero set of j is given by y = X. In view of Lemma 3 we have then (122)

Izc(x, t, y, s) -

(r+Tf)(x, t) =

(125)

and a cutoff given by

xl :s CClx - yl + (c)-IT

I

)

Iz - xl ""

T

317

on the support of the

l

This allows to exploit integration by parts with respect to z in the region (111). Outside of this region, the desired estimates follow from a simpler integration by parts in )" using (11 0). We would like to point out an interesting feature of the above arguments. For classical Fourier integral operators we can choose coordinate systems so that the non-vanishing of the Monge-Ampere determinant in small neighborhoods is equiv­ alent to the non-vanishing of the Hessian in x, y of Sex, t, y). This is done from the outset and the Monge-Ampere determinant itself no longer enters the analysis [32], [33]. In the above argument on the other hand it is crucial to introduce dyadic partitions away from the zeroes of the Monge-Ampere determinant itself, rather than from the zeroes of say the Hessian, since these approximate better the critical points Zc in (111). Now the critical points Zc are zeroes of a linear expression in S, while the zeroes of J are of course zeroes of a non-linear expression in S. The appearance of this non-linearity is due to the fact that the approximation (112) does not hold everywhere, but only in a region determined by the cutoff itself. It is very intriguing that the localization procedures required by linear analysis inevitably introduce some non-linearity and lead us exactly to a covariant expression. We conclude by sketching the relation between the Radon transfonns of (94) and singular Radon transforms. For the sake of simplicity we shall assume that the family Mp is semi translation-invariant in the sense that the defining function Sex, t, y) of (96) is independent of t. The operators T and TOT can then be expressed as

)"(S(x, t, z) - S(y, s, z»

(121)

yl :s

(c)-12- l

IS~(x, t, z) - S~(y, s, z)1 "" Ix - YIT I .

(124)

We can obtain approximations of Zc up to errors bounded by the right hand side of (112) by adding to the equation z II' = 0 any tenn bounded by 0 (Ix - yl) or O(C- 12- 1). In particular in azw we can drop (t - s - X)(x - y)-1(8 + (8), replace y by x, and s by t. The approximating equation is az(Y

'iIiiill;~:i·;!l"'l.i~;ilfll!l.:-'~

and we obtain the following lower bound for the phase variation

+ (8)

= 0 {::::::} z = zc(x, t, y, s).

(117)

-

SINGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS

(123)

Sex, t, z) - S(y, s, z) = (x - y)W(x, t, y, s, z) W(x, t, y, s, z)

;f~~~~"'-""'-~"g_-ina;;i'i!fi""''U'iii~;:tF30-'':'.~~;;';;":-''':''&:i!'~"''~;ii;l£''-;i!ii'~,i@~~~~~';i'·"-·w,";"B'~S"~iia:;j'i;j""i:<:b'~~%jik

I II

fey, t - sex, y»dy fey, t

+ sex, z)

- S(y, z»dzdy.

Fixing (x, t) we make the change of variables (y, z)

~ (u, v)

u=x-y (126)

v

= S(y, z)

- Sex, z) - (S(y, zc(x, y» - sex, zc(x, y)))

where zc(x, y) is the critical point S~(x, zc) - S~(y, zc) = O. Observe that IJacobianl = IS~(x, z) - S~(y, z)1

318

CHAPTER i4

""' Ix - yllz - Zc(X, Y)I (127)

Ivl ""' Ix - Yllz - Zc(X, y)e

SO that the operator T* T

SiNGULAR INTEGRALS AND FOURIER INTEGRAL OPERATORS

319

Nagel-Stein-Wainger theorem [5] (see also the discussion at the end of Section II. 2) for singular Radon transforms applies, giving the desired estimate.

corresponds to the case s = 112 ofthe following operators

Columbia University

(128) (Us f)(x, t)

=

II

f(x - u,t - v

+ S(x, Zc(x, x

- u» - S(x - U'Zc(X,X - u)))

dudp

lulslvl S This relation reduces the LP - L q bounds for T to L 2 bounds for singular Radon transforms. In fact, the estimates along the segment (11 p) - (llq) - (1/4) ~ 0 of Theorem 10 follow from the boundedness of T*T from L 4/3 (R2) to L 4(R2). By complex interpolation this is a consequence of the endpoint bounds

• Us is bounded from LIto L 00 for Re s = 0; • Us is bounded from L 2 to L 2 for Re s = 1. The first statement is trivial. As for the second, the operator Us can be factorized as Us = LsMs where (129)

I

(Ms f)(x.1) =

I

(Lsf)(x, t) =

f(x, t - v )dv-

Ivl s

f(x - u, t - S(x -

+ sex, Zc(X, x

- U»

du , c(x , x - u») ­ luis'

U Z

2

The boundedness of M s on L is the standard one-dimensional theorem on singular

integrals. As for L s we recognize that it is a singular Radon transform with phase

function

cI>(x, y)

(130)

== Sex, zc(x, y» - S(y, zc(x, y».

In general, this phase function is degenerate. However we have

Lemma 5. Let z(x) be the zero ofthe Monge-Ampere determinant S~yCx, z(x» =

O. Then (131)

III

cl>XYY

1 Sill ( -( (dz(y) )2 = 4 xyy y, z y» ----;ty"

+ 0(1x

- YI ) .

This shows that for two sided folds IcI>~~y I ~ c > O. Thus the Hessian of the phase (130) does not vanish to infinite order, and the refined boundedness Christ-

REFERENCES [I] J. Antoniono and G. Uhlmann. "A functional calculus for a class of pseudo-differential operators with singular symbols." Proc. Symp. Pure Math. 43 (1985) 5-16. [2] V. Arnold, A. Varchenko, and S. Goussein-Zade. "Singularites des applications differentiables." Editions Mir, 1986. [3] M. Beals, C. Feffennan, and R. Grossman. "Strongly pseudo-convex domains." Bull. Amer. Math. Soc. 8 (1983).125-322. [4] M. Christ. "Hilbert transfonns along curves I." Ann. of Math. 122 (1985),575-596. [5] M. Christ, A. Nagel, E. M. Stein, and S. Wainger. "Singular and maximal Radon transfonns." In preparation. [6] C. Feffennan. "Monge-Ampere equations, the Bergman kernel, and geometry of pseudo convex domains." Ann. of Math. 103 (1976), 395-416. [7] G. Folland and J. J. Kohn. "The Neumann problem for the Cauchy-Riemann complex." Annals of Mathematics Studies 75, Princeton University Press, 1972. [8] G. Folland and E. M. Stein. "Estimates for the ab-complex and analysis on the Heisenberg group." Comm. Pure Appl. Math. 27 (1974), 429-522. [9] D. Geller and E. M. Stein. "Estimates for singular convolution operators on the Heisenberg group." Math. Ann. 267 (1984), 1-15. [10] A. Greenleafand G. Uhlmann. "Non-local inversion fonnulas for the X-ray transform." Duke Math. J. 58 (1989), 205-240. [11] . "Estimates for singular Radon transforms and pseudo-differential operators with singular symbols." J. Funct. Anal. 89 (1990),202-232. [12] . "Composition of some singular Fourier integral operators and estimates for the X-ray transform I." Ann./nst. Fourier 40 (1990), 11,1991 preprint. [13] P. Greiner and E. M. Stein. "A parametrix for the a-Neumann problem." In Estimates of the Neumann Problem, edited by P. C. Greiner. Mathematical Notes 19. Princeton University Press, 1977. [14] V. Guillemin. "Cosmology in (2 + I) dimensions, cyclic models, and defonnations of M2 •1 ." Annals of Mathematics Studies 121. Princeton University Press, 1989. [15] V. Guillemin and G. Uhlmann. "Oscillatory integrals with singular symbols." Duke Math. J. 48 (1981),251-267. [16] L. Honnander. "Fourier integral operators I." Acta Math. 127 (1971), 79-183. [17] . The Analysis ofLinear Partial Differential Operators /-N. Springer Verlag, 1985. [18] A. Melin. "Lower bounds for pseudo-differential operators." Arkiv Math. 9 (1971), 117-140. [19] R. Melrose. "Equivalence of glancing hypersurfaces." /nv. Math. 37 (1976),165-191. [20] R. Melrose and M. Taylor. "Near peak scattering and the correct Kirchhoff approximation for a convex obstacle." Ailv. Math. 55 (1985), 242-315. [21] R. Melrose and G. Uhlmann. "Lagrangian intersection and the Cauchy problem." Comm. Pure Appl. Math. 32 (1979), 483-519.

,_

~II---===-="-==;======:::-"~-I,

___"_.':::.,,, .:'-::',",::_::::,":::,"

~·'·'_·,

..'""r,~,_

,,,,,,.'U~~-'=:::""'"

V"~".::,_

._:'::'::~:'~=~'M"""

"'~~_'·

.•T.~_~.'.'_.'~".'_~·'

"~_~~"

':'"" ,._ :-~~_" ..•.""""_,,,..

...'"

=.":~~':-'''~~~:'::=''''d'''''''.,

CHAPTER 14

[22] G. Mockenhaupt, A. Seeger, and C. Sogge. "Wave front sets, local smoothing, and Bourgain's circular maximal theorem." Ann. ofMath. 136 (1992), 207-218. [23] D. Milller. "Singular kemels supported by homogeneous subrnanifolds." J. Reine Angew. Math. 356 (1985), 90-118. [24] D. Muller and F. Ricci. "Analysis of second order differential operators on the Heisenberg group I." Inv. Math. 101 (1990),545-582. [25] A. Nagel, N. Riviere, and S. Wainger. "Hilbert transforms associated with plane curves." Bull. Amer. Math. Soc. 223 (1974), 235-252. [26] D. Oberlin. "Convolution estimates for some measures on curves." Proc. Amer. Math. Soc. 99 (1987), 56-Q0. [27] Y. Pan. Ph.D. dis~., Princeton University, 1989. [28] Y. Pan and C. Sogge. "Oscillatory integrals associated to folding canonical relations." Colloq. Math. 60 (1990), 413--419. [29] D. H. Phong. "On integral representations for the Neumann operator." Proc. Nat. Acad. Sci. 76 (1979),1554-1558.

15

Counterexamples with Harmonic Gradients in }R3 Thomas H. Wolff"

[30] D. H. Phong and E. M. Stein. "Some further classes of pseudo-differential and singular integral operators arising in boundary value problems I." Amer. J. Math. 104 (1982), 141-172. [31] - - - . "Singular integrals related to the Radon transform and boundary value problems." Proc. Nat. Acad. Sci. 80 (1983),7697-7701. [32] - - _ . "Hilbert integrals, singular integrals, and Radon transforms I and II." Acta Math. 157 (1986), 99-157 and Inv. Math. 86 (1986), 75-113. [33] - - - . "Singular Radon transforms." Duke Math. J. 58 (1989), 349-369. [34] - - _ . "Radon transforms and torsion." Int. Math. Res. Notices 4 (1991), 49-60. [35] - - - . "Oscillatory integrals with polynomial phases." Inv. Math. 110 (1992),39­ 62. [36] F. Ricci and E. M. Stein. "Harmonic Analysis on nilpotent groups and singular in­ tegrals, I. Oscillatory integrals." J. Funct. Anal. 73 (1987),179--194; "II. Singular kemels supported on submanifolds." J. Funct. Anal. 78 (1988), 56-94. [37] C. Sogge and E. M. Stein. "Averages of functions over hypersurfaces: smoothness of generalized Radon transforms." J. Analyse Math. 54 (1990), 165-188. [38] E. M. Stein. "Maximal functions: spherical means." Proc. Nat. Acad. Sci. 73 (1976). [39] - - _ . Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series 43. Princeton University Press, 1993. [40] E. M. Stein and S. Wainger. "Problems in harmonic analysis related to curvature." Bull. Amer. Math. Soc. 84 (1978), 1239-1295. [41] M. Taylor. "Propagation, reflection, and diffraction of singularities of solutions to wave equations." Bull. Amer. Math. Soc. 84 (1978), 589-611. [42J - - _ . "Fefferman-Phong inequalities in diffraction theory." Proc. Symp. Pure Math. 43 (1985), 261-300. [43J - - _ . Pseudo-differential operators. Princeton Mathematical Series 34. Princeton University Press, 1981. [44] A. Thompson. "A singular Radon transform in Rp3." Comm. Part. D!ff. Eq. 14 (1989), 1461-1470. [45] F. Treves. Introduction to pseudo-differential and Fourier integral operators. Plenum, 1980.

INTRODUCTION Many results about hannonic functions in 1R2 are proved using complex analysis explicitly or implicitly, so it is natural that some of them should fail in higher dimensions, or should at least need to be reformulated. The purpose of this paper is to work out three such counterexamples. Let a > 0 be small enough.

Theorem 1. There is a nonconstant harmonic function on the upper half space 1R~ C 1R3 which is Cl+ a up to the boundary, and whose gradient vanishes on a boundary set ofpositive measure.

Theorem 2. There is a harmonic function u on 1R~ such that (0.1)

sup [ lV'u(x, tWdx\dx2 < 1>0

JIR

00,

2

and such that V'u does not have nontangentiallimits almost everywhere.

Theorem 3. There is a bounded domain in 1R3 whose harmonic measure puts no mass on any set with Hausdorff dimension less than 2 + a. These results extend easily to IRn , n ::: 4. What they have in common, in addi­ tion to being counterexamples to the obvious generalizations of two-dimensional 'This research was supported by the National Science Foundation, under grant DMS 87-03456.

322

CHAPTER 15

theorems, is that the two-dimensional results may all be proved using the sub­ harmonicity of log lV'ul when u is a harmonic function on a domain in 1R2. For Theorems 1 and 2 this is well known, e.g., if u: jR~ -+ jR and V'u vanishes on a boundary set E with positive measure, then log lV'ul is -00 on E and, being subharmonic, is identically -00. I will now give a brief discussion of Theorem 3, which motivated this paper. In [12], Jones and I showed that harmonic measure for any domain in jR2 puts full mass on some set with Hausdorff dimension I. This completed a line of work due to Oksendal [19], [20] (who also made the conjecture), Kaufman-Wu [13], Carleson [6], and Makarov [15]. In particular, Makarov had proved the same result for simply connected domains in [15, Theorem 3]. It is sharp for any smooth domain, so the apparent generalization to jRn would be that harmonic measure always lives on a set with dimension n - 1. The relevance of subharmonicity of log lV'ul for Oksendal's problem is due to Carleson [6] who used the following argument. Consider a scale invariant boundary such as the von Koch snowflake. Harmonic measure at a fixed interior point is then mutually absolutely continuous to an invariant measure for the scaling transformation on the boundary. In ergodic theory it is known that the dimension of such an invariant measure and its entropy are equivalent quantities. Computing the entropy turns out to be a matter of evaluating

. 11

where nn is the nth stage in the construction of the snowflake and gn is the Green's function, and a is surface measure. Subharmonicity of log IV' gn I away from the pole implies the integrals in (0.2) are bounded below in terms of an upper bound for So (0.2) is certainly nonnegative, which is what is needed to conclude diam that harmonic measure is at most I-dimensional. This "argument is reversible, so the main step in proving Theorem 3 is to find a snowflake-type construction in jR3 for which (0.2) is negative. Theorems 1 and 2 are similar. Instead of needing to construct a domain and its Green's function, we have a fixed domain (the upper half-space) and we need a sequence of harmonic functions Un with suitable normalization, such that

nn'

(0.3)

(

JQOl

log lV'unldx\dx2 -+

en

inner functions on the ball in and by Lewis [14] and myself [31] to construct p-harmonic functions without almost everywhere boundary values. The method itself goes back to Men'shov-the Men'shov correction theorem and the fact that every measurable function is a.e. the sum of a trigonometric series. Consider (0.3). One shows first that there is no true obstruction to (0.3) such as the subharmonicity in the 2-dimensional case. This comes down to proving that if e is a unit vector in jR3, then there is a harmonic gradient V'u with suitable decay at 00 such that (0.4)

{ log

JR2

-00.

Here Q(I) is the unit square in jR2 and jR2 is the boundary of the upper half-space. The constructions for (0.2) and (0.3) are done by the same method. This method seems to be the canonical one for counterexamples involving boundary behavior;

e.g., it has been used by Aleksandrov, Hakim-Sibony, and Low (see [I]) to construct

Ie + V'uldx\dx2

< O.

Once (0.4) is known, one sets up a recursive construction as follows. Let V' g\ be any smooth harmonic gradient. On a small enough scale V' g\ is approximately constant, so by adding on dilates of the functions V' u one obtains V' g2 with

{ log!V'g2/dx\dx2 < { loglV'gIidx 1d x2 -I],

JlR2

JR2

I]

>

ofixed.

One then changes to a still smaller scale and repeats the procedure. (0.2) is treated the same except that the modifications are made in the domain and the initial step is to find a perturbation, n, of the upper half-space with the property that {

Jan

hm lV'gnllog lV'gnlda, n--+oo nann

(0.2)

323

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN JR.3

• IV' goo Ilog IV' goolda < 0,

where goo (Green's function with pole at

00)

satisfies goo(x) = 0 when x

E

an

and lim..--. oo goo(x) - X3 = O. Theorems I, 2, and 3 leave open a number of questions having to do with the size of the number a. Consider first Theorem 2. Stein and Weiss [25] found the analogue of subharmonicity of log lV'ul in higher dimensions, namely that IV' u !(n-2 l !(n-\) is subharmonic if u is harmonic on a domain in jRn. They introduced a the space of functions satisfying (0.1) (now called the Stein-Weiss H space) as a a generalization of H p theory to jRn, and proved that functions in H have a.e. One can ask whether Theorem 2 can be pushed boundary values if a ::: up to their critical exponent. The main obstacle to doing this by our method is that one would need the analogue of (0.4), Le., for each p < and unit vector e E jRn a harmonic gradient V' u on jR~ with rapid decay at 00 such that

:=i.

:=i

(0.5)

{

JRn-1

(Ie + V'ui P

-

I)dx\ ... dxn-l < O.

Such functions probably exist, although my argument for (0.4), which consists of explicitly writing down u and computing the relevant integral, does not show how to construct them. Similar remarks apply for questions like Theorem 2 where Riesz transforms are replaced by double Riesz transforms, etc., as in [5].

it-~"'~.7~;<>:Mh;;{~#.i~~i.i.'i~~i..;i>;~~~.$';'~~ii£<;~i".,;w~~';.,i;i~-,,=,,",;;,M;'~-iWI*1~.i'~imi~.~~.M:iQ~~';;W[~

324

CHAPTER 15

One can also ask whether C 2 or Coo can replace Cl+a in Theorem 1. This has to do with the rate at which (0.3) can diverge and could be a delicate question. A 2 C counterexample would lead (by taking one derivative) to a harmonic function u with u = ddu = 0 on a boundary set of positive measure, which is how the "'3 problem has sometimes been stated. Regarding Theorem 3, Bourgain [3] showed that any harmonic measure in lRn puts full mass on some set with dimension n - En where En > 0 depends only on n. It would be interesting if one could compute the best value of En. Both Bourgain's argument and the argument of this paper are too crude for that. I want to mention some previous work related to Theorems I and 2. Uchiyama [26], [27], [28] gave a new method for studying singular integrals on LP, p :::: I. His method does not use subharmonicity at all and applies to more general situations than harmonic gradients, but a special case of the result of [28] is another proof that Stein-Weiss H P functions have boundary values for p > 1 - E. In [27], he considered the small p case and gave counterexamples like Theorem 2 for related problems on discrete martingales. The proof of Theorems I and 2 uses some of the ideas of his method for the positive results in [26]; in particular, the functions satisfying (0.4) play the same role as his Lemma 2.3. The question of whether there are uniqueness theorems for boundary sets of positive measure in higher dimensions was posed by BeTS. The previous work that has been done on it appears to be concerned mainly with local properties near one boundary point; cf. Mergelyan [18] and for the strongest known result, Rao [21].

