Embedding and Multiplier Theorems for Hp (R^n)

in §5 show that the weights

,

f

Y(p)

s'

in

agree.

p<1 •

Then

Y(p) c Hp

f E Y(p)

n L1

and

when

f

CX>n(;- 1),

1

N = n(p - 1) •

1+ jkj

For

.!!!!!_

satisfies so this

The examples

f1

in (1.12) cannot be replaced by

essentially smaller ones. Theorem lb becomes false when

*

consider a smaller space

Definition

*

f E Y

if

Y

p

=1

•

•

f E L' ,

Jf

=0

,

(1.14)

THEOREM lc.

y

*c

H

1

.!!!!!

In place of

II fll H1 $

c

II fll y*

and

Y(l)

we have to

BAERNSTEIN AND SAWYER

10

This is the circle.

H1 (Rn)

version of Zygmund 's

theorem on the

We are surprised that no one seems to have discovered it before.

It includes the Taibleson-Weiss theorem for q>l

L log L

and

1 <X> n(l - -) • q

The examples

p

=1

in § 5

,

since

a

K' q

show that

*

q c Y

log+

for

lxl

cannot be replaced by any essentially smaller weight, and the examples show that

L log L

f3

cannot be replaced by any essentially weaker integrability

condition. Theorems la,b are about functions with possible non-integrable singularity at the origin. hold if the rings

~

By translation invariance, the conclusions still

are replaced by rings

I

2k ::; x- x0

I ::; 2k+ 1

•

would be interesting to find embedding theorems in which the one point singular set

!x0 !

is replaced by a substantially larger one.

It

2.

FOURIER EMBEDDING

In this section we use the embedding theorems of §1 to prove sharp

s.

analogues of Lip a

Bernstein's

Hp

theorem which asserts that functions of class

on the circle have absolutely convergent Fourier series provided

1

a>z The Lipschitz or Besov spaces be the set of all

F E La(Rn)

B~'b are defined in Stein's book (S] to

for which the norm or quasi-norm

\IF\1 a b

(2. 1)

Ba '

is finite.

Here

than a ,

and

l_sa_sao , u(x, t)

a2:0 ,

s

denotes an integer larger

is the Poisson extension of

Obvious modifications are made when of the

O
a

= ao

or

b

f

= ao

to

Rnx [O,ao) •

A brief account

B spaces appear in [S, Chapter 5] where they are denoted

a b Aa' •

More thorough treatments have been given by Taibleson [T] who denotes them A(a;a,b),

and by Peetre [P].

Stein and Taibleson consider only

but the results of interest to us are still valid for Let

~

O
denote the Fourier transformation.

TAIBLESON 1 S THEOREM ( T, III] • LP,

b2:1,

~

maps

1 1 Bn(-p - ~),p ,e. 2

continuously

~

O
as we 11 when

0 < p~2 •

lbe case

l~p~2

,

but his argument works just

p = 1 may be regarded as an n-dimensional

non-periodic analogue of Bernstein's theorem.

Lp

theorems of this type in

the periodic case had been proved earlier by Szasz [Sz] and Minakshisundaram and Szasz [MS] • 11

12

BAERNSTEIN AND SAWYER

Recall the spaces

A more precise version of

introduced in § 1 •

Taibleson's theorem is

(2.2)

To see that (2.2) implies Taibleson's theorem one uses the inclusion 1

1 2

n(-- -),p

K2 p

c Lp

of the K-spaces.

0 < p :=; 2 ,

which is easily proved from the definition

For the reader's convenience we will indicate at the end

of this section how (2. 2) follows from the definition (2. 1) • The ''homogeneous" version of (2.2), due to Herz [H) •

This. result furnishes the motivation for Peetre 1 s defini-

tion of the Besov spaces in [P). Returning to the non-homogeneous case, by (1.6) we have 1

1

n(-- z),p K2 P

(2.3)

1

n(p- l),p c K1

O
From (1.8) if follows that

1

1

n(p- z),p

Hence (2.2) and (2.3) imply that 1 N = [n(P - 1)] ,

B2

N

c c

,

where

and it makes sense to define 1

1

\ F E B2 P

2

n(-- -),p

(2.4)

THEOREM 2a.

.li

1 n(p - 1)

continuously ~ Hp •

f.

z,

: (Dt3F)(O)

O
0 '

0

s

It3l ::; Nl •

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn) PROOF.

This follows at once from (2.2), (2.3),

13

and Theorem la.