Acknowledgments. I am grateful to J. Goodman for verifying Lemma 1.2 numerically. This work was done while I was at the Courant Institute.

Added September, 1991 Almost all current research in Euclidean harmonic analysis owes something to the work of E. M. Stein and this paper is quite clearly not an exception. I dedicate it to him with appreciation and respect. Since it was first circulated in October 1987 there has been some further progress which I will now summarize. A. B. Aleksandrov and P. Kargaev [2] have given an alternate, less computational approach to the existence of the building block functions (0.4), (0.5). For given n 2': 3, let lR~ = {x E lRn : X n > O} and denote by SH(lR~) the harmonic gradients Vu: lR~ ~ lRn vanishing at infinity, smooth up to the boundary and with boundary values in the Schwarz space. Then Aleksandrov and Kargaev show that for any p < n~2 and unit vector e E lRn there is a harmonic gradient Vu E SH(lR n ) such that (0.5) holds. They also give several additional applications of this construction including among others the

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN

325

]R3

following: (1) given any continuous function f: lR - ~ lR , n 2': 3, there is a harmonic function u on lR~Cl up to the boundary and such that Vu coincides with f on lRn outside a set with arbitrarily small measure; (2) for p < n~2, any LP function f: lRn - 1 ~ lRn coincides with the a.e. boundary values of some harmonic gradient Vu belonging to the Stein-Weiss HP space; (3) for p < n~l there is a function in the Stein-Weiss H P space whose a.e. boundary values belong to a ray {e: r > OJ, for some e E lRn • J. Bourgain and the author [3] used the construction of Aleksandrov and Kargaev to show also that there are harmonic functions u on lR~ C 1 up to the boundary and such that u and ddu vanish on a common boundary set of positive measure. This "'. answers one of the questions mentioned above. W. S. Wang [30] has proved further related results, in particular, that examples analogous to those of Bourgain and the author exist on any domain with a Cl+ f boundary. On the other hand the question of higher smoothness in Theorem I remains open and no further bounds have been proved on the dimension of harmonic measure. It is also still unknown whethe~(0.5) can be pushed all the way up to the Stein-Weiss critical exponent :::::~ . I have made some changes in the preceding introduction mainly reflecting the fact that in 1987 I was unaware of relevant previous work-Men'shov's classical work as well as [17], [21]. I have also corrected some misprints and minor inac­ curacies in the subsequent sections but have not attempted a serious revision. It should therefore be pointed out that several arguments can be done more simply, and in particular, the appendix is now definitely unnecessary because of the work of Aleksandrov and ~argaev. I would like to take this opportunity to present the following observations which show that functions proving (0.5) will tend to depend strongly on p as p ~ First we have the following "rigidity" result for the Stein-Weiss subharmonicity inequality. n

1

n

:::::i.

Proposition A. Suppose u is harmonic on a domain Q C lRn • n 2': 3, and IV u I :=~ is also harmonic. Then u is either affine, or of the form blx - ej2-n + cfor some b, c E lRand e E lRn .

Proof Since u is real analytic we may shrink Q and may therefore assume that Vu never vanishes. If we examine the proof of the Stein-Weiss inequality [24], [25] we see that strict inequality must hold at a given point unless Vu is an eigenvector of the Hessian H u and, furthermore, all eigenvalues other than the one corresponding to Vu are equal. Then in the present situation we conclude that there is a function A: Q ~ lR, such that (i)

Hu(x)(Vu(x))

= A(X)VU(x)

___ "'-"".'-~~'=""c:~_","_-=,=:'=-.:-:o",.-

_

_.

•.-

--

- ':;,. .~i;,;;;;;;:-=

-~._._._.

.,.--

~

326

-

""'-.,; ..",-'"

..

-_~i""'-

··~_m'·'

CHAPTER i5

(ii) Hu(x)

r (VU(X)).1 = -

. ":·,·, .'-

-i<..."",-_

_X·~';"ii"d'ci'''':iiF;i;.i'C'''''''~~"",,"':,~.'''';'''''"'':.'·:''';·''''';;;';'·'''''''''''~'~~~''''~·'''~",;,<,;';~0>if;i;.O;;"';;""'''''~''''''''';;''''''-'''''''''''''~''''',,"~"

COUNTEREXAMPLES WiTH HARMONiC GRADIENTS IN

n~l A(X) . id.

where id is the (n - 1) x (n - 1) identity matrix. Now (i) implies that IVul is constant on level sets of u, while (ii) implies that the level sets are umbilical (all principal Curvatures at any given point are equal). By [23] Theorem 26, the level sets are contained in planes or spheres. The proposition now follows by applying the uniqueness statement in the Cauchy-Kowalewski theorem on one of the level sets.

•

A somewhat deer.er application of the same general principle is given in Theorem 2 of [8]. We now apply Proposition A to prove

~ lim inf k-+oo

~

L

:=~ + L

Proof. Let p =

l)dx =

Ln_, (IV(x. e + UkW -

The Laplacian is taken in the pointwise sense. It is well defined except at critical points and therefore a.e. The inequality is easily justified b1. a limiting argument starting from the smooth subhannonic functions (IV(x·tl~k!Jph€2)PI2 - 1 and using Fatou's lemma. Suppose now that (Vud converges in LP + L I • Then the maximal function estimates in [25] imply {VUk} also converges unifonnly on compacts of IR~ and the limit function is a harmonic gradient Vu. Then using the LP + L I convergence and the fact that lie + Vulq - 11 ;S min(IVul, IVuI P) for q ~ p, we get

+ VUklPk

1) =

-

= In particular, limk-+oo

J Ie + VUklP

{ (Ie + Vul JlRn-1

lim { (Ie k-+oo JRn-1

P - I)

+ VUklP

- 1).

- 1) ~ O. On the other hand t.(IV(x . e

+ u)IP) on IR~ except at critical points of x . e therefore a.e. By Fatou's lemma

UkW) --+ t.(IV(x . e

(

JIR~

xnt.(IV(x· e

+ uW)

~ lim inf k-+oo

(

JR~

1 BOUNDARY PROBLEMS FOR RIESZ SYSTEMS

xnt.(IV(x. e

identify 1R2 with IR~ \IR~. Squares in 1R 2 with sides parallel to the axes are denoted Q, and Q(N) = N, N] x N, N]. The standard basis is el, e2, e3. A hannonic gradient is a function /: IR~ --+ 1R3 which is the gradient of some harmonic function u, Qr else the boundary values of such an /. Both / and U will always be assumed to vanish at 00. If X is a function space on 1R2 , then X H = {f: 1R 2 --+ 1R3 : / is a harmonic gradient and the components of / belong to Xl. Thus SH is the harmonic gradients whose components belong to the Schwarz space S, etc. Before doing the constructions we summarize some properties of Schwarz space harmonic gradients.

[-!

l)dx

~ JRn { xnt.(IV(x. e + ukW)dx. +

lim { (Ie k-+oo JRn~1

O.

We work in 1R3 and often write points of 1R 3 as x = (x, X3) with x E 1R2 . The upper half space is IR~ = ~ E 1R3 : X3 > 0). We write IR~ for its closure and

:=i. We have

Ln-I (Ie + VUkl P -

{ (Ie + VUklP - 1) JRn-1

We remark that convergence to zero can take place only in a fairly weak sense, if at all. Variational calculations like those in Section 1 below imply that {Vu n } cannot converge to zero in LOa.

:=i

limit point o/the sequence {Vud is zero.

327

It follows that t.(IV(x . e + u) IP) must vanish identically, so Proposition A applies to x . e + u. Since Vu cannot be bounded below at infinity, the only possibility is that x . e + U = x . e + const and the proposition is proved. •

Proposition B. Suppose that n ~ 3, e is a unit vector in IR n, {V ud c SH(IR~){pkl --+ and JlRn-1 (Ie + VUkl Pk - l)dx < O. Then the only possible I

]R3

'''''''''~~~:''''''-=-''''''"''''''''''''''''''''''''' ~

+

+ u, and

+ UkW)

!

[-!

!

(i) Let /: 1R2 --+ 1R3 be a harmonic gradient such that h E S. Then / E SH if and only if XU hdx = 0 for all multiindices a = (ai, (2) of length

J

~

This follows by taking Fourier transforms: 11 = -i~I1~I-1 13 and 12 =-i~21~ I-I 13 are smooth at zero if 13 vanishes to infinite order there. (ii) (a) SH is dense in L~, 1 < P < 00. (b) For any k E Z+ there is e E Z+ making the following true: if / is a harmonic gradient such that h is a smooth function with compact support and with J XU / (x )dx = 0 when la I ~ e, then for any E > 0, there is g E SH such that sUPxER2(1 + Ixl)kl/(x) - g(x)1 < E. I believe these facts to be well known but will include a proof. The proof in my 1987 manuscript was incorrect; I thank W. S. Wang, who very tactfully pointed this out. Let 1/1 be a Schwarz function on 1R 2 such that ;j, has compact support and;j, = 1 in a neighborhood of O. Let 1/Ir (x) = t- 2 1/1 (t- I x). To prove

O.

~il;'t·'W

328

CHAPTER 15

(a) it suffices to approximate functions I E L~ such that h E S, since such functions are dense in L ~ by boundedness of the Riesz transfonns. However, if h E S, then by (i) and the fact that.(fJ = 1 near 0, we will have 1-1/11 I E SH. Moreover 1/11 I --+ 0 as t --+ 00 by Young's inequality since lEU for 1 < q < p. To prove (b) note first that the moment assumption implies /I (x) I ;S (1 + Ixl)-C£+2). It therefore suffices to show that if XU I(x)dx = O(lal :::: l) and I/(x)1 ;S (l + /xi)-C£+2) then sUPxEIR2(1 + Ixl)kl1/l, l(x)1 --+ 0 as t --+ 00. This is done as follows: Fix x and t ~ 1. On the one hand, since

*

*

f

*

1/IES 11/11

f

* l(x)1 ;S t- 2

(l

+ lyi)-CH2)

Ix I ) -C£+2)

< ( 1+ -

'"

t

( + 1

Ix - yl )-C£+2) -tdx

/1/11

* l(x)1 = t-

1

2

I(Y)(1/I(X

t-

<

t- CHI )

* l(x)l::::

((1 + 1~lrC2k+2)t-C2k+I)Y/2 + Ixl)-k,

SH then g is rapidly decreasing on the upper half space, i.e., Ig(x)1 :::: Ck(l + Ixl)-k, x E lR~. One expands the Poisson kernel P J as a Taylor series at 0 with remainder, scales to obtain an expansion of PI for t large, and uses (i). (iv) Hannonic gradients g E SH such that g =f. 0 on Q(2) are dense in SH. Choose finitely many ¢I ... ¢n E SH such that (¢j(x)} spans lR3 for each x E Q(2). This can be done by compactness since (x: ¢j(x) do not span} is closed and clearly SH(X) spans for each fixed x. If I E SH is given, then let 0 be small and consider :

Q(2)

~ ~<8j <~

f

(x)

+

Lemma 1.1. For each unit vector e E lR3 there is q E SH such that q(x)ldx < O. We can choose q so that e + q never vanishes on lR2 .

Iri

2

log Ie

+

= [

d/ dt d 2I dt 2 Taking

t

JlR2

= [

(e + tq, q) dx

Ie + tq 12

Iql21e

+ tql2 - 2(q, e + tq)2 dx. le+tql4

JIR2

= 0,

dI I = [ (e,q)dx = 0, dt 1=0 JlR2

E

1 J[

329

The first step in proving Theorems 1 and 2 is to distinguish the 3-dimensional case from the 2-dimensional case by means of the following fact.

~ Y) - p(X ~ Y))dY )

which suffices.

Yk3Ck

f~!Ii!illi

T

The inner integral is finite for each fixed x since the zero set of

.

:::: t- I / 2 (t

(iii) If g

rrh~fiij.hll~Mi

'.

Eo j ¢ j (x) has codimension 3, and is bounded by a compactness argument on the minors of the matrices (¢j(x)}. On the other hand, if f + EOj¢j has a zero for a certain 01, ... , On then f Q( 2) II + EOj¢jl-2dxJdx2 = 00. If this happened for all choices of the o's, Fubini's theorem would be contradicted.

= 2k we obtain 11/11

!nC:fiHI!.6C"iEll"~'

COUNTEREXAMPLES WlTH HARMON1C GRAD1ENTS 1N 1R3

'i-;

f I/(Y)II f If-I dy

;S '" Taking l

f

2

'

It is natural to try to prove Lemma 1.1 by perturbing off the case q = 0, and we now look where this leads. Fix q E SH and let I(t) = fIR 2 log Ie + tqldx. By calculations which are justified (for t small) by the rapid decay at 00,

.

On the other hand let p be the degree l - 2 Taylor polynomial of 1/1 at then

m

II + EO j ¢jl-2do J ... dOndxldx2.

2 d 2I

I

dt 1=0

=

i

(lql2 - 2(e, q)2)dx.

]R2

If e = e3 (or -e3) it follows, e.g., using the Fourier transfonn that ~~t 11=0 = 0 for any q E SH. But if e has a nonzero tangential component, then for similar reasons ~ 11=0 > 0 for any q E SH, i.e., 0 is a local minimum.

Proofof Lemma 1.1 when e = ±e3 We can assume e = e3. Differentiating once more, 3

d I I = -3 dt 1=0

i(

8q33 - 6q31ql 2) dx

lR2

= 2 [ (qj -

JlR 2

3q3(q~ + qi»)dx.

Temporarily drop the requirement that q -Ix + e31- 1• Then

q3

=

Ix

+ e31- 3•

qf

E SH

+ qi

and try q

xf + xi = Ix

+ e316

V¢, ¢(x)

I

330

CHAPTER 15 00

L2 (qj - 3q3(qr + q;) )dx

=

21 (r 2 +

0-9 / 2 - 3r 2(r 2 + 0-9 / 2 )rdr

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN 1R3

space, we obtain qn (i), J qn = O. Also

E

SH such that qn -+ q in L ~ and e + qn never vanishes. By

log Ie + qnl - (e, qn) ~ C1qnI

2Jr

= - - < O. 35

Approximating this q in L 3 nonn by one in SH we obtain q E SH with ~ /1=0 < O. Then for small positive t, fn~.2 log Ie + tq I < 0 and e + tq has no zeros.

331

Sinceqn -+ qinL 2 andlogle+ql-(e,q}

E

2

•

L1,wecanpasstoasubsequence

and assume that log Ie + qnl - (e, qn) - (log Ie + ql - (e, q}) ~ /,

Proof ofLemma 1.1 when e has nonzero tangential component We can assume e = (a, 0, .JI=(2),- 0 < a ~ 1. Again we temporarily drop the requirement that q E S. To emphasize the +el-direction it is natural to consider q = Y'/ where in polar coordinates on 1R2 , /(XI, X2, 0) = ¢J(r) cosO. Next observe that if ¢J(r) = -ar for r < I, then (e + q)1 = (e + qh = 0 when r < 1. Having defined ¢J this way for r < I we decide to extend it to 1R 2 keeping the L 2 nonn of q as small as possible. That leads to the choice q = a Y' g, where g is the hannonic

function with boundary values

g(Xb X2, 0) =

-r cosO,

ifr < I

{ -r- I cosO,

ifr > I

def

1

r

lim sup { (log Ie + qn I - (e, qn) )dx n->oo

JlR2

~

{ (log Ie + ql - {e, q} )dx < O.

JlR 2

Since J(e, qn) = ONn will satisfy the conditions of Lemma I when n is large enough.

•

Lemma 1.1 leads immediately to it refinement ofitself where instead ofa constant vector e we consider a constant vector plus a small error.

We then need to show

Lemma 1.2. 1(a) = 2JT JlR2(log Ie + ql - (e, q})dx < O/or each a

with / ELI. By Fatou's lemma,

E

(0, I].

The integral is proved absolutely convergent at 00 by expanding the logarithm to second order and using that q E L 2 . It is easily seen to be absolutely convergent locally, e.g., this follows from fonnulas (A 1) and (A2) and statements A3.1 and A3.5 in the appendix to this paper. Lemma 1.2 could in principle have been proved by computer work and error analysis, but I decided to treat it as a calculus exercise instead. This argument is messy and is therefore postponed to the appendix. It turns out that if one uses polar coordinates the O-integral can be evaluated explicitly in tenns of certain elliptic integrals (cf. Lemma AI.I and the subsequent remark). Jonathan Goodman offered to integrate the resulting function of r numerically, and found that 1 is decreasing in a with 1 -+ 0 as a -+ 0 and 1(1) ::::::: -0.28, so the reader can believe the lemma without going through the technicalities in the last three sections of the appendix. We now show how to obtain Lemma 1.1 from Lemma 1.2. If e = (a, 0, ~), andq isas in Lemma I, thenq E L 2 and (e + qh is everywhere nonnegative. So by first convolving q with a strictly positive approximate identity and then approximating the resulting harmonic gradient by one in the Schwarz

Lemma 1.3. There are finitely many harmonic gradients qj E SH and numbers K > 0, P > 0,77 > 0 making the/ollowing true. 1/ N is sufficiently large, then/or each v E 1R 3 there is j such that if g: Q(N) -+ 1R3 and IIg - vll oo < KN- 2 Ivl, then I

plgl ~ Ig + Ivlqjl ~ p-llgl

1

log Ig + Ivlqjl

Q(N)

~

-11 +

on

Q(N),

1

log Igl·

Q(N)

Proof. Lemma 1.1 and a compactness argument show there are finitely many qj and numbers p, 11 such that if e is any unit vector then for some j, 2p ~ Ie (1.1)

(

+ qjl

~ (2p)-1

log Ie + qj I

JlR 2

~

-311·

Replacing 377 by 277, the integral in (1.1) may be taken over Q(N) provided N is large enough. When v is a unit vector, the lemma follows by easy estimates with absolutely convergent integrals. The general case follows by properties of the logarithm function. • We now set up a recursive procedure with the basic step provided by Lemma 1.3, which will prove Theorem 1 and the following result:

332

CHAPTER 15

=

Lemma 1.4. Suppose p > 0 is small enough, and let k k p < 00 be large Claim. For any d ::: enough. Then for given E > 0 there is a harmonic gradient g which is continu­ -3

ous on lR+ and satisfies: Ig(x)1 < Elx\-k when Ixl > 1,

flR 2 g(X)XadXldx2 =

o for

all a = (aI, a2) with lal ~ k. SUPt fJR2 Ig(X. tWdxldx2 ~ C, UJR2Ig(X,OWdxldx2)I/p < E, and I{x E lR2 : Ig(X. a)1 ::: C-I}I ::: c- I fora

certain a

= a€ > O.

qf(x)

= qj(N8- n(x -

QEA. ly-aQI~d

L QEA.

Proofof(1.5) Sinceqj E SH we have Iqf
L

~ CA

-

QEA.

ly-aQI~d

~

CAN-484nlgn_lCY)I(

~

¥N- n + A(n log n)I/2, 2

with 1/ as in Lemma 1.3. And

= gn-I +

L

L

Iy - aQI-4 +

L

r(n-I)IY - aQI- 3 ).

ly-aQI~m8Jl(d.a·-I)

IVQlqf
QEA.

where vQ = gn-I (aQ) and j(vQ) is the index provided by Lemma 1.3. We now make some estimates and show in particular that (1.2), (1.3) are again satisfied.

Cr2nd-2,

ifd < 8n ­

1

C8- 2n r(n-l)d- 1 •

ifd::: 8n -

1

I

and (1.5) follows. For (1.6), use IVqf
= gn-l(Y) +

L

IVQlqf
y¢Q QEA.

Taking d (1.7)

gn

max(l, r(n-I)IY - aQDlgn-ICY)lly - aQI-4N- 48 4n

Since the sums are comparable to integrals, the expression in brackets is

Ign-I(Y) - gn-l(x)1 ~ BNr(n-I)lgn_l(x)llx - YI.

~

l

IVQIIVqf
ly-aQI~d

Ign-I (Y)! ~ A max(l, r(n-I)lx - yJ)lgn-l(x)l.

mgx 10g(lgn-i1/lgt!)

IVQllq~vQ/Y)1 ~ CAN-482n lgn_ 1CY)lr l max(r l , 8-(n-I»)

E SH

We will also have a subset An-I C G n- I, called nonstopped squares at stage n - 1. Take gl as above and take A 1 to be those squares in G 1 which do not touch the boundary of Q(l). Then (1.2), (1.3) are satisfied when n = 2, provided 8 is small enough. We now define An and gn' A square Q E G n belongs to An, provided every Q' E G n- 1 which contains or touches Q belongs to An-I and, furthennore, (1.4)

[0, 1],

a·- 1 >ly-aQI>d

If Ix - YI < 8n- l , then (1.3)

4on and Y E Q(2) x

aQ))

with qj as in Lemma 1.3. At stage n (n ::: 2) we will have constructed a harmonic gradient gn-I obeying the following estimates when x E Q(2), Y E Q(2) x [0, 1]. (1.2)

L

(1.5)

(1.6)

Here C is independent of E. The last two properties are the significant ones. Theorem 2 follows easily from Lemma 1.4 as we explain at the end of the section. Choose three large -enough constants A, B. N in that order. They depend on properties of the functions q j and will be kept fixed throughout the proofs. Next choose an initial harmonic gradient gl E SH with IgII :f= 0 on Q(2), and then a large number A and, finally, a small number 8 > 0 with 8- 1 E Z. Constants will initially be independent of A, B, N. gl, A. 8. Let G n be the grid on Q (1) consisting of squares with side 8n . For each Q E G n, let aQ be its center point. For each j, let

333

COUNTEREXAMPLES WfTH HARMON1C GRADfENTS fN JR3

= 48n and using (1.5), 4 18n(Y) - gn-l(y)1 ~ CAN- Ign_Iwl.

when Y E Q(2) x [0, 1]. We make N large enough that C AN-4 < ~ K N- 2 with C as in (1.7) and K as in Lemma 1.3. Now restrict temporarily to x = x E JR.2. If x E Q E An then jgn-I(X) - gn-l(aQ)1 ~ BN8Ign_l(aQ)1

,----------,-.--

.."

-,;:",,,,,,-,-,--=---.---.-:......!,.;-'----------,...

----------.------~".-----"-_--.-.--

~'";;--.,','i=_~..=,'~~~~,;;;;:~,.-------------.",-

334

CHAPTER 15

by (1.3),andwe take 8 small enough that BN8 < ~KN-2. Then x implies

.:s

Ign(x) - gn-l(aQ)1

+ Ign-I(x)

[gn(x) - gn-l(x)1

I

.:s 3

2

E Q E An

.:s ~KN-2(2[gn_l(aQ)1 + Ign-I(x)

- gn-l(aQ)J)

2 < KN- Ign_l(aQ)I.

Ign(y)1 < Co Ign-l(y)1

(1.8)

2p-

on Q(2). Moreover if Q

Ign-Ii :::

E An

Ign I

:::

If Ix - yl > 8n- I / 2, (1.2) for gn now follows from (1.2) for gn-I provided 8 is small enough. If Ix - yl .:s 8n- I / 2, (1.2) for gn follows from (1.10) by choosing 8 small enough, provided A > 2Co . To prove (1.3) for gn, write

.:s

Ign-I(x) - gn-l(y)1

-T/8

N-

2

..

+ ~ log [gn-II + CAN-6 82n

~ log Ignl .:s - ~ T/N- 282n + ~ log Ign-Ii,

provided N is large enough.

Ign-l(y)1

.:s

(I

.:s .:s .:s

+ BN8 1/ 2)/gn_l(x)1 Ign(Y) - gn(y)1 C1gn-l(aQ)1

C(I

if

Ix -

+ Ign(Y) -

I(gn - gn-I)(X) - (gn - gn-d(y)1

q~vQ)(Y)1

.:s

n CAN8- lx - yllgn-l(x)1

if N is large. If neither x nor y belongs to any Q E An' the same estimate is valid and slightly easier. For general x, y with Ix - y I < 8n , draw a straight line from x to y and then from y to y. That gives tE[O,I)

yl < 8n- I /2.

+ Ign(Y) -

and the right side is

gn-l(y)1

+ BN8 1/ 2)lgn_dy)1 + CAN- 4 Ign_ICy)I,

.:s C(lgn-ICY)I + Ign-I(Y)I)

.:s

Ign(x) - gn(y)1

gn-l(y)1

using (1.7), (UD), so if N is large and 8 small, Ign(y)1

-

l(gn-gn-l)(x)-(gn-gn-I)(y)l.:s CANo-nlx-yl max Ign-l(tX+(I-t)y)I,

Now ify E Q E An (the case where y ¢ U{Q: Q E An} is slightly easier), Ign(Y) - gn-l(y)1

Ign-l(aQ)llq~vQ)(x)

The first term is .:s C1gn_l(aQ)IN8- n lx - yl where C depends on Lipschitz bounds for the qj' The second term is .:s CAN- 4 Ix - yl8- n sUPQ Ign-d by (1.6) (with d = ~ 8n ) at the points of a straight line path from x to y and the mean value theorem. By (1.2) for gn-I,

Next we prove (1.2) for gn' Observe that (1.3) for gn-I implies (1.10)

.:s

+I(gn - gn-I)(X) - (gn - gn-I)(y)l·

by (1.7). So (1.9)

- gn-I)(X) - (gn - gn-I)(y) I·

The first term is .:s BN8-(n-1) Ix - yllgn-l(x)l. To bound the second suppose first that x and y belong to the same . Q E An. Then

2 plgn-II,

then

2n

+ I(gn

I

~ log Ign I .:s -T/8 2n N- 2 + ~ log Ign I .:s

+ Ign-l(y)1 .

Ign-l(x)1

Ign(x)1 -

J(gn - gn-d(x) - (gn - gn-d(y)1 I

335

and by (1.8),

Ign(x) - gn(y)1

Therefore, Lemma 1.3 applies to g = gn and qj = qj~vQ)' after scaling by N8-n, and we get p-llgn I ::: Ign I ::: plgn I on U(Q: Q E An). Using that gn = gn on Q(2)\ U {Q: Q E An}, (1.7) implies

r~-----

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN]R3

- gn-l(aQ)1

+ Ign-l(aQ)1)

KN - (Ign-l(x)1

..-;;----;:;..-- -----------··;;;----;;;-·,-·-----:"",""";,·';::,;....

CA 2N8- n lx - Yllgn_l(x))lby(1.2). So

.:s (BNo-(n-l) + CA 2N8- n)lx - yllgn-l(x)1 .:s 2p-I(BNo-(n-l) + CA 2No- n)lx - yllgn(x)I,

which proves (1.3) for gn provided B > 2Cp-1 A 2 and 8 <

~ pB- I (B ­

2Cp-1 A 2).

That means that the construction makes sense, and now we show that it con­ verges. From here on, A, B, N are fixed. Constants can depend on them. Let E = U{Q: Q E An}. Let r: Q(2) --+ Z+ U (00) be the stopping time in the construction: rex) = 00 if x E E and otherwise rex) = min(n: x ¢ Q E An). We use the usual notation gr(x) = gr(x)(x).

nn

336

CHAPTER 15

-3

Lemma 1.5. The gk converge uniformly on lR+ to a harmonic gradient g. This g is Holder continuous with g = 0 on E and C-llg~ I s Igl s C1g~ I on Q(2)\E. If a > 0, E > 0, k < 00 are given then the following estimates hold if 8 is small enough: Ig(x) - gl(x)1 < Elxl- k when Ixl > 1 and Ig(x) - gl(x)1 < E when x E Q(2) x [a, 1].

Proof. Fix x E Q(2)\E. If k 2: r(x) + 2 and Q E A k then Ix - aQ) 2: 8~(x)+1 by the rule about touching squares in the definition of Ak. By (1.5), with d = 8~+1 2: 8k - 1, Igk(X) - gk-l(X)1

s

C1gk_l(x)18k-~.

Calculus then shows Igk-l (x) I S 21gHI (x)j for k 2: r + 2 if 8 is small. By (1.8), Igk - gk-Ji s C8k-~lg~1 whenk 2: r + 1. This implies C-llg~1 S Igkl s Clg~1 whenk 2: r,andthat{gk(x)} converges uniformly on {x E Q(2): r(x) s n}for any given n. However if k S r(x) then the stopping rule (1.4) implies Igk(X)1 s Ce- ak (here a and C dependongl and),,), so it follows that {gd converge uniformly on Q(2), and clearly the limit satisfies g = 0 on E and C-llg~ I s Igj s C1g~ I on Q(2)\E. To estimate when Ixl > 1 note the preceding argument gives a bound IIgj IIL""(Q(2» S M where M = M(gl, ),,) is independent of 8 for 8 small enough. Using that qj E SH and that there are at most 8- 2j squares in A j, (1.11)

Igj(x) - gj-l(x)1 S C kr

337

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN]R3

Proof. Wedothet < 8 case, leaving the other to the reader. By (1.2), !gk(X, t)1 C1gk(x)l. By (1.5), (1.8), if j 2: k + 1 then Igj(X, t) - gj-I (X, t)1 S C1gj-l (X)I82 j -k-l-max(k+1.j-l) j S C1gj-l (X)18 - k­ 1 S C(

and the result follows if 8 <

28 )j-k-l P Igk(X)I,

t p.

•

We still have to show why g can be taken to have the special properties in Theorem 1 and Lemma 1.4. Basically, this is because of (1.9). Let r 1\ k min(r, k), and if Q is·a square, ~ a function, let Q(
f

Lemma 1.7. On Q(2), log Igkl Slog IgJi -

1

t N-2'7r 1\ k + h k where h k satisfies

ephk _< C 1eClnp2

xEQ(2) ; ~ I\k=n

for all n E Z+, p E (0,00). Proof. Let
t

2j (r j lxl)-kM,

for any given k if Ixl > 1. This implies uniform convergence on 1R 2 \Q(2) (hence on IR~) and also the estimate Ig - gJi < Elxl- k when Ixl > 1. The estimate Ig - gJi < E on Q(2) x [a, 1] also follows in this way; replace Ixl by a on the right side of (1.11). We prove the Holder continuity on Q(2) only since on 1R2 \Q(2) it follows easily from (1.6). In this argument, constants depend on all parameters including 8. By the stopping rule (1.4) we know Igki S for some '7 < 1, when k < r, and therefore Igi S C'7~. Also (1.3) together with boundedness of the gk implies IV gk I s C 8- k. If j 2: r (x) + 2, then using (1.6) and the rule about touching squares, IV(gj - gj_l)(x) I s C8j-2~(x). Taking k = r(x) + 1 and summing a series gives IVgl S C8-~, i.e., IVgl S C1gl-W when g ::j:. 0, where w = (log 8) (log '7)-1. This estimate implies g is Holder with • exponent (1 + w)-I.

c,l

Q(eP!n)

= ePQUn-l) Q(eP(Vrn+fn-1-Q(!n-l») S epQUn-lleCp2 S ecp2 Q(eP!n-,)

{ Lemma 1.6. Fix t E (0, 1]. If t > 8, then Ig(X, t)1 S C(lgl (x, 01 + Igl (x)l)for all x E Q(2). If t < 8, then Ig(X, t)1 S C1gkWI where k is the largest number with 8k 2: t.

s

]u/Q:

eP!n < ecp2 { QEAnl

]u/Q;

eP!n-1 QEAnl

< e cp2 { ]U(Q: QEAn-d

eP!n-l.

t

338

CHAPTERi5

1

eP!,

~ eC(n-l)p 2.

U[Q: QEA.}

Now we have, using Lemma 1.5, log Igkl ~ log IgrAkl

+

C2

L

= log Igil +

¢>j

+

C2

2::oJ~rAk

~ log Igil -

1

2" N-

2

+

Tl(. 1\ k - 1)

frAk

+ C2.

The set {x: • 1\ k = nl is contained in U{Q: Q E An-d, so the lemma follows from (1.12) if we take h k = frAk + 4 N- 2Tl + C2. •

Proof of Theorem 1 Fix gl E SH with gl # 0 on Q(2). We claim that lEI> 0 if ).. is large and 0 small. For n 2: 2, let Fn be the union of all squares such that Q C Q' E An-I and (1.4) fails on Q at stage n. Then. 1\ n = non Fn. Also, if Q is one of the squares comprising Fn then h n- 1 2: )..(n log n)1/2 somewhere on Q by the fact that (1.4) fails. So h n- I 2: )..(n log n)1/2 - C everywhere on Q by (1.3), and then by (1.8), h n 2: )..(n log n)1/2 - Cion Q. SO h n 2: 4 )..(n log n)1/2 on Fn provided)" > 2CI(2 log 2) -1/ 2 . Lemma 1.7 implies

(iv). • Let no be a large enough integer. By taking ).. very large we can guarantee that if .(x) ~ no then x E Q(2)\Q(1) or x belongs to the double of one of the squares in G I \Al. In either case, Ixl 2: Let a > 0 be such that I{x E Q(1): IgI(X, a)1 > C-I}I > C- I and make 0 smallenoughthatthe last statement of Lemma 1.5 is valid with this a. If g is as produced by the construction, then I{x: Ig(x,a)1 > 4C-lll > 4C-I. Moreover,

!.

J

Q(2)

•

IglP r(x)=n IgIIP

f

Accordingly 1; IFn I may be made arbitrarily small by taking).. large. If x E Q (1) \ E, then x belongs either to the double of one of the squares used to define one of the Fn , or to the double of one of the squares in G 1/ A h so IQI\EI

~

f

+

+

f

Ig/P L IgII P ­ p n>no r(x)=n Igil

f

C L IglP n>no r(x)=n IgIIP

By Lemma 1.7,

-_IA2 16CI

IglP IgII P - p n~no r(x)=n Igil

< lOP L IglP n~no r(x)=n IgIIP

= (4C I )-I)..n- 1/2 (1ogn)1/2 gives IF,n I ~ C In

f

Igl P = L

IF,n I <_ C2 eC2p2n- 4Ap(n log n)I/2 , for any p. Taking p

339

lal ~ e, and with I{x: 1¢>3(X)1 > Clill > Cll. This is simply the existence of compact support functions with many vanishing moments. Let 1/1 be a Schwarz space approximation to ¢> obtained by convolution as in (ii). We now modify 1/1 so as to obtain the first mentioned property of gl. For a given E, let t be small enough and hex) = L z 1/1(t- I x - z) where the sum is over lattice points z E 71} with Iz I < (8t) -I. The rapid decay of 1/1 implies that Ih I ~ C, and (1 + Ix I)k 11/13 (x) I may be made arbitrarily small when Ixl > ~ as described in (ii). So there is no significant cancellation among the different functions 1/13 (t -I X - z) and it follows thatl{x: Ih(x)l2: 4Cll}12: 4Cll. Iflxl2: !,thenlt-Ix-zl > 4t- I lx lfor all z, so Ih(x)1 ~ C3t-2( 4 t-Ilx 1)-(k+2) which is < E(4Ixl)-k if t is small enough. To obtain gl, approximate h by·a harmonic gradient without zeros on Q(2) using

Iterating n - 1 times, (1.12)

COUNTEREXAMPLES WiTH HARMONIC GRADIENTS IN]R3

o<

•

and the proof is finished.

Proof ofLemma 1.4 We need p < (2C I) -I N- 2 Tl with C I as in Lemma 1.7. Let k be large enough and fix E > O. We need gl to satisfy Igil < lO(4Ixl)-k when Ixl > JJR2 xagt(x) = 0 when lal ~ k,lgll ~ C and I{x E Q(1): IgI(x) > C- I II > C- 1• C as well as the constants appearing below may depend on k, but not on lO. Such a gl is easily constructed using "atoms": let e be large and fix a andJ xa¢>dx = owhen smoothharmonicgradient¢>with¢>Jsupportedinlxl <

!,

!

= Cle- fJn , where

4(LlFnl +40), n:;:2

~ Cle-pn~/2N2 eCIP2n

fi

> 0 since p < Tl/2CIN2. SO J Q(2) IglP < CEP if no is large enough. If k as in Lemma 1.6,

t ~ I, then choosing

J

Q(2)

Ig(X, t)IP

~C+C

J

+C

J

< C -

Q(2)

Igk(X)IP

Igk(X)IP Q(2) IgI(x)IP

~ C +C +CL n

~

C.

e­ fJn

340

CHAPTER 15

The next to last line here follows from Lemma 1.7 by conditioning on 'r /\ k. Estimates when Ix I > I follow from Lemma 1.5 if 0 is small enough, and the vanishing moment condition is built into the construction. •

Proof of Theorem 2 This is a very standard argument; probably it is well known that Lemma 1.4 implies Theorem 2. It g is a harmonic gradient, then we call the function whose gradient is g the primitive of g and denote it Ug' The example will have the following additional properties which willpe used in extending to higher dimensions: ug is continuous -3

f

on ~+ and C I for large Ix I, and for suitable l, IR 2 x a Ug (x) g and Ug vanish at 00 faster than Ix I- t .

= 0 when laI ~

Fix k large enough. Let gE be the function in Lemma 1.4. Note that ME ~ ~ CElxl l - k forlarge lxi, and IgE(X, tWdxldX2) lip < E

unless

The intervals ( :;;, ]

t E ('rE, 2),

2 N ) ]

341

are disjoint ' " __ ME·I < 00. L.J Nj

(1.14)

These are obtained recursively by choosing Ej < B-j and then N j > j max(2 ME]' 2Nj_1 'rE~~I)' Let g = Lj gE],Nr Property (1.14) implies that the series converges unifonnly on compact sets in lR.~ and the limit has a primitive which extends continuously to R~. For any t,

f

l and

lIu g,lloo and IUg,1

(L2

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN 1R3

Ig(X,!W

~L ~

f

IgEj,Nj(X, tW

C,

since at most one tenn in the sum is ~ B-jp. Moreover,ift fj. Uj('rEj N j- I , 2Nj l), then

f

Ig(X, tW <

L B-jp,

with 'rE > O. Let

L gE(Nx -

gE,N(X) =

which can be made arbitrarily small by taking B large. In particular, we can guarantee that for such t

m)

m

1 1 I{x: Ig(X,t)1 > '3 C;I}1 < '3C31,

!

where m runs over all points (2z, 0) with Z E ~} and Izi ~ N. Consider Lm gE(Nx - m) where we now restrict to points m as above with INx - ml > 1. In view of the estimate IgE(X)1 ~ Elxl- k when Ixl > 1, this sum will be ~ CE for all x and in fact ~ CEd-(k-2), where d = d(x) = minm(INx - ml: INx - ml > 1). It now follows that Ifx: IgE,N(X,

(1.13)

with U E =

U

f

~)I

m

f

lIu ,Nlloo

2

Hx: Ig(X, NjIUE)1 > '3C31}1 > '3 C31 . !gE(Nx -

m,

NtW dx l dx2

which is ~ C for all t and ~ CEP if t fj. (~, ~). Let UE,N be the primitive of gE,N. Since U E vanishes rapidly at 00, an argument like the one for (1.12) bounds lIu E ,N 1100; due to the different scaling we obtain E

1 1 Ifx: Ig(X,t) -gE],Nj(X,t)1 > '3C31}1 < '3C31,

2

~L

E ('rEjNjl, 2N j l),

and, therefore,

> C;I}I > C;I.

as in Lemma 1.4. Also, !gE,N(X, tW dx l dx2

with C3 as in (1.13). In the same way, we guarantee that for t

~ CMdN,

Now let {Ej}'f and fNj}'f converge rapidly enough to 0 and Specifically, let B be a large constant. We require Ej < B-j.