The examples following the statement of Theorem la show that 1

1 X n(-p - -2)' p

cannot be replaced by any larger

2

1

xn(p -

1 z),q

with

2 1

n(p - 1) E small

q

q> p , 1

xn(p-

z

and the examples

1 z),q

2

X space, for instance f1

in §5 show that when

does not transform into

Hp ,

no

matter how

is.

Suppose next that

1

0:> n(p- -

1 z) '

Then (1.6) and (1.4) show that 1

n

- l},p Ko:,q c Ko:- 2• q c Kn(-p 1 2 1

(2. 5}

and we may define

THEOREM 2b.

0
If

B~'q

as in (2.4} with

0 < q ~"'

~

O:>

in place of

n(~

-

t>

then :J

maps

X~'q

continuously 1E!2 Hp •

PROOF. If

When

1

z

n(P- 1) l

this follows from (2.2), (2.5) and Theorem 2a.

n(~ - 1) E z then K~'q c Y(p} for

p

<1

'

Ko:' q c y* 2

for

p=1 ,

so the result follows from (2.2) and Theorems lb,c. A particularly interesting case of Theorem 2b is obtained when q = 2 • From (2.2) it follows that

Moreover, if JIF
0:>

1

n(p -

z)l

and

0: 2

F E B2'

then the condition

<"" implies F E x~• 2 , Consequently, we obtain the

14

BAERNSTEIN AND SAWYER

following simple sufficient condition for

COROLLARY 1 •

F E Hp

Then

1

Suppose that

1

to belong to

O
CX>n(jj-2),

llFII

and

F

Hp •

and that

p ~ C ta(F) • H

It is also possible to prove an homogeneous Besov spaces analogue of (2.2) for to the

Ba, a b ,

B and

HP

Fourier embedding theorem for the

a

denoted

K •

Aa, b by Herz, who proved the

Johnson [JO 1] gave an alternate approach

B spaces and Flett [F] gave another proof of Herz's theorem.

Janson [Ja] has recently written an interesting paper on this subject. Suppose that

0 < p< 1

and

eZ

n(p1 - 1)

1

•

1

B'n(-P--2),p 2

The elements of

are equivalence classes of tempered distributions modulo polynomials of degree

~

N •

[H, p. 289] •

By [H, p. 313] we have

and the representing distributions in this last space are in fact functions whose derivatives of order order

1 n(p- - 1) - N •

a unique representative

N satisfy a Lipschitz condition of 1

1

So, each equivalence class in

G E eN

CN

which satisfies

•n(--

B2 P

2),p

contains

(D~G)(O) = 0 ,

It follows from Taylor's integral formula that

(2.6)

1

THEOREM 2c.

ll

1 1 Bn(p - 2), p

contains .! unique function

2

0 < p < 1, ,!.!!!!

n(p - 1)

~

G

z

then each equivalence sl!!!!,

.!2!: .!h!.£h G E Hp

•

Moreover

i!l

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(R 0 )

nan HP s c ncn "n(-1 1 -) B

P

2

PROOF.

Consider the function

G

p

2 '

satisfying (2.6).

Bernstein-type theorem [H, Theorem 1], 1 1 tence of g E Kn(p- ~),p for which

15

Then G E

or [F, p. 547],

s' .

Herz's

asserts the exis-

2

(2. 7)

for every

gl E s I

~

E S which vanishes in a neighborhood of the origin.

by

(2.8)

(gl' cp)

g 1 E Hp • S1

=

Jg(cp- PtfP) '

1 1 1 •n(-- !) p "n(-- l),p K2 P ' c K1 P

Since

of

cp E

by (1.6),

s .

Theorem la shows that

G= g 1 ,

Theorem 2c will follow once we show that

(g 1 , cp)

= (G,

E S which vanishes in a neighborhood of the origin.

support at the origin, and Since

[ TW, does

as elements

•

From (2.7) and (2.8) it follows that cp

Define

g 1 E Hp ,

p. 105] • P •

is a polynomial

for every

Hence

..,

g 1 - G has

p

we have

By (2. 6) ,

Since

cp)

1 n(p - 1)

G also satisfies this bound, and hence so

f.

Z ,

we

must have

p =0 '

and hence

g1

=G ,

as required, The reader will note that Theorem 2c generalizes Theorem 2a, since it 1

implies that i f F

is any function in

has distributional Fourier transform in

1

B~(p- z),p Hp ,

where

then PI!

G

=F

- PI!

is the N'th

16

BAERNSTEIN AND SAWYER

Taylor polynomial of

F

at

x

=0

•

We have been unable to find a substitute result for Theorem 2c when 1

n(p - 1)

is an integer.