00

respectively.

It follows that there is a set of x with measure at least ~ C 3 1 on which lim SUPHO Ig(X, t)1 > ~ C;I and lim inf Ho Ig(X, t)1 < ~ C;I, so Theorem 2 is proved. From the construction, U g has as many zero moments as we like, and it is easily checked that it is C I for large Ixl and that g and u g die faster than a given power of lxi-I. • The extension to higher dimensions is achieved by adding dummy variables. For Theorem 1 this may be done in a trivial way, but for Theorem 2 an argument is required. Denote variables in i:(n ~ 4) by z = (x, y, t), x E lR. 2 , Y E lR.n -3, t ~ O. Let y,: lR. n - 3 ~ lR. be a smooth function with y,(y) = 1 when IYI ~ 1, y,(y) = 0 when Iyl ~ 2, and with sufficiently many zero moments.

342

CHAPTER 15

Let u = u g where g is as in Theorem 2, and let F be the harmonic extension of 1{r(y)u(x, 0) to ~~. By the reflection principle, F(x, y, t) - u(x, t) extends smoothly across the set{(x, y, 0): Iy) S I} C ~n-l, so V F doesn't have radial limits almost everywhere. We will show that V F is in the Stein-Weiss H P space. 1 Since ¢(y)u(x, 0) is C for large Ixl + IYI and rapidly vanishing at 00 and has lots of zero moments, it is enough to show

r

sup

IV F(x, y, tWdxdy <

00

0
for fixed R. Fix Yo and write 1{r(y)u(x,O)

= q(y)u(x, 0) + (1{r(y) -

q(y»u(x, 0)

where q(y) = 1{r(yo) + (V1{r(yo), y - Yo). The function Gyo(x, y, t)

q(y)u(x, t) is harmonic (compute its Laplacian) and

1

Ixl
IVGYo(x, Yo, tWdx S

~3 7rR3I1ull~IV1{r(yoW + 11{r(yoW

1

lu(x, tWdx S C

Ixl
uniformly in Yo· Let Hyo(x, y, t) be the harmonic extension of (1{r(y) q(y»u(x, 0). Since 1(1{r(y) - q(y»u(x, 0)1 s Cjy - YOl2 locally and S k CjYllxl- at 00, the formUla for the derivatives of the Poisson kemel applies and shows that IV' Hyo(x, Yo, t)1 is bounded. We conclude

r

IVF(x,y,tWdxdy

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN]R3

show that for scale invariant boundaries the dimension of the harmonic measure should be determined by the quantity

r

lim ~ IVgnlloglVgnlda n-->oo n Jan'

(2.1)

where nn is a natural smooth (enough) approximation to our domain n and gn is its Green's function with fixed pole p. (We normalize Green's function to be positive and satisfy IV g I = ~~ where (J is surface measure and w is harmonic measure at p.) In particular, in ~3 the sign of (2.1) determines whether w is > 2­ or < 2-dimensional. In contrast to the planar case considered in [6], the sign of (2.1) may be chosen freely by adjusting the parameters of our construction. The situation is similar to section I in,that once we know how to "improve a constant" in a suitable se'hse, we can set up an iterative procedure to construct fractill boundaries with control over (2.1). Now the role of a constant is played by the upper half space. Suppose ¢: ~2 ~ lR is a Lipschitz function with supp ¢ C Q(l). Let U = {x E ~3: X3 > ¢(X)}. Then there is a unique harmonic function goo: U ~ ~+ ("Green's function with pole at 00") such that goo vanishes on au and limx --> 00 (goo (x) - X3) = O. One shows that for x E lR 2 , IVg(x)1 = I + O(lxl- 3 ) as Ixl ~ 00. This is done as follows: letwbeharmonic measure for the upper half-space of a disc centered at the origin and containing supp ¢. Then using the maximum principle, X3 -

Cw S goo S

sf <

I(¢)

[IVG,(x, y, IW

+ IVB,(x, y, IW]dX) dy

Cdy

00.

X3

+ Cw,

x

E

U

n lR~

for suitable C. And IVwl is O(/xl- 3) at 00 when x E ~2 by the Poisson integral formula, so the statement follows from l'Hopital's rule. The integral

Jrnax(lxl,IYI)
"1(1

343

=

r IVgoollog

Jau

IVgoolda

is then easily (given [7]) seen to be absolutely convergent. At the end of the section we will find piecewise linear functions ¢ with support in Q (l) and arbitrarily small Lipschitz norm, such that I(¢) has either sign. For now, fix any piecewise linear ¢ with supp ¢ C Q(l). Next fix b > 0 and small. If Q is a square (Le., a square in some plane in ~3) with center point aQ and given unit normal e, define "pyramids" PQ and PQ by

2 DIMENSION OF HARMONIC MEASURES

PQ

= cch(Q U (aQ + bf(Q)e})

The purpose of this section is to prove Theorem 3. We will do the main construction in somewhat greater generality than is needed for this since it doesn't require any extra effort and tends to show that there is no reason at all for a harmonic measure in ~3 to be 2-dimensional. As explained in the introduction, ideas of Carleson [6]

PQ

= int cch(Q U (aQ - bf(Q)e})

where cch denotes the closed convex hull. Let N be a large number. In particular, N should be large enough that (x : X3 = N-1¢(Nx)} is well inside PQ (1) U PQ (1). Constants will be independent of N

344

CHAPTER 15

provided N is large enough. Define (with e = -e3) A = {x

E

PQ(l) U PQ(l) : X3 > N-I¢>(Nx)}

= {x

E

lR3 : x

a

E

Q(l), X3

Then a· C PQ(I) U P Q(1). a is a polygon with lots of faces. Fix a Whitney decomposition W of each face into squares Q with side g-k, k = 1, 2, .... Choose a distinguished edge for each square. These choices may be made in an arbitrary way, but once made, they are kept fixed. The following is easy to see if N is large enough.

= g-k. Andfor

each k ::: 1, U{Q E W: f(Q)

::s

345

directly descended from Qj+1 for j = 0, ... , n - 1. If Q' is descended from Q we write Q' -< Q. We will write m(U, Y, z) for the harmonic measure of the set Y (Le., Y n aU), relative to the domain U, evaluated at z E U. Let w= wen, " 0).

= N-I¢>(NX)}

Lemma 2.1. There are at most Cg k squares Q E W with f(Q)

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN R 3

g-k} c U{D(aQ, Cg- k ): Q E W, f(Q) = g-k}.

Suppose next that n is a domain and Q c an is a square with P Q n n = 0, PQ c n, and with a distinguished edge y. Let e be the normalinto P Q. Form a new domain n as follows: Let T be the affine map with T(Q(l)) = Q, T(O) = aQ, T(-e3) = e, and T({~} x [-~'~] x {OJ) = y. Let A Q = T(A) and aQ = T(a). Then n n (PQ U PQ) = A Q and n\(PQ U PQ) = n\(PQ U PQ). We call this construction "adding a blip to n along Q." Transferring W by T, ann (PQ U PQ) = aQ has a natural decomposition into squares with distinguished edges together with a set of a-finite length. The domains we want are obtained by iterating this procedure. Let no be the unit cube. If nn-I has been constructed, then its boundary will be given as ann-I = En-I U (UQ.,_, Q) where En-I has a-finite length and each Q E Q,,_I is a square with a distinguished edge and, with PQ n nn-I = 0, PQ C nn-I, and, moreover, if QI, Q2 E Q,,-1' then intPQ, n intPQ2 = 0 and PQ, n PQ2 = 0. To

form nn, add a blip along each Q E Q,,_I'

Thenann = En-IU(UQEG aQ)isdecomposedasEnU(UQEG Q')where ---II-I ---n-I the Q' are obtained from the preceding decomposition of the aQ • If Q E Q,,_I' Q' E Q" and Q' C aQ then we say Q' is directly descended from Q. If N is large enough then it is clear that for Q' directly descended from Q, PQ' U PQ, C PQ U PQ and in fact by properties of the Whitney decomposition, dist(PQ, U PQ" (PQ U PQ)C) ::: C-If(Q'). Moreover, PQ, C A , PQ, C A Q, Q and if Q'I' Q; are directly descended from the same Q then PQ, n PQ' = 0, - , 2 P Q; n PQ; = 0. So the induction hypothesis is satisfied and the construction can continue. Let n = limn->oo nn. Also let G = U:I Q". If Q, Q' E G we say Q' is descended from Q in n stages if there are Qo = Q', QIo ... , Qn = Q with Qj

Main Lemma. Suppose ¢> has sufficiently small Lipschitz norm and I(¢» =f:. O. Then

if N

is large enough there is a number d such that

lim r->O

10gw(D(x, r))

log r

=d

for a.e.

(dw) x E an

where D(x, r) is the intersection of an with a ball of radius r, centered at x. Thus w puts full mass on some set with dimension d and no mass on any set with dimension less than d. If I(¢» < 0, then d > 2, and if I(¢» > 0, then d < 2.

We will prove only the I(¢» < 0 case explicitly but we have set up the lemmas so that the I(¢» > 0 case follows by changing the notation. Certain estimates of harmonic measure will have to be made. Fortunately the whole construction takes place within the class of NTA (nontangentially acces­ sible) domains of Jerison and Kenig [9] and most of the necessary estimates are proved in [9]. Let n be a bounded domain. If diam n = 1, then n is an NTA domain with NTA constant A if it has the following properties: • For each x E an, r < A -I, there are points Ar(x) E n, Br(x) E n c with IAr(x) - xl ::s Ar, IBr(x) - xl ::s Ar, and dist(Ar(x), an) ::: A-Ir, dist(Br(x), an) ::: A-Ir. Existence of the points Ar(x) (resp. Br(x)) is called the interior (exterior) corkscrew condition. • If x, yEn then there is a path y from x to y with length(y) ::s A Ix - y I and dist(y(t), an) ::: A-I min(ly(t) - xl. Iy(t) - yl). This is called the Harnack chain condition. In general n is an NTA domain with NTA constant A if a dilation of it with diameter 1 is such. This definition is easily seen to be equivalent to (say) the one in [11]. If n is an NTA domain, t > 0, then we say n is Lipschitz on scale t with Lipschitz constant M if D(x, t) n an is the graph of a function with Lipschitz norm ::s M for each x E an. Now suppose r c G is closed under the descent relation, i.e., Q E r and Q -< Q' imply Q' E r. Then we can form a domain n r by adding blips only along squares in r: n~ = no, n~+1 is obtained from n~ by adding blips along squares Q Ern Q" -it is clear by induction that such squares are contained in an~ and satisfy PQ c n~, PQ n n~ = 0--and n r = limn->oo nr'

___________ ._ .. _ . _ _

.

_

~

~:

346

CHAPTER 15

Lemma 2.2. n r is an NTA domain with NTA constant independent of Nand r. If r contains no squares with length < t then n r is Lipschitz on scale C- 1t with C and the Lipschitz constant independent of N, t, and r. Proof We can assume r is finite. understood to belong to r.

Squares denoted Q, Q', Q, etc., will be

For each Q, define VQ = A Q\ U {PQ': Q' directly descended from Q} and Q W = (PQ U PQ)\(A Q U U{PQ,: Q' directly descended from Q}). Then VQ and WQ are Lipschitz domains provided b is small enough, and V C n , r Q WQ n n r = 0. Moreover, VQ n VQ, = 0 if Q :f:. Q', and n r = S U (U V ) Q Q where S = no\ U {PQ : Q E f1J n r} is a "trivial" set. For each Q, let b be Q a point of VQ with dist(b Q, aVQ) ~ diam VQ ~ i(Q). There are three things to check in order to know that n r is an NTA. Exterior corkscrew condition. Fix x E an r , 0 < r < 1. We will ignore the easy case where x E ~ E f1J \r. Jiggling x slightly to avoid edges, it follows that x E aQ for some Q E r. Let Qx be a smallest such square. Let Q be a smallest square such that Qx -< Q and i(Q) 2: r. Since x E P Q U P it Q

follows that dist(x, W Q) S Cr. Let z be a closest point to x in awQ. Since W

Q is Lipschitz with diameter 2: r there is Br(x) E WQ with IBr(x) - zl S Cr anddist(Br(x). aWQ) 2: C-1r. Then IBr(x) - xl S Cr anddist(Br(x), nr) 2: dist(Br(x), aWQ) 2: C-1r.

~.--

.•

~

""-

~

.-"0_

-, .. _-_.

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN 1R3

Suppose x

E

anr\S, Qx is the square with x

E VQ,'

347

and Qx -< Q.

Claim. There is a path Y from x to b Q with length(y) dist(y(t), anr) 2: C-1Iy(t) - xl·

S

Ci(Q) and

Proof. Let Qx = Qo -< QI -< ... -< Qn = Q with Qj directly descended from Qj+l. Using the Lipschitz domains VQo and VQjul U VQj we can find paths Yo from x to b Qo and Yj from b Qj _1 to b Qj (j = I, .... n) such that length(Yo) S f:i(Qo), dist(Yo(t), anr) 2:

c- I min(IYo(t) -

xl. IYo(t) - bQol)

length(Yj) S Ci(Qj), dist(Yj(t), anr) 2:

c- I min(IYj(t) -

bQj-L I, IYj(t) - b Qj I).

Now Ib Qj -xl S diam(PQj U P Q) S C dist(bQj' aVQj ) S C dist(bQj' anr) for any j. It follows that Iy - x I S Iy - b Qj I + C dist(b Qj • an r ) for all Y E n r , and, therefore, that dist(Yj(t), anr) 2: C-1IYj(t) - xl. Let Y be Yo followed by YI followed by ... followed by Yn. Then dist(y(t), an r ) 2: C-1Iy(t) - xl and _ length(y) .::: C Lj i(Qj) .::: Ci(Q), since i(Qj) S i(Qj+l).

k

Now fix x, y

E

n r . We assume neither of them belongs to S. So there are

Qx,Qywithx E VQ"y E VQy.LetQbetheminimalsquaresuchthatQx ~ Q

Interior corkscrew condition. Choose Qx and Q as before, let z be a closest point 2: C-1r.

o)

and Qy ~ Q.

toxinaVQ,andletAr(x)satisfYIAr(x)-zl S Cranddist(Ar(x), V

Harnack chain condition. It is here that we use that 1I¢IILip(1) is sufficiently small.

We use this assumption only to assert that domains (2.2) below are Lipschitz and

therefore NTA.

We show first that if Q, Qjil, i = I, 2, j = I, 2, are squares with Q(1) directly descended from Q and Qj2) directly descended from Qjl) then } 2

(2.2)

VQ U VQ(I) U VQ(2) J

J

and

VQ U U(VQ(I) U V (2)) QJ j=1 J

are LipSChitz domains. Consider, for example, the first domain. It is obtained from PQ by adding a blip, adding two more blips, then deleting certain pyramids all of whose bases make a small angle with the plane containing Q (provided 1I¢IILip(1) is small). resulting (if b is small) in a figure whose boundary is aPQ\ Q together with a Lipschitz graph over Q.

Case 1. Each of Qx, Q y is either equal to Q or descended from Q in at most two stages. Then we simply use the Lipschitz domains (2.2). Case 2. One of Qx or Qy is descended from Q in more than two stages but not the other. We suppose Qx is descended from Q in more than two stages. Let Q satisfy Qx -< Q and Q is descended from Q in exactly two stages. Then Ix - b Q I S

C Ix - y I. To see this, let Q' satisfy Q -< Q' -< Q. Then Ix - y I 2: dist(x, (PQ' U PQn 2: dist(PQ U PQ' (PQ' U pQ,n 2: C-1i(Q) 2: C-11x - bQI, proving the assertion. Now connect x to b Q by a path Yl as in the claim. The length of this path is S C1x - y I. Connect b Q to y by a path Y2 with length(Y2) S C1b Q - y I and dist(Y2(t), anr) 2: c- I min(IY2(t) - bQI, 1Y2(t) - yl) using the Lipschitz domains (2.2) associated to Q. Since Ib Q - yl .::: Ix - yl + Ib Q - xl .::: C1x - yl,

348

CHAPTER 15

we have dist(Y2(t), aQ r ) :::: C- I min(IYj(t) - YI, !Yj(t) - xl) and length(Yj) C/x - YI. Now let Y be YI followed by Y2.

:s

Case 3. Both Qx and Qy are descended from Q in more than two stages.

Choose Q~I), Q~I) so that Q~l), Q~l) are descended from Q in exactly two stages and Qx -< Q~I), Qy -< Q~I) Let bx = bQ~I), by = bQ~l). It follows as before that Ix - bxl :s Clx - YI, IY - byl :s C1x - yi. Use the claim to connect x to bx by YI and Y to by by Y3, and (2.2) to connect bx to by by Y2. Then YI followed by Y2 followed by Y3 is the desired path. To prove the last statement of the lemma, let:E Q = A Q6.PQ (6. = symmetric difference)-the change in Q r made in adding the blip along Q. Note dist(:E Q , (PQUPQy):::: C-If(Q).lfx E aQr,letQ E rbeminimalsubjectto D(x, C-It) n:E Q =j:. 0. Then D(x, C-It) C PQ U PQ. By minimality it follows that D(x, C-It) n aQ r = D(x, C-It) n aQ, which is a Lipschitz graph. • We now give some general estimates for hannonic measures. We will assume n that Q C IR is NTA and a basepoint pEn has been fixed; a lower bound for (diam n)-I dist(p, aQ) is incorporated into the NTA constant A. All constants C, Ct, etc., depend only on A. The basic result proved in [9], Lemma 4.10, and Theorem 7.9 is the "rate theorem": if u and v are positive harmonic functions on n, x E an, r > 0 and u and v vanish continuously on D(x, r) n aQ, then on D(x, ~), ; :s C ~i1;g;;. Here D(x, r) denotes a ball of radius r, centered at x. Moreover; extends to a HOlder continuous function on n n D(x, ~) and satisfies the following estimate when Y, z E an n D(x,

0

(2.3)

I

u(y) v(y)

u(z) v(z)

I < C( Iy -

ZI)/3 u(Ar(x)) , r v(Ar(x))

f3

>

f?

if

°

< r < R, x

E

Q then

Wj

=

w(Qj, " Pj). Suppose E C aQI n aQ 2 satisfies D(x, Cilr) n aQI C :s rand dist(E, aQI6.aQ2) :::: Cilr. Suppose Y C E. Then

E, diam E ~ < C ",,(Y)

-

W7.(E) "'1(E) .

Proof. The assumptions imply that E is contained in the union of a bounded number of discs of radius Cilr whose doubles do not intersect aQI6.aQ2. The statement now follows from the rate theorem, Lemma 4.8 of [9], and the doubling property. •

i

•

Lemma 2.5. Let Q I and Q 2 be two NTA's with a common base point p. Suppose theHausdorjfdistancebetweenaQlandaQ2is:s Cit, and that x E aQ I naQ 2 with D(x, t) n ani = D(x, t) n aQ2. Then w2(D(x, t)) ~ wI(D(x, t)) where the constants depend on A and C I.

Proof. We will show w2(D(x, t)) :s CWI(D(x, t)). By the doubling property we can assume D(x, 2t) n aQ I = D(x, 2t) n aQ2. Let D = D(x, t), and let D j = D(x, 2 j t), j = 0, 1,2, .... We have (2.4)

r

w2(D) - WI (D):s

lnlnan 1

W(Q2, D, {)dwi (0­

To see (2.4), regard the function w(Q 2 , D, z) as extended to Q I U Q 2 by setting it to zero on QI \Q2. Then W(Q2, D, z) - W(QI, D, z) is subhannonic on Q I and zero on aQ I \Q2 and (2.4) follows. Let k~ be the kernel function d"'C;:~"{) • If { E (D j + 1\D j ) n Q 2 n aQI, then, estimating k~ by [9], Lemma 4.14, we have

max k~(Y) YEDnan2

w2(D) (dist({, aQ2))a . , W2(D j + 1) 2f t


-

< -

cr ja

Ct>O

w2(D) w2(D j +l)

by the assumption about the Hausdorff distance. So, W2(D) - wI(D)

Proof. If we required only Ct > 0 this would follow from the doubling property, as does a corresponding inequality in the opposite direction. To get Ct > n - 2, normalize Q so that diam Q = l. By "localization" ([9], Lemma 4.11; see also [lID we can assume R = l. Lemma 4.1 of [9] applied to the Green's function g of Q with pole at p on the domain n/ D(O, C-I) gives g(Ar(x)) :s Cr/3 with f3 > O. Lemma 4.8 of [9] then implies w(D(x, r)).::: Cr/3+n - 2 = Cra. •

349

Lemma 2.4. Let Q 1 and Q2 be two NTA's with base points PI. P2, and

w(n 2 , D, {) :s w2(D)

O.

Other results that will be used, such as the doubling property of harmonic mea­ sure, are in Section 4 of [9]. We also need some further estimates which are not in [9] but follow easily from the techniques used there.

Lemma 2.3. There is Ct > n - 2 such that w(D(x, r)) .::: C( )aw(D(x, R)).

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN 1R3

:s

C

L f

.

wI(Dj+1 n Q2) w2(Dj +d

r

ja W2(D).

We make two estimates on WI (Dj+1 n Q2). By [9], Lemma 4.2 we know w(n 2, Dj+2' {) :::: C- I if { E Dj+1 n ani n Q2. It follows by the maximum principle and doubling property that WI (Dj+1

n Q2)

:s

CW2(Dj+2)

:s

CW2(Dj+I).

====-~-:-~-=..-=-="::=-~~::..-==-==-~-=~,:::-::_.=::::::::::.~=-~-=-=~,~-=="=-"==~=:='--

350

CHAPTER 15

On the other hand WI (Dj+l) ~ Cj+IWz.(Dj+d :~~i by the doubling property. So for any jo < 00,

Wz.(D) - wI(D) ~ C LTjotWz.(D) j>jo

The lemma follows by taking jo large enough that C Lj> jo 2- jot < 1.

•

Suppose next that n is also Lipschitz on scale t. Then ([7]; see also [10]) harmonic measure (with basepoint p) is absolutely continuous to surface measure and the density ~~ is·the normal derivative of Green's function. Moreover IVgl satisfies an A2 condition with respect to w on discs of radius ~ M t for any M < 00, i.e.,

rn~1

(2.5)

1

D(x,r)

IVgl 2da < ­

c(

w(D(x, r)))2 r,,-I

if r ~ Mt, where C depends on the NTA constant A, the Lipschitz constant and M, Actually only the case t = diam n is stated in [7], [10] but the localized version follows using Lemma 4.11 of [9].

Lemma 2.6. Let n be NTA and Lipschitz on scale t. Let x let).., > 0 satisfy C

w(D(x, r)) w(D(x, r)) I rn-I -<).., -< C I r,,-I

E

an, r

~

Mt, and

-I

.

Then (

JD(X,r)nan

Ilog()..,-IIVgl)l

IVgl w(D(x, r))

a blip along Q, and let g and g be the Green'sfunctions with pole at p, wand W the harmonic measures at p. Then {_ IVg\log I'Vglda

j::;jo

da~C.

Proof. This is a formal consequence of (2.5). One can quote the fact that the logarithm of an A2 weight is BMO, or prove it using Jensen's inequality and • boundedness of the function x log_ x.

Lemma 2.7. Suppose n is NTA and Lipschitz on scale t. Let Q c an be a square with f(Q) ~ Mt, and with PQ n n = 0, PQ c n. Let be the result of adding

n

~

2 ( IVgllog IVglda - TJN- w(Q) Jan

Jan provided N > No is large enough. No and TJ depend on the Lipschitz and NTA constants and on M, and of course on 4>. Proof. We will use [9] as a reference for estimates although often only the smooth or Lipschitz case is needed. We scale so that Q = Q(l) and PQ C JR~. Let I q = be3 and let goo be the Green's function of {x: X3 > N- 4>(Nx)} with pole

!

at 00. I lt is easy to see that w(D(O, t)) ~ w(D(O, t)) for all t > N- . (If we want, we can use Lemma 2.5 and the doubling property for this.) Likewise I 2 fO(o,l) IVgoo Ida ~ t for all t > N- . Alsog(q) ~ w(Q(1))by[9],Lemma4.8,soby[9],Lemma4.4,g ~ w(Q(l)) on PQ(l). By the reflection principle and Hopf lemma, IVgl ~ w(Q(l)) (2.6)

on

1

Q( -)

2

IVg(x) - Vg(y)1 ~ Cw(Q(l))\x - yl

if

x, y

E

1 Q( 2:)'

Similarly goo (q) ~ 1. Let m = (Cd N)e3 where C I is a constant which is > 2 II 4> II Lip(l), so that dist(m, an) ~ ~' Theng(m)-g(m) ~ CNw((D(O, ~)) by [9], Lemma 4.8. So by [9] Lemma 4.4, we have g - g ~ Cw(D(O, ))gm on n\D(O, where gm is the Green's function ofn with pole atm. So IV(g - g) I ~ Cw(D(O, ))\Vgm! on an; D(O, ~) and by [9] Lemma 4.14, it follows that (for

*

i)

*

some fJ > 0) IV(g - g)1 on n\D(O,

The next lemma is one of the main steps in the proof. We are assuming 1(4)) < 0, i.e., adding a blip to a half-space decreases the value of f IVgoollog IVgool, and we want to conclude that adding a blip to an NTA-domain decreases the value of f IV g Ilog IV g I. This will follow in a standard, if slightly complicated, way using the rate theorem to compare with the half-space case, provided N is large enough that the blip can be regarded as far from the nonflat part of the boundary.

351

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN JR.3

C j 2- jotw I(D).

+C L

..__.'---- -_.._-_..__.._--------­

(2.7)

i).

~

Cw(D(O,

~ ))(Nlzl)-.8 w (D(O,

!zl))-IIVg\

Using the preceding bounds,

IV(g - g)(z)1 ~ CN- 2w(Q(1))w(D(O, Izl))-I(Nlzl)-.8jVg(z)l.

If in addition Z E Q(!), this implies by the formula preceding (2.6) that (2.8)

IV(g - g)(z)1 ~ C(Nlzl)-(2+.8)w(Q(l)).

In particular, IVg(z)! ~ w(Q(l)) when Z E Q(!) and Nizi is large enough. Likewise, comparing goo with X3, we get \Vgoo(z) - e31 ~ C(N!zl)-(2+.8),

352

CHAPTER 15

if z E ]R2 and Nizi is large enough (actually, we could take f3 = I here). By the rate theorem, lV'g(z)1 '" w(Q(I» lV'goo(z)! "" woo(Q(l»

~

COUNTEREXAMPLES W1TH HARMON1C GRAD1ENTS 1N 1R3

353

using (2.9) and)'. ~ w(Q(l). The expression in the claim is I + II + III + IV where

f_

1=

w(Q(I».

[A-IIV'gllog(A-IIV'gl) - )'.-IIV'gllog()'.-IIV'gJ)]da

JannD(O,N-Q)

II

Moreover, by (2.3) and L'Hopital's rule,

=

f_

[)'.-llV'gllog()'.-IIV'gJ) -1V'goollogIVgool]da

JannD(O,N-.)

lV'g(Y)1 I/V'gCl')(Y)/

(2.9)

-

lV'g(z)1 lV'goo(z)1

I::: Cjy -

zltlw(Q(I»

f_

III =

.

lV'goollog lV'goolda

JannD(o.N-·>.

if Y, z E graph t/J with y, Z E Q(!). Now let A = lV'g(O)I and fix a < I with a(2 + f3) > 2 and (for notational convenience) a > (l - a)(2 + f3).

IV = -

f

A-IIV'gllog(A -IIV' gJ)da.

JannD(o.N-.)

If N is large enough, then expression III is ::: -7/N- 2 by a change of scale and the fact that I(t/J) < O. We will show the others are error terms.

Claim.

f_

A-IIV'gllog(A-IIV'gJ)da -

JannD(O,N-.)

f

111:::A-Illog(A)'.-I)1

A-IIV'gllog(A-IIV'gJ)da

f_

lV'glda+I)'.-I-A-II

J annD(O.N-·)

JannD(o.N-.)

::: -7/N- 2

f_

lV'glllog()'.-IIV'gJ)lda.

J annD(O,N-·)

for N large enough.

To prove this, fix

z

E ]R2

with Izi

N-ot. Let)'.

IV$(z) I IV$oo(z)I •

w(Q(l» and

Then A

~

The first integral is ::: CN- 2a w(Q(l». Lemma 2.6 applies to the second integral showing that it is ::: Cw(D(O, N- ot » ::: CN- 2a w(Q(l». Since (2.10) implies the constants in front of the integrals are ::: C N-(l-ot)(2+/l)w( Q(l»-I, we get

III ::: C N-(l-ot)(2+tl)-2ot ::: C N- 2-tl(l-ot).

I)'. - AI ::: IlV'g(O) I - lV'g(z)11 + IIV'g(z)1 - lV'g(z)l/

lV'g(z)1 I IV'- ()I + lV'goo(z)1 goo z ::: C(N- ot

Expression II is

II

­

+ N-(l-ot)(2+tl) + N-O-ot)(2+tl»w(Q(l»

:::

1

aflnD(O.N-:-.)

j,-IIV'-' j,-llV'gllog _ g da 1 (,V'goo,)1

+ f_

1)'.-IIV'gl-lV'goollllog/V'goollda

J annD(O,N-·)

(2.10)

I)'. - AI ::: CN-(I-ot)(2+ tl )w(Q(l»

::: Cj,-I N-ottl

where we used (2.6), (2.8). Also if Ix I < N- ot , X EOn,

!)'.-IWg(X)1 (2.11)

--IIIV'g(X)1 _ IV'goo (x) I

IV'goo (x) II = A

::: N-ottllV'goo(x)/

lV'g(z)j

IV'goo(z) I

f_ JannD(O,N-.)

I lV'goo(x) I

Wglda + N-ot/l

f_

lV'goolllog lV'goollda.

JannD(O,N-.)

Lemma 2.6 also holds for the Green's function at 00 of a Lipschitz domain, so the second integral is ::: C N- 2ot . Thus 1111 ::: C N- ot(2+tl). Finally, IIVI ::: C N- 3ot by (2.6); the claim follows.

356

CHAPTER i5

The doubling propeny implies [

JoQ!

J'V'gkjlog IVgklda.:s [

J00,1-1

J'V'l-lllog J'V'gk-llda - P3,

and the lemma follows by iterating on k.

•

ForeachQ E G,letf Q = ann(PQUPQ). ThenQ' -< Qimpliesf Q, C f Q and the f Q are otherwise disjoint. Lemma 2.9. If 71 > - 0 is small enough, then for each sufficiently large k there Q. with f(Qj) = 8- k, such that EJ=I w(f Q) :::: 71 and w(r Q .) < .

log l(Q j 12 _ -TJk for all J.

are squares QI '"

Proof. Fix 71 > 0 sufficiently small. We first find such squares with Wk- I (Q j) replacing w (f Q). Let Q I ... Q, be all squares of length 8- k, with Q I ... Q.

being thosefor which log

w;~~~~f) : : :

-TJk. Let D j = D(aQj' C8- k). By (2.13),

there are pairwise disjoint regions R j with Qj C R j C D j such that (Jn k- I = Uj Rj . Take)., = w;(Q~~f) and apply Lemma 2.6 on D j , restricting the integral to R j' This gives , Wk-I(D-)

LWk-I(R j ) log ;::::: CLWk-I(D j )

f(Qj) j

j=1

+

4: i J

357

We now follow [6] and derive the main lemma from Lemma 2.9 using elementary results in ergodic theory. Fix a square I; E fb. The shift T on infinitely many symbols acts on f I;' Namely, for each Q E G I with Q -< I;. the restriction of T to f Q is the conformal affine map which takes Q to I; and preserves distinguished edges. This defines T except on a set with a-finite length and therefore a.e. (dw). Letting ~ be the partition {rQ: Q E Gd, the nth partition V~:~ T-j~ is {rQ: Q E ~}. For x E fE, let Q,,(x) satisfy x E fQ.(x) and Q,,(x) E ~. D~fine a,,(x) = f(Q,,(x»2 and h,,(x) = w(fQ.(x)' Lemma 2.10. w is mutually absolutely continuous to a certain invariant measure /-L for T. T is ergodic with respect to /-L and ~ is a generating partition. The limits

lim,,~oo .!.n log ---L-( ) and lim,,~oo n .!. log" h (I ) exist a.e. / (d/-L) and are constant. an .t .t

=

~ E'j:~ w(T-jy). Let /-L be a weak· limit point of{/-L(")}. Then /-L is an invariant measure [22]. Let E, = {x E f E : dist(x, B) < E} where B is the boundary of the square I;. Then

Proof. Let /-L(")(y)

n E,

= 0, then C;IW(Y) .:s /-L(Y) .:s c.<.V(Y).

(2.14)

If Y

(2.15)

/-L (,,) (E,) ::::: CEil

with

f3 > O. independently of n.

E ~. Let S be the affine map with S r f Q = n l = n, PI = 0, n2 = Sn,P2 = S(O), E = fE\E"

For let Y be as in (2.14). Fix Q T". By Lemma 2.4 with

we have Ivl-Illog IVl-llda.

(2.16)

1

By Lemma 2.8 and the doubling property

C;lw(Y)w(f Q) ::::: <.V(T-"Y

n f Q)

:::::

C,<.V (Y)<.V (f Q ).

. So C;lw(Y) ::::: <.V (T-" Y) ::::: C,<.V(Y) and (2.14) follows. To do (2.15) fix again Q E ~. Then T-" E, n fQ can be covered with CE- 1 discs ofradiuSEf(Q). By Lemma 2.3 and the doubling property,

, wk-I(Q-) "Wk-I(R·) log J < -pk - C < -£>-ok. ~ J f(Q .)2 f"'<' J=I

COUNTEREXAMPLES WiTH HARMONiC GRADiENTS iN]R3

J

k I If EJ=I W - (Qj) .:s 71, then using Wk- I (R j) .:s k C- , we get

cd- I (Qj) and Wk- I (Qj)

w(T-"E,

n fQ) :::::

CEIl<.V(fQ).

::::

-CTJk(log C - log 64) - TJk ::::: -f>2k,

which is a contradiction if 71 is small enough. Now, for each j E {I, ... , s} let n Qj = n r where f = {Q E G: Q i Qj}. Since n Qj is obtained from nk - I by adding blips along squares of side ::::: 8 -k , the Hausdorff distance from (Jn Qj to an k - I is ::::: C8- k and Lemma 2.5 shows w(n Qj , Qj, 0) ~ Wk-l(Qj)' A symmetry argument (or [9], Lemma 4.2, and the doubling property) and the maximum principle imply w(f Q) ~ w(n Qj • Qj, 0). So w(f Q) ~ w k- I (Qj). The lemma follows. •

with f3 > O. (2.15) follows. By (2.14) /-L and w are equivalent on each fE\E, and since (2.15) shows lim,~O/-L(E,) = lim,~o<.V(E,) = O,theyareequivalent. Given (2.16),thefact that T is ergodic and ~ is a generating partition follows by a standard argument as in [6]. Next, define T' by T'(x) = f(Q)-1 if x E f Q and Q E G I . Claim. I/(log T')

0

T"I/L'(dw) ::::: C independently ofn.

Fix Q E ~. The set {x E f Q: (log T') 0 T" (x) > ).,} is identical with U{rQ': Q' directly descended from Q and f(Q') < e->"f(Q)}. By Lemma 2.1,

0.__

358

CHAPTER 15

the latter set can be covered by Ce A discs of radius e- A • By Lemma 2.3,

w({x

r Q : (log T')

E

0

~ ~_"

"",.

~

=~"-'"

._~._.~-

,~-

..-

"..~

~

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN 1R3

359

Likewise,

10gw(D(x,r)) , I 1m sup S 2fJ

Tn(x) > AD S Ce- IlA w(r Q ),

with fJ > O. The claim follows. From the claim we conclude that log T' E L1(dJJ,). Since ~ log = ~ L~:~ log T' 0 Tj, the ergodic theorem shows limn-+ oo 1.n log...!... exists a.e. dJJ, and is constant. Now fix QI E _G . By the I doubling property, there is a constant r such that

t

r->O

log r

•

and the proof is complete.

~

w(T-nr Q , for all Q that

E

fin.

nr Q )::=:

C£(QI)'w(f Q ),

Itfonows thatJJ,(rQ,) ::=: C£(Qd r and, sincelogT'

L

Q, EG -I

1 JJ,(f QI) log - - < JJ,(r Q,)

E

L1(dJJ,),

00.

t

Proof of Main Lemma Note that any squares Q E G with £(Q) = 8-k must belong to fin for some n S k. By Lemma 2.9, we can choose E so that E

I . hn(x) }) ::=:­TJ rI;: - rnflog - - S -TJ k nsk an (x) 6

for a sequence of k tending to therefore that (2.17)

00.

It follows that limn->oo 1. log !!..a. < 0 and n

fIn

I I !!.. , 11mn-+ 00 Ii og fIn , log h n II. hm - - = 1 + , 1 =p n-+oo log an hmn-+ oo Ii log an

> 2 log hn(x) log an (x) as r -+ O' , Since l'1mn -+ 00

log fIn±! (x) IOgfI.(X)

+

0((1

I

og an+l(x)

Proof. Let gn, g be Green's function at 00 for Qn = (x: X3 > tPn(X")} and A = (x: X3 > tPCX)}. WecanassumesupptPn C Q(1). There is no difficulty at 00 since lV'gn(x)1 = I + 0(lxr 3 ) uniformly in n as x E 1R2 -+ 00 and similarly with IV' g I. It follows that it is enough to prove (2.18)

[

JQ(2)

)-1)

[

JQ(2)

liminf 10gw(D(x, r)) > 2fJ > 2. r-+O log r ­

as

n -+

00.