PROOF OF (2.2).

Suppose that

F E B~,q •

Write

f =F •

By the definition

(2.1) and Parseval's formula, (2.2) is equivalent to proving that the expression

(2.9)

llfll2

+It'

tsq-aq-1 dt(J n1x12s lf<x>l2 e-tjxj dx)q/211/q

0

is equivalent to

R

11 fll

a

K ,q 2

.

According to [F, Theorem 1], for h: (O,m) .. [O,m)

O
O<~<m

and

we have

Apply this result with

I f( x ) 12

h(r) = rn- 1 + 2s Jr

d a(x ) ,

lxl = r

k = q/2 ,

~

= 2s - 2a ,

da

s

normalized surface measure •

The second term in (2.9) is easily seen to be equivalent to

tr 0

<J 2 t lf<x>l 2 lxl 2 s dx)q/ 2 tqa.-qs-ldt} l/q t~lxJ~2t

""'1r 0

<S 2 t

t~lxl~2t

1f<x>1 2 dx)q/ 2 tqa.-l dt1 11q,

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn) and this last expression is equivalent to

as required.

17

3.

f E HP

If p I~ /r<s>

then m f

c IIfl!

is a continuous function satisfying

1

1 P I~~"., I n [ TW, p. 105) • H

defines an element of

is well defined from Hp

f

then

HP

-t

if this mapping takes The function

where

s

MULTIPLIERS

s'

S' ,

and thus the mapping

We say that

Hp

-t

(m

'f)¥

m is a Fourier multiplier of

continuously into

m is said to satisfy a

f

Hp •

~ormander

condition of order

s ,

is an integer, if

O
(3. 1)

for all

1~1 ~ s •

Fix a function

0~1J~l,

Define, for

..,

1] E

c0

1

on

1]=

n

(R )

with

1/2~1~;1 ~2'

supp 1J c ll/45\sl ~41 •

6>0 ,

In §1 we introduced certain spaces

K and in

§2 the Besov spaces

B

It is easy to check, via Plancherel's theorem, that (3.1) is equivalent to

(3.2)

18

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn)

19

which, by (2.2), is equivalent to

(3.3)

The following multiplier theorem is due to Calderon and Torchinsky [CT, p. 167]. 9. 39] •

A different proof was given by Taibleson-Weiss [TW, 9.45,

Related results have been obtained by Peral and Torchinsky [ PT] •

CALDERON-TORCHINSKY THEOREM.

sup 5

£!.,

~

1 1 CX> n(P -

z) ,

If

l!m511

a 2 <

CD

O
and i f

1

K2'

equivalently,

m ~~Fourier multiplier

.2f HP ,

It follows in particular that

0 < p ::;1 •

m is a Fourier multiplier of

satisfies a lformander condition of order

s > n(~ -

t>

The case

HP

if

p= 1

m is

due to Fefferman-Stein [FS, p. 150] • We are going to prove a sharp generalization of the Calderon-Torchinsky theorem which corresponds to the endpoint case for multiplier conditions in the same way that Theorem 1 corresponds to the endpoint case for embedding theorems.

1

Note that, by (1.4) and (1.6), we have when CX> n(p -

continuous embeddings

rv

K""'

2

2

- 1) - 1) c Kn(!. P 2 'p c Kn(!. P 'p

2

1

z)1

the

20

BAERNS TEIN AND SAWYER

THEOREM 3a.

Suppose that

0

M=

~

m

< p < 1 and !h!!:_

sup n.n6n
is !. Fourier multiplier of

1

. t he sense t h at Th e t h eorem i s s h arp 1n by any larger space of the K-type.

, .!!:!.!!

- l),p Kn(-p 1

The proof of the theorem is in §6 and the

examples showing sharpness in § 7 •

1

From (2.2) and (1.4)- (1.6) it follows that either

~

= n(lp - !) 2

and

q~p

or

cannot be replaced

~

1

> n(p- -

n(-- l),p

K p 1

z>1

~

::> :Ji(B '

2

q

)

if

Thus we have the

following corollary, which furnishes a practical sufficient condition for multipliers.

COROLLARY 1.

suppose

!!!.!!.

0

.!!!!!. .!:.!!!!. either

sup llm611Kc:,q 6

~ =

1

n(p -

z>1

and

< ""

2.