V'gn CX, tPnCX)) - V'8nCX, tPnCX))12dxldx2 -+ 0,

1

as n -+ 00. We can replace Q(2) by Q(8) in (2.19) by standard theorems on continuous dependence of solutions of the Dirichlet problem, since both tPn and 1{fn are smooth outside Q(1). Ifwe now replace Q(1) by Q(4) andtPn by 1{fn + En we have a reduction of(2.18) to the case tPn ::=: tP. This case will follow from the regularity for the Dirichlet problem on Lipschitz domains due to Verchota [29]. That theorem implies

+ 0(1)

= I . S0

lV'gCX, tPCX)) - V'gnCX, tPnCX))12dxldx2 -+ 0

In proving (2.18) we may assume tPn ::=: tP· To see that, let En = IItPn - tPlioo. Let 1{fn satisfy 1{fn = tPn on Q(3) and -En S 1{fn - tPn S 0 outside Q(3) with 1{fn - tPn = -En outside Q(4). Let 8n be Green's function at 00 of X3 > 1{fn ei), i.e.,8n = 0 when X3 = 1{fnCi) and 8n - (X3 + En) -+ 0 as x -+ 00. The rate theorem applied to 8n - gn and gn implies that lV'gn - V'8nl S 8(En)lV'gnl on (x E aQn : X E Q(2)}, where 8(E) depends only on E and a bound for IItPn IlLip(I), and 8(E) -+ 0 as E -+ O. So (2.19)

exists a.e. and is> l. Suppose x is a point where (2.17) and limn->oo ~ log an (x) exist. For r > 0, let n be the largest number such that £(Qn(x)) > r. By the doubling property

10gw(D(x, r)) 2 log hn(x) log r ::=: logan+l(x)

Lemma 2.11. Suppose {tPn} is a sequence of Lipschitz functions on 1R2 with common compact support and tPn -+ tP in Lipschitz norm, Then l(tPn) -+ l(tP),

Since; is a generating partition, this implies that T has finite entropy with respect to JJ,. Let hn(x) = JJ,(Qn(x)). Then limn-+ oo ~ log exists a.e. (dJJ,) and is constant by the Shannon-McMillan-Breiman theorem (the version we need for infinite partitions with finite entropy is in [16]). By (2.14), limn->oo 1. log -hi n • exists and is constant. •

w ({ x

We still have to construct suitable variations tP. First we get rid of the piecewise linearity requirement.

[

JQ(2) as n -+

00.

lV'g(x, tP(x)) - V'g(x, tPn(x))l2dxldx2 -+ 0

In particular, by the Lip(1) convergence,

[

JQ(2)

/V'Tg(X, tPn(X))12dxldx2 -+ 0,

360

CHAPTER 15

where VT is differentiation along ann' Using regularity for the Dirichlet problem again, this time on the domain nn, we obtain

(

~ C laQ. ( ---+0

n ---+

00

~

dlVgl

•

Now fix l/J E Ccf(JR.2). Let Q t = {x E JR.3: X3 > tl/J(X)}. Let get, x) (t E JR., x E t ) be the Green's function of Q t with pole at 00. Then get, x) is smooth in t and x jointly. Denote I(tl/J) by I(t). We claim that I vanishes precisely to third order at zero for suitable l/J, as in the first case of Lemma 1.1.

n

t=O

----;j"( (2.22)

=

dg '

d U3 '

d21VgI dg" -d 2 = -d t U3

d 2g '

-

+ 2l/J-d 2 + IVl/J1 u

3 lll 2 d31VgI dg d g" -= + 3l/J3 dt dU3 du~

+ 3l/J

2

2

,

d 3g' du~

dg' -

+ 3-IVl/J1 dU3

2

.

Also, for t arbitrary but U3 = 0 and Ii outside the support of l/J we have

Lemma 2.12.

=

'

When t = 0, g is linear and the last term in each of the last three expressions drops out. If U3 = 0, then IVgl = n 3) ~' so when U3 = t = 0 (2.21) gives

and the lemma follows.

I

3 ' d dg ) dg dg" d ( = gil + l/J - ' + l/J = ­ + 2l/J du~ + l/J2 d g d u3 dU3 dU3 dU3 dt dU3 3 3 4 d 3 dg dg" l 2 d g' 3 d g d 2g 11 - 3 -- = +3l/J- +3l/J +l/J - . dt dU3 dU3 du~ du~ duj

IVT(gn - g)1 2dx, dx 2 as

361

]R3

d 2g

IVgn(x, l/J,,(x» - Vg(x, l/Jn(x))l2dx, dx 2

lQ(2)

dl -d t

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN

2 d 21 -d t

I t=O

=0

and

1[( -d~ )3 -

3 d 1 = -3 dt]R2

dX3

- 2-d~ ]dx,dx2

31Vl/J I

dlVgl

dt

dX3

Here ~ is the harmonic extension into the upper halfspace and V is the ]R.2 gradient.

d 21VgI dt 2 d 3/Vgl

(2.23)

dg ' = dU3' dg"

dU3 ' Proof We also let t::. be the ]R.2 Laplacian. Let g', gil, gill be the first three t­ dg lll derivatives of g with x fixed. Then g'(t, .), g"(t, .), glll(t, .) are harmonic on Qt> ~ = dU3 vanish at 00, and vanish on ant outside the supportofl/J. Moreover, g!(0, .) = _~ In computing It> Itt, 1m we can differentiate freely under the integral sign. For ("Hadamard variational formula"); to prove it on JR2, which suffices, differentiate there is no problem locally since all functions are smooth, while if Ii is outside the equation g (t, x, tl/J (X» = 0 with respect to t. the support of l/J, then the first three t-derivatives of IVgllog IVglda are sums We do the calculations using coordinates (t, u) E JR x R~ and the map (t, u) ---+ d of terms each of which (by (2.23» involves either log IVgl, E.8~d or
beginning of the section and the same proof works in general. So we have uniform 2

3 integrability. dn)/ (d n)1 2 (d )1 n (2.21) ( dt' e3 t=O = 0, dt 2 ' e3 t=O = -IVl/JI, dt 3 ,e3 t=O = 0 Keeping in mind that IVgl = I when t = 0, and using (2.20), (2.22), dg I dg = g + l/J (ii) ­ dg ' dt dU3 du 2 -dl I = -dUl dt t=O ]R2 dU3 dg d dg d ( I dg ) dg' d 2g 2 2 -dtd -dU3 = -dU3 -dt = -dU3 g +l/J(ii)- = +l/J(ii)­ d 1 dg" d g' dg' (2.24)-2 = + 2l/JdU3 dU3 du~ ,

+ ( dU3 + IVl/J1 2 du,du2, dt ]R2 [ dU3 2 2

d d g d d ( I _ dg ) d dg 3

2 - 2 -- = - 2 = g + l/J(u)­ 3 d I dg' d g ' 2 d g' (d gl )3 dt dU3 dU3 dt dU3 dt dU3 ~ = 91Vl/J1 2 --.£ +6l/J- +3l/J ­ 3 (2.2S)dt It=0 ]R2 [ dU3 dU3 du~ du~ dU3 I

-I

1 1 Lo 1

du~

)2

]

,

/I

,

11/ •

362

CHAPTER

15

J

d 2g" dg' dg" dg'" +34>-- + 3 - + dUldu2. du~ dU3 dU3 dU3

: jR~ ~ jR is harmonic and vanishes at 00 then fJR2 ~ 2 0, f JR2( -%;)2 = fJR2 IV 11 • Therefore, It/I=o = O. Also, the first term on the right side of (2.24) is zero, and, using g' = -¢,

Now we use that if I

1 d ¢ d¢ d 1 -2/ = [-24>-2 + (-d dt 1=0 dU 3 u3 ) 2

2

JR2

= LJ24>£:,.4> +

= LJ(:~

r

(:~

2

+ IV4>12Jdu1du2

r

2 + IV4>1 Jdu 1du 2

2 -IV4>1 Jdu 1du 2

The last term in (2.25) is zero. The preceding two terms are

d2g" d¢ dg" [4>-- - -JdU 1du 2 JR2 du~ dU3 dU3

1

which is zero by Green's identity for the harmonic functions remaining terms in (2.25) are

1[ - 2 1[ 1[ - 2 JR2

2 3 d¢ d¢ d ¢ d ¢ ( d¢ -9IV4>1 + 64>- - 34> + dU3 dU3 du~ duj dU3

2

¢ and

)3J dUldu2

JR2

JR2

d¢ 2 + 34> 2d¢ + ( -d¢ -3IV4>/ -d¢ - 3-£:''(4)) £:,.dU3 dU3 dU3 dU3

=

=

~. The

)3J dUldu2

d¢ d¢ 2- d¢ ( d¢ -9IV4>j - 6¢-£:"4> + 34> £:,.- + dU3 dU3 dU3 dU3 2

)3J dUldu2.

The middle terms vanish by Green's identity for £:,. since 4> has compact support.•

Remark. Similar calculations may be done with any initial domain in place of the upper half space and for most domains one does not have to compute so many derivatives since curvature terms enter into the first variation. However, this is useless for proving Theorem 3. The curvature effects scale out as N ~ 00.

5#; )

t > 0 small in Lipschitz norm by a piecewise linear function, we have a proof of Theorem 3. Taking t < 0 instead of t > 0 we obtain a domain whose harmonic measure is less than two-dimensional. Domains homeomorphic to a ball with this property (infact, with I-dimensional harmonic measure) may be found much more easily by other means (a closely related result is in [32]), but it may be new that there is a scale invariant and/or NTA example. It is also amusing to note the following: the {froof of the main lemma works just as well if Q is replaced by the complement of Q, except that 1(4)) must be replaced by f IVgllog IV' gl where g is the Green's function at 00 on the domain below the graph of 4>. It follows that if Q is the is < 2­ domain just used to prove Theorem 3, then harmonic measure for dimensional. It is probably possible to make similar examples where the interior and exterior harmonic measures are both > 2-, or both < 2-dimensional. This wotild involve working with a function 4> as in Lemma 2.12 for which ~ 11=0 = 0 and ~;! 11=0 has the appropriate sign. The calculations for the fourth derivative are rather complicated and 1 have not worked this out. Another remark is that some of the technicalities in the proof of the main lemma could be avoided by working in a special situation where it is possible to use a finite partition. However, one would then have to compute sgn 1(4)) for special 4>, which (unless it were done numerically) would appear to be a difficult problem. Finally, we show how to generalize to jRn, n ::: 4. An unbounded domain with dim w ::: n - 1 + E may be obtained by crossing the domain in Theorem 3 with jRn-3. To get a bounded domain, intersect domains Ul ... Un -2 of this type with different choices of the dummy variables. If U is the resulting domain and E c with dim E < n - 1 + E, then

au

w(U, E) ~

L w(U, En aUj) ~ L w(Uj, En aUi) = O. i

i

APPENDIX: PROOF OF LEMMA 1.2 This argument uses a lot of high school math; the details of some of the calculations will be omitted. The last part is a proof-by-cases and a hand calculator is used to evaluate logarithms and the like. The reader who wants to check through it all will need a calculator and scratch pad. While 1 was working out the proof, it often seemed to be a huge waste of time. It would be desirable to have a conceptual proof of Lemma 1.1.

5#;

3

2 The expression f [( Jalso came up in the proof of Lemma 1.1 - 31 V4> 1 and was shown to be negative if 4> = 4>0 = -Ix + e31- 1. Approximating 4>0 in 1 the W ,3 Sobolev norm by a function t/t, and then approximating tt/t with

Co

363

rr

=0.

3

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN]R3

Added September, 1991 See the remarks after the introduction, and reference [2].

-="-= __

-=.-------~---~":'S~.=:;::::::~~~~~~1'..::;:~"":"--"'~~,~-~-::::2'~~'?_::::=:=..':":"".-~~~":~~::::~~~::;~~':~;:;:.;_:.:~,~s~.s'£~:~''::,',,·~"~.:-~"fZ!"''':2.ti~''d::tJ::::::

364

CHAPTER 15

A.l The setup

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN R

and t!,.Tg

dg + -ddzg 2 = -t!,.Tg = -(dr+

1::1< (e, q (r, &») g~ = 0 for all r. (b) Let 1/I(r) = ~ 1::1< (J4-rL~;:~n<m\1/2

drjJ.

i

Thenfor r < 1,

When

r . d& 1-1< log Ie + q(r, &)1 21T !log(

~ (~ +

2

2

1/I(r) S

a 1/1 (r)

log-­

~ a

1/I(r) :::: Jl - a

2

1_ ~ r

Thenfor r > 1, 1<

Ie

1 + q(r, &)1- S log( - (.JC d&

2

21T

-1<

+

+

Ie + ql2 =

11<

- b 2, and also 1/I(r),

(2) It will be clear from the argument for (c) that we could also obtain a formula for the integral appearing there. This formula is too messy to be useful in the proof, but was used by J. Goodman to evaluate lea) numerically.

Proof Recall e

=

(a, 0, ~), g is the harmonic function on IR~ with

boundary values

((x - cosrjJ)2

+ (y -

sinrjJ)2)1/2

1 + a21Vgl2

+ 2(a 2gx + a~gz)

= {(~ + a1/l(r) cos &)2,

r < 1

a cos 2 & + b cos & + C,

r > 1.

1

log I cos& -

d&

pl- = log Ip + JP2=l1 21T

.

where the sign of the radical is chosen so that IP

-log2,

pEe,

+ JP2=l1 ::::

1. When

PI , P2 E IR we obtain

1

s

IpI! Ip21 \ log /P21 - log 2,

jr--­ 10g(lpII + V pr - pD - log 2, Ipil :::: Ip21·

d&

1<

loglp2·cos&-pJ!- = 21T

-1<

Part (b) of the lemma follows from this. For (c) we use g(r, &) = { -r cos &, -r

-I

cos &,

if r < 1

Lemma A.l.2. Suppose pet) = at 2 + bt

if r > 1

Then

d&

1<

. We wnte VT

=

(d d) ,t!,.T dx' dy

=

(-I, 0)

Vrg

2z cos rjJdrjJ

+ (y - sinrjJ)2+ Z2)3/2'

cos rjJdrjJ

-1<

-1<

+ c)2

((x - COSrjJ)2

We record the formula

(A.3)

(1) It is easy to check that 4ac - b2; therefore, (a are :::: 0 for all r.

(A.I)

= -2 cos &d& Ir=1 .

with 1/1 as in the lemma. Part (a) of the lemma is now obvious. Also

1<

= aVg.

r=1

= aR.~,

= 1/I(r) cos&,

~. r

Remarks.

and q have

I

36S

0, the fundamental theorem of calculus gives

dg 1 = ­ dz 1T

2

.J(a + c)2 - b2)).

1-1<

-

a

= a 2( r 4- + 1/1 (r)2), b = 2a~1/I(r), c = log

z=

(A.2)

Jl - a - a 1/1(r)2)) ,

r

d 2g 1 dz 2 = - 21T

(a)

1

dg -)d& dr_

3

If (x, y, z) E R.~ then by the Poisson integral formula

Lemma A.l.l.

(c) Let a

1. Thus as distributions on 1R2

= 0 when r 7'= 2

In this section we reduce the problem to estimating an explicit integral. We will use cylindrical coordinates r, &, z as well as rectangular coordinates x, y, z on ~~.

=

2j!"-"-::..""~~~:T;zg;?:;::_~"~~'·~';li:..':<-:J!2;:(.':''Oc''S0''~~'~t~~£:..c~~:?:'

=

d fiX!

+

2 dd 2'

y

if r < I

1 1 { (2" cos 2&, 2" sin 2&) r

2

r

if r > 1

When z

= 0 we

log p(cos&) -

21T

/ -1<

Proof Let D i..{ij). Then

r

= 4ac -

s

+ C is nonnegative on the real axis.

1

210g( - (.JC

2

b2 :::: O. Write p(t)

1-1< log p(cos &) d& 21T

+ J(a + c)2

= aCt

= log a - 2 log 2

- {)(t -

- b 2)).

I), { =

fa (-b +

+ 2 log I{ + ~I,

366

CHAPTER 15

with the sign of the radical as in (A.3). Now

I~

+

~12 = :.

2

Proof ofLemma A.2.1 It is understood that r < I in all fonnulas.

1

+ '!'J(a + c)2 - b 2 + .!.re-J"E, .a a

a

y,~~/cx

2

where E = 4c - 2b + 4ac + 2ib...{ij and../E is taken to have nonnegative real part. Estimating er../E by ..;rET, we obtain

Is

+

~12 s :.a

+

~ /(a

a

+ c)2 - b 2 +

~ .JC-t(a +

a

=

~

c)2 _ b2

27r

log[ ~ (~ + /1 - a Z - a Z1/fZ)]rdr

2

{

1y,~JI-cx2/cx

= {

d8

Ie + q l - rdr

log

(

1y,~JI-cx2/cx

~2 10g(1 -

.!. (.JC + j(a + c)2 _ b 2)2, a

S -Q 10g(1 - a Z) -

and the lemma follows.

• •

Part (c) of Lemma A.U is immediate from this.

A.2 Simplifying the formulas We want to get rid of the roots and logarithms. Define lin(a) = 1,<1 log Ie q(r, 8)1 ~~ rdr, lout(a) = 1,>1 log Ie + q(r, 8)1 ~~ rdr. Also define d =

f

1/f2r dr,

r<1

1/f2r dr,

Q =

r<1

y,~CX-1

+

2

Q - 2

< -log(l-a z)

1

Z

1 aZ 4 I - a

- - --x Z

d8 logle+ql-rdr =

y,>~/cx

27r

a

1

f

'

JT'=t

1( I

s 2: 2: -

r<1

(I_cx 2)'/2

! t. Also

2:

a1/f

logl(-)Irdr

y,>JI-cx2/cx

-1 ( -1 - Q)

rdr.

z)

-I ( I - /I - - - z 1/f rdr

2 1 - a

y,~JI-cx2/cx

where we used 10g(1 - t) S -t, 1 -

1

(I + VII - ~ 1/fZ)])rdr I - a

a Z) + 10g[.!. 2

2 2

f

x =

367

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN]R3

1

2

y,>~/cx

(

Z a 1/fz rdr log )­

4

i-

Q

Z

)

[ a (d - x) ] Q log 4(! _ Q)

y,~a-I(I_cx2)'/2

•

by Jensen's inequality.

Lemma A.2.t. lin S ~ 10g(1 - a

Z

) -

! 1~:2 x + i (i -

Q) log[ :(Y=-~~].

Proof of Lemma A.2.2 Here it is understood that r > 1. We use the following fact.

k

Z (1 - a ) log( :~~ ) + V where V satisfies the following

Lemma A.2.2. loul S two estimates. Let

i,

E [-1, I] and let y = min( 2- I / z3- 3/ 4 (1 - e Z)-1/4). Z Then (1 + 2et + t )I/4 S I + Y min(t, (t + 2e)+)for t E [0,00).

"Lemma A.2.3. Let e

2 (2a Z - I + a Z( 2' 2 + 4'))]+ 1 ) . r(r) = min ( 1/f(r)z, [1/f(r)z + 2' a r r

°

= (1 + 2et + t Z)I/4 -

1. Then f(O) = f( -2e) = 0, and f < on (0, -2e) if e < 0, so it is enough to show I'(t) s y when t > max(O, -2e). We have I'(t) = (t + e)(1 + 2et + t Z)-3/4, f"(t) = (- !t Z - ~t + %e Z)(1 + 2et + t Z)-7/4, so the critical points of I' are -e ± J2 - 2e z . If e > J2/3, then I' is decreasing on [0,00); hence I' S 1'(0) = ~. If S e < J2/3, then I' has a maximum at -e + J2 - 2e z ; its value there is 2- I / z3- 3 / 4 (1 - e Z)-1/4. One computes ~ S Y for all e while 2- I / z3- 3 / 4 (1 ­ e Z)-1/4 = y when e < J2/3. The e < case follows the same way: if

Proof Let f(t)

i

Then

I z V(r) S _a

4

f""( 1 I

z

(1

z a Z Z ) r(r)rdr,

+a

)r

V(r) S 12- /\4a (1 - a Z))-1/4 a Z 3

11f""( I + r1Z )-I/Z r(r)rdr

i­

°

°

-----

368

-.;,

CHAPTER 15

-,J213, then f' is decreasing on [-2e, 00) and if -.../2/3 < e < 0 then f' • has a maximum at -e + ../2 - 2e 2.

e ::::

=

a2

2a2

I+-;r+~

=

e

'

2a

2

-I +a2(~.+ ;.) 2a2 ~ I + "72 +

E

< -1 - 2

2

2

a + 4" r

min(t, (t

+

2e)+)

Z

+ -2aZ + -a4 )-I/Z r(r)a z r

Z

a )-I/Z

( 1+ -z + 4" r r

c)2 - b 2)

r

~

(

a r

I+

2

2"

)-1

~

1 -..

a2

,

r > 1.

a2 r 4 min(t, (t

+ 2e)+)

~,~

The other estimate comes from

2a

2

2 r4

a

r2

2a

2

2 a r4

r2

J1+

2 2 ) a 1+ -2a + -1 t 2 + 2et + I r2 r4

/

/

2

I

"2 T I / 2 r 3 /\1 -

II(r) ~

2a- +a­ ) 1+ r2 r4

because (1 2 2a -2

r

J1 - + - J 1+

min(t, (t

y 2

+

Z

+ -2az2 + -a 4 )-I/Z r(r)a z r

+ ~ + ~)2 = r r

r

4a 2(1 - a 2)(1

+ -4)z. r

+ 2e)+)

•

A.3 Estimates for 't/J

2 2a 2 r

q I -dO ~ [ -1 2Jr 2

= I(r)

eZ )(1

2

a + 4" r

/ 2a z 2 e )-I/\.; 1 + ?2

= 12-3/\1 - eZ)-1/4 ( I

2

where we used A.2.3. So, since log x ~ x - I,

f log Ie +

4and

and the first stated estimate for V follows using

(J1 - - + - + (J1 - - + - +

+ -2y

2a r

-2

2

1( I 4

r4

4,

JI +

= -

2az

= -21

369

We set V = 11 II(r)rdr and make two estimates corresponding to y :::: y ~ 2- 1/ 2 3- 3 / 4 (1 - e 2)-1/4. Using y :::: we get

[-1,1),

and define y relative t~ this e. Then

~ (...;c + 1(a +

·==~~~-=-=~~-~~=~~~-~~7

00

II(r) :::: -I 4

a 21{!2

- .._·_'<=....

~._-====---==-=,-=---=====--==--===-,==--==.===----===--===--========

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN 1R3

Now, working from A.Ll (c), let

t

.. _--='--"._­

.'-.'- -

-----,,-----~ --_._-~--=-----=="'"'==-,==-----=---_._---..,....--,_._

We give some formulas for 1{! and then some estimates.

2 2 + -(i2 + -I / I + -2a2 + -a 4 - 1] 4

r

2 2a -2 r

+

r

2

2 a 4" mm(t, (t r

1{!(r- l ) = r1{!(r)

(A3.1)

1

r

00

(A3.2)

+ 2e)+)

1{!(r)2r dr = 2

Proof On the one hand 1 g; ;~ rdr = 4 1 1{!(r)zrdr, and on the other hand 1 g; ;~ rdr = 1 IVr gl 2 ; ; rdr = 101rdr + 11 r- 3dr = 1. •

+ II(r).

00

Now

.."

I(r) =

~ [ (I - ~:)

1+

2 4 a - a (r 2 - a 2)2

+

(

2 a 1+r2

)J1+

1 a2 - a4 < -4 - ­ - 2 r - a4

2 4 a - a ] 2 2 (r + a )2

-1

(A3.3)

2 1{!(r) = ; r

t

t l / 2dt (1 _ t)I/2(1 _ r2t)I/2

10

if

r < I.

Proof The function ( (!+r 2)z':'r(l +z2) ) I/Z has a single valued branch on the unit disc slit from 0 to r. Accordingly,

since .Jf+X ~ 1 +

f I

4x. It follows that oo

I(r)rdr

~

1{!(r)

1 I+ 2 - (1 - a ) Iog(--2)'

a2

8

I - a

= e ( -Ii Jr

=e ( - I

Jri

Izl=1 (1

+ r2 -

1( + Izl=1

(I

Z

r(z

+ z))I/Z

dZ) -

Z

r 2 )z - r(7 z

+ ])

iz

)1/2dz )

6!I,~iQ<\~l'-

370

CHAPTER 15

( -21'( ni (1 + r

=e -

0

X

2 )x

- r(x 2

+ 1)

)1/2 dx )

,

,('If

__

.-·-.)boljo>~t,.*'''''_'lfn~-'''''''''''''''--'''''''~''''·~,'''''''''L''''>~·_'·"'~_·." ~

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN Ii3

d For r < l, dr ((1

(A3.8)

and now let x = r 1.

•

Proof. Lett

- - IS

dl

r

di

This follows from A3.3. Note it implies 1/1 is also increasing.

The first statement follows from A3.4 since 1/1' (0) = 1. The second statement follows from the first.

2

(A3.6)

d

7

- < d < 3 - 8

Proof· Let v(r) = tp. Then d fd v2rdr. v2 is increasing, so

1

o

(as

.

In

= fol v 2r 3dr

1 v 2(r - 2r 3)dr = -I (r 2 - r)v 4 21 1 2

0

A(t)

=

-2 n

B(t)

= ­I

jJr

1

cos 2 9

-:n;

jJr -Jr

2n

---d9

l-tcos9

2 cos 9 d9 (1 - t cos 9)2

+

(1 r 2 )2

A(t) = t- 2((1 - t 2 )-1/2 - l) = - ' - - - ­

2

-I (r 2 - r 4 ) -dv dr 2 dr

2(1 - r 2 )

B(t) = 1- 2(1

+ (I

- t 2)-3/2 - 2(1 - t 2)-1/2) =

It follows that d ::: ~. Also, 2 v (r - r 3)dr:::

by A3.5, and this implies d

::s

t 10

J

(A3.7)

(1

~.

•

+ 2" )-1/21/1 (r)2 r dr

::s

(I

+ r 2)-1/2(log -I- + log(l + r) I - r

:r ((1

= Iv 211 0

-

11

+ r 2) 1/21/1 (r» ::s 0 -

ft (1 + J

~ )-1/21/1(r)2r dr =

statement follows by A3.I, A3.2 as above.

when

- arctan r)

2

dvd r I0 dr

fdo + r 2)-1/2 rv (r)2dr,

+ r 2)2 + 2(r 2 -

r 2)-I«(I

< (I - r 2)-I(I -

+ r2) ( 1 +

1

1

I- r

1+r

r2 (1

.

r 4 ).

•

r < 1.

-

and the •

(A3.10)

d1/l < -4 r -1/2(1 dr

-

n

_ r 2)-1

when

1+r

r 4»1/2 4

­ r ) + r 2 )2

1

- - - + - - - - ­2

< O.

By A3.I,

+ 4r 2 -

Proof. A3.8 implies

4

o

(1

(A3.9)

Proof· Let P = 2(.,,12 - I). Let v be as in A3.6. Let I(r) = (1 + r 2 )1/2 ­ 2 pr - l. Then 1(0) = 1(1) = 0 and 1 > 0 on (0, I) (this follows because 1 is concave down at the critical point r = (p-2 - l) 1/2) so ~ - pr v(r)2dr 1 r2

2(1 - r 2 )'

::: 2(.y2 - 1)(2 - d).

r

) 11( r +

+ r 2 )2

4

I

1

(l

The statement follows from these formulas and the fact that -ddt, = 2(1(1+,-~21 )

(r - r 3)dr = _,

1/I(r) OO

~ 1(1) •

foo cosIJd9 -00 (I-t coslJ)I/2

may be evaluated using residues to be

::s O.

t 10

r 4 )1/2.

where

and by A3.l, A3.2, 2 - d =

0

+ 4r 2 -

cos2 9d9 (1 - 1 cos 9)3/2

A(t)1/2 B (t)1/2,

A2).

II

/:n; -:n;

1 = 21T

::s

1/I(r)::: rand(withx, QasinA2)x ~ Q2.

(A3.5)

(1 - r 2)-I(1

2 = 6. =! 1+, Then (1 + r )1/2,'r(r) 'I':n;

1/I(r) . increasing . . on [0 , I) .

(A3.4)

+ r 2 )1/21/1(r» ::s

371

•

r < l.

•

...__

__-- -

.~.___

372

CHAPTER 15

l 2 Proof Let f(r) = r / (1 - r 2) ~. We claim f(r) will follow if f(r) > f(r 2). From A3.3,

21 21 I( ) - -;; 1(r ) -r

2

-

7Z'

1

0

-

1

0

_

-

l 2 / (1

2

.:s

Hm

SUPHI

f(t). This

)t l / 2

r - r dt (1 - 1)1/2(1 - r 2t)3/2

_~,"'-_._._

'"

__... _

.n__--.'-

P~oof j} r1{t(r)2dr

.r. (1{t(r) 2

.~

__ .,____

-

[I

=

_

373

~ (1 - s2)1{t(s)2 + 21{t(s) j}(1{t(r) - 1{t(s»rdr +

(1{t(r) _ 1{t(s»rdr

=2

[I [T

1/I'(t)dt rdr

4

21 ¢J(r, u)r l / 2(1 - r 2)u l /2 du 0 (1 - U)I/2(1 - r 2u)3/2

= [1(1

1

(u

_ t2)1/I'(1)dt I

2 _

< -4 [ t -1/2dt - 7Z' s

- t),

1

=

2u 3/ 2 r l / 2 (1

(1

+

=1(s)"2(1-s2)

r 2)

[I

+ u)I/2(1 + r 2u)3/2 .

(1/I(r) - 1{t(s»2 r dr =

One computes ~ > 0, so ¢J(r, u) < ¢(r, 1) = ( ~ )1/2 < I, and the claim follows. To evaluate the limit, write d1/l

-

dr

s

-

1 17< 1 - r

2 2 3/2 d(} 7Z' -7< «1 - r) + r(} )

= -

1

00

= =

1

7Z'(1 - r)

2

7Z'(1 - r)

(1

-00

1

16 l-s'/2

.:s ~ (1 2

_

where 1(s) = Jr J=Sr' K(s) -

U 2)-ldu

dt

(1

+ V)-I(1 + V2)-ldv

.fi

.fi

(1

+

v)-3dv

•

A.4 Final calculations

- s2)(1/I(s)2

128 (l+s'/2)-2_ l- 2 s

Jr2

128 7Z'

-2

/1

/1

t- I / 2U- I / 2(1 -

s

1

+

1(s)1{t(s)

+

K(s»

*

Remark. 1 and K are decreasing on (0, I). For K, this may be seen by letting t = s 1/2_then ~ = (1 + t 2 ) 0+1)3 and calculus shows that (1+1)3 is 128K (s) 3+1 ' 3+1 increasing.

7Z'

+ lower order terms,

r1/l(r)2dr

264

s

= K(s)"2(1-S2).

•

s
=

r 1/2(} ) 1- r

and A3.10 follows.

(A3.1l)

1{t'(u)1/I'(t)(1 _ t2)du dt

16 7Z'

.:s =

2

[I [I .:s 2 [I [I

+ lower order terms

+ t 2)-3/2dt + lower order terms (t

- (I - max(t, u)2)1/I'(t)1/I'(u)dt du

s

=

117< (cos(}-r)cos() d(} 7Z' -7< «(1 - r)2 + 2r(1 - cos (}»3/2 .

-

[I [I 1 s

We show this is -: (1 - r)-I + lower order as r ---+ 1. The denominator is ~ max«1 - r)3, (}3), so we can ignore O«(}2) terms in the numerator and O«(}4) terms in the denominator. This leads to

dr

....

1{t(s»2r dr and

r(1 - r )t l / 2 dt (1 - t)I/2(1 - r 4t)3/2

'7Z'

¢J(r, u)

d1/l

.__.....,

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN 1R3

with

-

-,__ -

The argument will proceed roughly as follows. Suppose a function I, involving logarithms or whatever, can be shown by sections A2, A3 to be an upper bound for 1 (a) on the interval [0.2, 0.27). Suppose 1 is increasing. If 1 (0.27) < 0, then Lemma 1.2 is proved when 0.2 .:s a :::: 0.27. The fact that 1(0.27) < 0 will be proved by evaluating 1 (0.27) on a calculator. With V as in A2.2, we can use r :::: 1/12 and A3.1, A3.2, A3.7 to get 1 1 + 2a 2 V < min(-a 2(2d) 2(21/2_1)12-3/4(4a2(1-a2»-1/4a2(2-d»)

-

4

1 + a2

'

•

Another frequently used fact will be that all Taylor coefficients at 0 of ¢ (a) = 2 - a 2) log( ~~:~) + ~ 10g(1 - a 2) = (1 - a 2) log(l + a ) + (1 +

i (l

i

i

~

.,

- -

374

_.~~

---.-----~

""'-.:.~''.;,;;~

__...

-,-~T~~~~~'~.;.~,~,'_'-.;.,-;,',_.._,_"

a 2)

log(l are nonpositive, and in particular (look at the first nonzero

coefficient) q, (a) :::: - a 4 •

2

1 ( 1+a ) J :::: - (l - a 2 ) log - - 2 8 1- a 2

I

au

I +a ) < -I (I - a 2 ) log ( 2 -

+ -4I a 2 ( 2 -

I - a

8

1· < -Q log(l - a 2) 2

m -

+ -I ( -I 2

2

Q) log (

I

+ 2(2 1/ 2

2

+ 2a I + a2 2

a d

4( 4 - Q)

d)

2

I 2 ( I +a ) - (I - a ) log - -2

1- a

8

1 I

+ 2: (2: -

(A,4)

+ -I a 2 + -Q 2

1+a

Q) log [ ,",_11

I

2

- 8

2

,",_.7J·

4

- a 2)

+

1- a

+ -1

e( - a

2)

4

2

log(l - a 2 )

a d + --­

2

2 1+a a ) log -1---a-2

I

"8 (l -

1

+ 4"

2

log(l - a )

+

7

a

2

+ 0.173a

-64-e -1---a-2

The calculator shows this is negative, as follows: Evaluate

4.

+

2

a d -8-1-

i,

J ::::

2

+ -1 log(l

+

2 1 -4 log(l - a )

using that 2(2 1/ 2 - 1)12- 3/ 4 (4a 2 (l - a 2 ))-1/4 is increasing and a calculator for the value when a 2 = 0.85. The bound is increasing in d when a 2 :::: so by A3.6 we can take d = 7/8 getting

64e(;-a 2)

2

.

+ 0.173 at

a~ = 0.85, ai = 0.82, ar = 0.79, obtaining values :::: 0.442 = f3,:::: 0.397 = f2,

k

Let aJ = 0.75. By calculator, (l - a 2 ) log( :~~~ ) + ~ log(l ­ a 2)+fja 2 is:::: -0.01 < owhen a = aj_1 (j = 1,2,3). The statement follows using monotonicity, as with (A5) in Case I.

= fl.

:::: 0.365

1+ a ) = -1 (l - a 2) log ( -2 8 I-a

375

+ 0.153a 2(2 - d)

Q -- !. 2' (A,S) (AA)

"""""",",,,"---""''-'~''''-~~

1)12- 3/ 4 (2 _ d)(4a 2(l _ a 2))-1/4 a 2 2

)

log(l - a 2 )

This is linear in Q so we need only check the endpoints, 0 or

_

1 ( 1+a ) < - (l - a 2 ) log - -2

,

where we have overestimated in Lemma A2.1 using x :::: O. Adding the right sides, then maximizing over d E (0, (0) gives

J::::

""'__....,_,.,"'-"""::;_~_,.._ .. "A'i!~....""'"""_.

Use the same bound for Jin and maximize over Q for fixed d. The worst Q is

4 - 4e(~~a') leading to Jin :::: ~ 10g(l - a 2 ) + ge(~~a2)' The second bound for V leads to

i

Case 1. 0.85 :::: a 2 :::: I.

1

..;,.""'......__",. ~

3 COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN 1R

CHAPTER 15

a 2)

""3i,"".~"'·,[,;

I 2 . _a 2

Case 3. 0.5 :::: a 2 :::: 0.75.

Q =0: (A.6) (AA) =

2

1

"8 (l -

2 ( 1+a ) a ) log I _ a2

2

+ 2:1 a 2 + 4"1 log [ 2e(l1 ++a2a2) ] .

SubcaseA. Q::::

(A.5){a 2 is decreasing in a by the remark about negative coefficients. The

J(s)::::

J(2- 1/ 2)::::

3(2 - x)

+ log[ 2 (3 -

f ::

2x)

1( -2x1 + 2: x log 2 + 0.32x + 6 _x 4x )

Case 2. 0.75 :::: a 2 :::: 0.85.

..!la-a

K(2- 1/ 2)::::

2

Thens :::: 2- 1/ 2 andinA3.11, 1.2. InthenotationofA2,

•

+ 1.7 ~ a + 1.2)(12: -

a2

: : 2( 1:2

+ 1)(~

)

Q

_ Q).

Using this to bound the last term in A2.1 and A3.5 for the second term, we get

2

J. < -Q 10g(1 - a 2 ) - -1 -a -2 - -1 ( -1 - Q ) log 2

f :: 4"

1.7,K(s)::::

l-a d - x:::: ( ~

]).

4a log ~ + 4(t - a) (log ~ - 1) = By concavity, if a > 0 then 4flog ~ flog -!e + ~ a. Taking a = 2 log ~ and evaluating log -!e on the calculator, we obtain ~ flog 0.32f + log ~; hence, using log x :::: x - I, (A.6) ::::

Lets satisfy !/J(s) = 2

calculator shows (A.5) < 0 when a 2 = 0.85, so it remains to consider (A.6).

Letting x = 1 - a 2 , we get 1 e l l 1 (A.6) = - (log - - 2x + -2 x log(2 - x) + - x log 4 3 2 x

~.

In

-

2

64 1 - a

2

2

2

) log(2(l - a 2 )) -1 log(l - a 2 ) - - 1 -a -2 - -1 ( 1 - - Q 4 64 1 - a 2 2

:::: -0.2x. (A.7)

1 - 4

2

2 I a - -2 64 1 - a

< -log(l-a) - -

log 2 2

+ --(2a -1) 4

=",===.=."~.,,,=

....... "'-~~""-...,."

•..

--""""""""'''''-~''''-''''''-''''''''~~~'''''~'='''''~' =''''.~-''''''''='.-,~=.====~=..,..

~""""",.",,,,,,,",,,~=-,,,=,-=-==--,,,==,,,"=,==,,,,

.'~ "".--

- •..'