£!.., equivalently

sup llm611Bo:,q < "" ' 6 2 then

m is !. Fourier multiplier of

Hp • 1

The corollary is sharp in that larger space

K t3,q

2

•

1

- -2), p Kn(2 P

cannot be replaced by any

EMBEDDING AND MULTIPLIER THOEREMS FOR Hp(Rn) When

p= 1

Theorem 3a becomes false.

0 1

The space

K1'

21 L

1

has to be

replaced by slightly smaller K-type spaces with appropriate weights Suppose that

w: IO, 1, 2, ••• } -+ [ 1,..,)

Define the space

f E L1(Rn)

K(w) to be the set of all

llf!IK(w) =

1 ~ w(k) ~ w(k + 1)

satisfies

...

J1 I < 1 1£1

+

E k= 0

X

<J

w(k) •

<.., •

for which the norm

jfj) w(k)

~

A critical role is played by the condition

is finite.

... (3.4)

E w(k) k=O

THEOREM 3b.

If

w satisfies (3.4)

(3.5)

M =

sup 6

~ m is A Fourier multiplier of

-2

<• •

.!!!!.!! g

!lm6 11K(w) < oo ~

H1 ,

00

where

w(k) - 2 ) 112

C = C(n) ( E

--

k=l

In particular, we may take

ex

Since K2'

2

ex-~

c K1

2

&. '

multiplier theorem for

e 1

when

c K1'

p= 1

w(k) = 2 0 <e

ek

,

e

e, 1

>0 •

<ex- 2n ,

Then K(w) = K1

the Calder6n-Torchinsky

follows from Theorem 3b.

Theorem 3b is sharp, at least within the context of reasonable

w(k)'s.

00

We show in §7 that if then there exists

w(k)

f

,

w(2k)

~

C w(k) ,

m satisfying (3.5) and

and if

f E H1

E w(k) - 2 k=O

for which

(m f)¥

=""

E H1

These examples will also show that in the calder6n-Torchinsky theorem K~' 2 cannot be replaced by

no matter how small

q

is.

BAERNSTEIN AND SAWYER

22

If no growth restriction is placed on rather chaotic. that

w(k)

the situation becomes

There exists a positive nondecreasing sequence

~(k)- 2 = ~,

but the only

mE L=(Rn)

w(k)

satisfying (3.5) is

such

m

=0

We will not give this construction, but only point out that the main fact g E L1

used is that if

entire function in Suppose that m E L~ , 1< p< 2 Thus

Lp

w satisfies (3.4) and Thus

2 S p <= •

LP

g(x+ iy)

is an

jz E en: lim zl ~M}

m satisfies (3.5).

Then

m is a Fourier multiplier on H1

is an interpolation space between

m Fourier multiplies

true for

then

en which is bounded on sets

K(w) c L1 •

since

II xl < 1\ ,

supp g c

and

for

l
and

L2

L2

For

([FS, p. 156]) •

and by duality this is also

Thus we arrive at the following refinement of

Kormander's multiplier theorem.

COROLLARY 2.

m .!!!!!, w

If

.!!_.!Fourier multiplier£!!

satisfy~

Lp,

hypotheses of Theorem 3b, then

m

l
HUrmander's theorem, as stated in [S, p. 96, Cor] is implied by the special cases

w(k)

= 2ek

,

sufficiently small.

e>O

As an application of Theorem 3a, we shall use it to prove a recent

theorem of Miyachi [M] about "strongly singular" multipliers. 0 < p< 1 ,

that

m(~)

a> 0 , b > 0

and that

s

is an integer larger than

satisfies

0 ~

(3.6)

m( ~)

for

0

I~

I<

The prototypical example is the function

where

t E C., ,

t

0

on

1~1

<1

,

•

1

1131 ~

s '