376

--"--''';';;'~~.;;;:~~~~~j.''=''''''''''';;'-~'''';;;'''=:ii''~=:;;;;;:.&i~;;;;';..•;,;

2

2 1 I + a ) +2(J2-I)12-3/4(2-d)(4a2(1_a2»-1/4a2. IoU[::S g(1-a )10g ( l-a 2

::s

2

ou -

8

1 - a2

-

2

+ -3

16

a

2

+

2 log 2

(2a - 1 ) - ­ 4

Estimating

4

lin

::s

Q) log (

i, Q ::s !'

a2

2

a d 4(1 - a2)O _

Q)

) .

2

In

(a d)

8

log

I +a I - a

2

--2

I a 2

--2

+

I-a

I, and A3.9 shows

I I [ 7a 2 ]

+ - 10g(1 - a 2 ) + - log 2 + 0.100 2 4 8 8(1 - a )

0.175, so using

2

+ 0.175a . < I, so if 1{f(s) >

1{f(2-1/2)

»

I < - 10g(1 - a 2 ) 4

-

I a2 -­ 64 I - a 2

-

2

2

ford -< 1 so , evaluating 2(2 1/ 2 - 1)12-3/ 4 (2 _ 1)(4.1. !)-1/4 < 0.16 , we 8' 8 4 4 get

::s

then s ::: 2- 1/ 2 . So we can bound lin as in the argument leading to (A.7), the only difference being at the last step where log(2(1 - a 2 is now positive so the worst case is Q = We obtain j.

2 Using that (4a (1 - a 2 »-1/4 is largest at a 2 = ~ (and a calculator for the coefficient of d in the last term when a 2 = ~) this is seen to be increasing in d

)

2

- a ) log

!.

.::S ~' so

I +- ) + -I log(l - a 2 ) + -I log _ _ 10g(1 - a 2 ) log ( 8 I - a 2

4 8 I - a2 1 2

3 4 +2(2 / - 1)12- / (2 - d)(4a 2(1 _ a 2»-1/4 a 2.

2

k

v'~-a2

2

a

::s -8I (I

::s !, then v'~-a2 :::

(a d)

- a

-1 log ( -7 ) + 0.16a 2 8 32

0.36, then ~ (J2 - 1)12- 3 / 4 (4a 2 (1 - a 2 »-1/4 ~' we have as before

Ii < -1 log( 1 - a 2 ) + -I log - In 4 8 1- a2

::s -I (I

»+

::s a 2 ::s 0.5.

as in case 1, then rearranging some terms,

2 2

This is increasing in Q since d

I

+ -I log(4a 2 (1 - a 2 8

·--ilj;··!¥~··'jtr1.~-ijt·'ii'''~~;i;

377

=

::s

2 - d

If a 2

::s -I

-.-.• --•• - '"

- '~'7 'Jm!~I~wi1-:#it;:Pⅈ'ril&-"it"i_TaE!iij;j;"jj;;?

I I 2 2 2

I < - - - - (2a - I) + 0.16a 10 8

lout

],. < -I log( 1 - a 2 ) + -I (-I -

I

.~-.,<, :'ijj'"

2

Case 4. 0.36

we find that

-

·'···"'::,r' 'ii'5if"'"'i~:

< O.

where the last line is easily checked since the preceding line is a quadratic polynomial in a 2 •

In

{'4i!~

using log x ::s x - I and evaluating log -lz. The function (1 - a 2 ) log( :~:~ ) is concave down in a 2 and so has a unique maximum. Checking values at a 2 =

0.55,0.56,0.57, we see that the maximum occurs between 0.55 and 0.57. At the maximum0.43 < l-a 2 < 0.45, I +a 2 < 1.57, so the value is < 0.073 < 0.08, and

If a 2

!.

'..'"

1 +a ) - -I (2a 2 - I) 2 - 0.18 + 0.100 2

- a 2 ) log ( I - a2 8

4

<0

::s

':'-i~r.'·M

log 2

1)­

3 4 I 2 < - - a - - a 8 32

Subcase B. Q

IF-

k

•

16

2 I +a ) + -I 10g(1 - a 2) - - I -a I < -I (I - a 2 ) log ( + -3 a 2

- 8 I - a2 4 64 I - a 2 16

+ (2a 2 -

::s -8I (1

~

1 +a ) + -3 a 2

< -I (I - a 2 ) log ( -

], t

··--'ri'·*'~:J"'=:i"·_-i~!'.

-I (1 - a 2 ) log ( + 8 I - a

[!'

1)12- 3/ 4 (4a 2 (1_ a 2 »-1/4

....

2

I -2 a ) -

where the last line follows by concavity of the logarithm function on the interval I]. Also

(J2 -

'~',~':'~

COUNTEREXAMPLES WITH HARMONIC GRAD1ENTS IN IR3

CHAPTER 15

By A3.6, 2 - d ::s ~. By calculator ~ when a 2 = 0.75, so

"'liI'i'-~

3 4 I a 2

I < --a - - - -2 +0.175a. 8 64 I - a lfa 2

:::

0.4, this implies I

::s (- ~ (0.4) -

(64)~O.6) + 0.175)a 2 < O.

2

If 0.36 ::s a 2 ::s 0.4. we modify as follows: If a 2 ::s 0.4, then v',;;a ::: 1.22, and if r = 0.8, then 1{f(r) < 1.19 by A3.9. So if a 2 < 0.4, then Q > (O.~)2 = 0.32, Q2 ::: 0.0256, and we can improve the second term in the bound for lin in obtaining

!

2

3 4 a 2

I < --a - 0.0256--2 + 0 175a 8 I - a •

378

CHAPTER 15

3

0.0256 0.64

8" (0.36) -

5 (-

+ 0.175 ) a 2

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN

(b) If

4 log Ca + Ca

> O. then

],. - -1 10g(1 - ( In 4

=0.

2

)

d -_ !4 ) ( -1 log C < ( _

Aa

-

Now we make estimates that will be used for all the remaining cases.

We always assume a 2 5 0.36. Then .J~-a2 2: A3.9 shows 1fr(0.83) 5 1.30 < and in A3.11, 1(0.83) 5 ~ and K(0.83) 5 l. It follows that if a 2 5 0.36, then

1.

1

I --a+ 3 ~ +1 ) ( --Q. I ) 2 2

d-x~ (

We set A a

=

2

l-a --;:;r-

+

a

"23

a

2

~ +

-a-

2

I and Ca

=

2

a A A2 .1 then Imp . I'les ~'

2

where we know x 2: max(Q2, d - A a ( ~ Claim. ~ (~ - Q) log Ca

-

Proof The critical point is Q

Q)).

= QcritZ(a 2) = -

=

~'

Aa (

4-

lin -

-I

4

log(l - a 2) 5 0::;

~

max -Q::;(d-

4-

~ )/(A.-I)

Q it implies

4-

1 < max (0::; ~ -Q::;(d- ~ )/(A.-I) 2 I

-4

Q 5 ~-JI' so

.

I

"2 log Ca

+ Ca

< 0, then

Case 5. 0.2 ::: a 2

- Q) (~ log Ca + Ca)

:::

.

A I_I

(Ca +

4log Cal is increasing in

log ( -I ( y2 9

+ -I y + I )) • 4

a2

¢(a)

0.36

50.223

0.27

50.152

0.2

50.088

0.13

2:0

0.12

:::0

0.36.

Subcase A. 0.27 ::: a 2

a2

1· -

--d I - a2 '

In

2

lin -

1:~2

In particular, ¢(a) ::: 0.223 < 0.25 for all a,.which finishes Lemma A4.l.

LemmaA4.1.

If

) - I a­ 6 I_ a2 .

-

and now we have the following:

(a)

=

= -9yI (i + 4-1 y + I) + -2yI

-1(1- - Q) log Ca

2 2

a

+ Ca

~ ; then

it + 4. •

- 4~ ~ (d - A(~2 Q)) I - a 2

2

( -10 I 2 g Ca

.,

...., l-a 2

we can restrict to the case where Q2 5 d ­

Q). Writing this as a quadraticfor

a

2d 4II og(l - a 2 ) 5 - 4l a l-a2 5 - (;l a l-a 2

'

•

We use the calculator to make the following table.

...., l-a 2

Then QcritZ = 4 / 4 • This is decreasing in t as long as ~ + ~ t 2 < 1, as will be the case since a 2 < 0.36. So QcritZ 2: Qcrid0.36) > lin

2

2

- -1 -a - d 4 I - a2

which is increasing in y since y 2: 1 and the logarithm is negative.

- log( 1 + 1 t+ 1 t 2 )

So in using (A.8) to bound

=1+ ~

¢(a)

(l:~2) log Ca' Let t

I

+ Ca )

Proof We have everything but the second inequality in (b), and that will follow if we know that the quantity on the right side of the preceding inequality is decreasing as a function of d. For this we show

Proof Let y

~ 1~:2 Q2 is increasing in Q on [0, ~].

-

-

5 < - . -, .

Lemma A4.2. Thefunction ¢(a) a.

],. - -I 10g(1 - a 2) < -I ( -I- ) Q log C - -I -a- x In 4 - 2 2 a 4 I _ a2 '

(A.S)

379

]R3

I

:::

0.36. By A4.1 (b), A4.2, and ¢(,J0.36) ::: 0.223,

2

I) -a ­

( (5)(0.223) - 12 6

- 10g(1 - a ) <

4

2

I - a2

< (5)(0.223) _

~) ~

-

6

12

I 2 <--a

to

0.73

•

380

CHAPTER 15

When a = 0.27, ~ (..;2 way we get 2

- ( 2»-1/4 ~ 0.182, and in the usual

1) 12- 3 / 4 (4a 2 (1

2

lout ~ 8"1 (I _( 2) log ( 1I + _ (a 2 ) 3 4 I <- --a 8

+ 0 . 182a 2 -

+ 0.182a 2

1 2 10a

The last line was obtained as follows: The integrand on the previous line was increasing in a-this uses a 2 ~ 0.2 which implies 2a 2 - 1 + a 2(~ + ~) < 0­ so we could replace a 2 by 0.2. We then changed variables r --+ I/r and used A3.I. If the (last) integrand is nonzero, then 2

fz -

Subcase B. 0.2 ~ a 2 ~ 0.27. Exactly the same argument, but using ¢ (JO.27) ~ 0.152 and ~ 1)12- 3/ 4(4(0.20)(0.80»-1/4 ~ 0.192, gives

(v'2 -

4:1 Iog(l

6 r

2

1/t(r) ::: 2" - 2r - 4.

(A.9)

< O.

lin -

- (

2)

~

Since 1/t is increasing and 2r 2 - 4 decreasing, the set where (A.9) happens is an interval [R, I). By A3.9 and calculator, R ::: 0.87. So V ~ IT 3/ 4 (4a 2 (l - ( 2»-1/4 a 2

1

1

~ IT 3/ 4 (4a 2(1 _ ( 2»-1/4 a 2(1

-

8

~

3 - -8 a 4

4:

+ 0 . 192a 2 -

1

I 2 -a

1

8

0.87

I -1/t(r)2dr ~

r

1 1

1 -1/t(r)2dr -

2)

~

(

~ -0.147a

lout

~ ~8 (l -

(

2

I)

(5)(0.088) 12

6

1og (I +-a ) 1 - a2

V ~ 12- 3/4(4a 2(1 - ( 2»-1/4 a 2

+ 2r 2 +

4J + drr .

rdr

(0.87)2 2

3 I < -0.I47a 2 - _a 4

-

2

a

0.88

8

+ 0.14a 2

< O. Case 7. ¢(a) < O.

+ V'

1

00

where by Lemma A2.2, I

Jl +

r­ 2

1

1°

(I

Then a 2 < 0.13. By Lemma AA.I,

[1/t(r)2

j. 10

:2((2a2-1+a2C~ + r~))Lrdr

~ 12- 3/ 4 (4a 2 (l - ( 2»-1/4 a 2

r

2

1

+

0 87 .

1 -1/t(r)2dr.

If we use a 2 ::: 0.12 to estimate (4a 2(1 - ( 2»-1/4, the calculator now gives V ~ 0.I386a 2 ~ 0.I4a 2 and, therefore,

2

)

1 1

oro

4

0.12 and ¢ (a) ~ 0.088. We have

Iog(l - (

1

+ (0.87)2)-1/2

Now by (A3.5), (A3.I), (A3.2), (A3.6)

Case 6. a 2 ~ 0.2, but ¢ (a) ::: O.

lin -

r

0.87

< ­ - 3

1

1

+ r 2)-1/2 -1/t(r)2dr

2

< O.

:::

(1

0.87

2

6 0.8 I

I

I) a

(5)(0.152)

(

< --a

By the table, a 2

381

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN JR3

1 < - Iog( I - ( 4

2) -

1 a 2

- --. 6 1 - a2

We estimate lout using the first alternative in A2.2 and dropping the term, i.e.,

6

+ r 2)-1/2[1/t(r)2 _ _2

lout ~ -I (l - ( 8

r

V

~ ~ a2 4

2

2

)

I + ( ) log ( -2 I - a

11{00[1/t(r)2 + a~2 (2a

2 - 1 + a 2(

+V

~2 + ~4 ))J r

r

+

rdr

0+:22)r 2

382

COUNTEREXAMPLES WITH HARMONIC GRADIENTS IN 1R 3

CHAPTER 15

[1

1 a2

Jo

= 4:

2 [t/r Cr)2+ a2(

2a 2

I

r 2-

1 2 +a (2+ r2 »))L-;dr.

Nowa 2 < 0.13 implies that

2 2" a

(2a

2 -

2

1

r

+ a 2C2 +

r

2

)

)

2

< 2" C5a 2 - 1) < -5.38

a

and

1 V::s -a 4

21 [t/rCr) 2-5.38]+-. 1

dr

r

0

Let s satisfy t/rCs)2 = 5.38. By A3.9, s > 0.96 so

[1

1 V ::s 4:a2CO.96)-2 s Ct/rCr)2 - 5.38)rdr

::s

2 4:1 a CO.96)-2 (32" C5.38) 1/2 + 1) 2"1 (l - s2)

by A3.II and the estimates J V

::s

::s

i, K

::s

1 that we have been using. So

2 4:1 a CO.96)-2 (32" C5.38)1/2

+

1) 2"1 (l - CO.96)2)

::s 0.05a 2. We conclude that

1

J < -

6"

a2 1 _ a2

3 4 Sa + 0.05a 2

< 0,

and no more cases remain. University of California, Berkeley

REFERENCES [I] A. B. Aleksandrov. "Inner functions: results, methods, problems." ICM Proceedings.

Berkeley, 1986.

[2] A. B. Aleksandrov, P. Kargaev. Letter to author, 1988. To appear. [3] 1. Bourgain. "On the Hausdorffdimension ofharmonic measure in higher dimensions."

Inv. Math. 87 (1987),477-483.

[4] J. Bourgain, T. Wolff. "Note on gradients of harmonic functions in dimension ~ 3."

Colloq. Math. 40/41, I (1990), 253-260.

[5] A. P. Calderon, A. Zygmund. "On higher gradients of harmonic functions." Studia Math. 24 (1%4), 211-226.

•

383

[6] L. Carleson. "On the support of harmonic measures for sets of Cantor type." Ann. Acad. Sci. Fenn. 10 (1985),113-123. [7] B. Dahlberg. "Estimates of harmonic measure." Arch. Rat. Mech. Anal. 65 (1977), 275-288. [8] S. Gleason, T. Wolff. "Lewy's harmonic gradient maps in higher dimension." Comm. Part. Diff. Eq. 16 (1991),1925-1968. [9] D. Jerison, C. Kenig. "Boundary behavior of harmonic functions in nontangentially . accessible domains." Adv. Math. 46 (1982), 80-147. [10] . "Boundary value problems on Lipschitz domains." In Studies in Partial Differential Equations, edited by W. Littman. MAA Studies in Mathematics, Vol. 23, 1982. [11] P. Jones. "A geometric localization theorem." Adv. Math. 46 (1982), 71-79. [12] P. Jones, T. Wolff. "Hausdorff dimension of harmonic measures in the plane." Acta Math. 161 (1988), 131-144. [13] R. Kaufman, J.-M. G. Wu. "On the snowflake domain." Arkiv for Math. 23 (1985), 177-183. [14] J. L. Lewis. "Note on a theorem of Wolff." In Holomorphic Functions and Moduli, edited by D. Drasin, et al. MSRI Publication, Vol. 10-11. Springer Verlag, 1988. [15] N. G. Makarov. "Distortion of boundary sets under conformal mappings." Proc. London Math. Soc. 51 (1985), 369-384. [16] N. Martin, J. England. "Mathematical Theory of Entropy." Encyclopedia of Mathe­ matics and its Applications, Vol. 12. Addison-Wesley, 1981. [17] V. G. Maz'ja, V. P. Havin. "The Cauchy problem for Laplace's equation." Vestnik Leningrad Univ. 23, 7 (1968),146-147. [18] S. N. Mergelyan. "Harmonic approximation and approximate solution of the Cauchy problem for Laplace's equation." (In Russian.) Uspekhi Mat. Nauk (N. S.) 11 (1956), no. 5 (71), 3-26. [19] B. Oksendal. "Null sets for measures orthogonal to R(X)." Amer. J. Math. 94 (1972), 331-332. [20] . "Brownian motion and sets of harmonic measure zero." Pacific J. Math. 95 (1981),179-192. [21] N. V. Rao. "Uniqueness theorems forharmonicfunctions." Math. Notes USSR 3 (1968), 159-162. [22] Ya. G. Sinai. Introduction to Ergodic Theory. Translated by V. Scheffer. Princeton University Press, 1976. [23] M. Spivak. "A Comprehensive Introduction to Differential Geometry." Vol. 4, Publish or Perish, Berkeley, 1979. [24] E. M. Stein. Singular Integrals and Differentiability Properties ofFunctions. Princeton University Press, 1970. [25] E. M. Stein, G. Weiss. "On the theory of harmonic functions of several variables, I: the theory of HP spaces." Acta Math. 103 (1960), 25~2. [26] A. Uchiyama. "A constructive proof of the Fefferman-Stein decomposition of BMO(Rn)." Acta Math. 148 (1982), 215-241. [27] . "The singular integral characterization of HP on simple martingales." Proc. Amer. Math. Soc. 88 (1983), 617~21. [28] . "The Fefferman-Stein decomposition of smooth functions and its applica­ tions to HP(Rn)." Pacific J. Math. 115 (1984), 217-255.

~":~~~;'''.'~~",-",==~~'''_'''''C'''''''-'''-,,"~'';;'''~~-=-~-_

384

:=-__-_-:-::=~~~~~_=_".,:_~._=::::::::=__=:::.:::_ ..,.~.-.:='O;~.0,,~;;:;;..;.,,-

..

'_·"'''C';;;·'="=-':-'i'''''c'''''~0'',~~~~~..,'-H,:i'''''""",~"",~·"·,_ -·,,,~~-,...,~~:~._~,,"·:v,.

CHAPTER 15

[29] G. Verchota. "Layer potentials and regularity for the Dirichlet problem for Laplace's

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,

.

.'' ',","'_:''_.'' ,

C"'-~"'-~-":;;-'-~"'""'-"'"'"_'''''_-''='

__·" - ..

;'-'""'.c-,-''''' • _ _ "'''''"~:""_._'"'':'''''~'".rc.,,,-

__ -"',--

""c~.

._-_._'-.,.~-~""---