1 •

ma,b

on

defined by

~~~ ~2 •

Suppose that

~

Assume

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn) Miyachi proved that (3.6) implies that HP

23

m is a Fourier multiplier of

provided

1 1 n(- - -)

(3. 7)

For

m b a,

with

a

and

b

<-b

2 - a

p

satisfying certain restrictions this had The case

been proved earlier by Sjolin (Sj]

settled by Fefferman-Stein [FS, p. 160] • argument that the results for O
when

1~p<2

n(p -

b

2) > -;; ,

Suppose that

0
does not Fourier multiply

and that (3.6) holds.

Hp

\lmJ 0: u B ,p

sup 6

and write

Set

0:=.2..

a

By

Miyachi's theorem will follow from the estimates

(3.8)

6,

Miyachi shows by an interpolation

0
Corollary 1 and (2.2) ,

Fix

had been

are consequences of those for

His Theorem 3 shows that 1

O
l~p<2,

g(s)

so we assume from now on that

<

m

2

= m6 (s) = m(6s)~(s) 1

6 2:4

•

Then

g

=0

if

1 6~4

Hypothesis (3. 6) implies that

(3.9)

C is independent of

where

Assume temporarily that non-negative integer and

6 • 0:

l

0
z

and write Then

a:

v<s •

=v + a

and

o ~ 1131

where

By [S, p. 153],

equivalent to the estimates

( 3. 10)

,

~ v ,

v

is a

(3.8) is

24

BAERNSTEIN AND SAWYER

(3. 11)

where

h

denotes a partial derivative of

g

of order

~

and

(l'.th) (x) = h(x+ t) - h(x) • Since

g

has support in

21 ~ Ixj

~ 2 , (3.10) follows from (3. 9) •

for (3.11) ,

(3. 12)

and also

(3.13)

For fixed

t,

~h(x)

12

and (3.13) are true with ltl ~ 6-a

since. If

b

a

E

z

norms in place of

measure~

L•

norms.

C.

Hence (3.12)

Using (3.12) for

it follows that the integral in (3.11)

C times

~=-=~+a ~

except on a set of

jtl ~ 6-a,

and (3.13) for

is dominated by

= 0

write

•

This proves Miyachi's theorem when ~

=

~

+ 1 ,

~ ~0

•

~ ~

Z •

Then (3. 8) is equivalent to

(3.10) and

where

(!'.! h)(t;) = h(t; + t) + h(t; - t)

- 2h(t;) •

The estimate (3.14) is

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn)

25

accomplished as above, the only change being that (3.12) is replaced by

The Calderon-Torchinsky theorem implies that an Fourier multiply critical index

p

H

p0 ,

1

for

n(p -

1

n(p- -

1

b

2> < ~ ,

1 2> = ~b ,

0

theorem for the endpoint case

1

but to get boundedness at the

it seems essential to have a multiplier 1

Kn(p- 2),p 2

m satisfying (3.6) will

4.

1 n(p--1)iZ,

Take Vx, t(y) = t

l' -n

fEY(p)

E t(

1

O
Suppose that

C~(Rn) ~

t

) .

PROOF OF THEOREM 1

f

EK~(p--l),p

and

1 N=n(P-1)

and that if

with

p<1

supp 'f c !lxl
if

O
Jt=1.

*

if

and p=l

Set

Define

and

Y*

By the definitions (1.9), (1.10) and those of

Y(p)

we have

Our aim is to show that

with appropriate bounds.

(4. 1)

sup jg(x,t)j E Lp, t>O

We claim that

(4.2)

sup t>O

Ig(x, t) I ;:;

C (J 1 (x)

+

J 2 (x)

+

J 3 (x))

where

sup t

tI

<

t xl

-n

J

\f(y)j dy,

Iy- xI ;:; t

26

27

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn) 1 N = n(p - 1)

For

we also need two other estimates.

1

sup lg(x,t)j ~ C (J 1 (x)+J 4 (x)+J 5 (x)), t>O

(4.3)

N = n(p - 1) ,

p

<1

,

where

J4(x) =

IYI

Js(x) =

J

jf(y)j lxl

dx'

>2jxl

(hl'N

jf(y)j lxl)

I Yl < 2 1xl

Ig(x, t) I ~ C

sup t>O

(4.4)

(l.J)N- 1 lxl -n

J

'

lxl

(J 6 (x) + \If\\ 1 > , L

-n

dx,

p= 1 ,

where

J6(x) t

sup t-n <1

J

lf(y)l dy. jy- xj


The proofs of (4.2) - (4.4) are at the end of the section. Assume now that J 1 E Lp,

i=1,2,3,

Suppose that

1 n(- - 1) > N • p

We shall prove Theorem 1a by showing that

with appropriate bounds.

x E ~ =

Ix

: 2k ~

Ix I ~

Consider first

2 k+ 1 I

, -.. < k < ao

•

(4.5)

we have

r tl

.

J (x)p dx ~ C E m~ 2jNp 2-kp(N+ n) 2kn ~ 2 j = k+ 1 J ,

Then, writing

BAERNSTEIN AND SAWYER

28 ClD

J

I: k

=

Since

-CD

j- 1 ~

J 2 (x)P dx ~ C

I: j

1

n(p - 1)

>N

we

,

= -co

I: k= -..,

mP 2 jNp 2k[n- p(N+ n)] j

deduce

(4. 6)

A completely analogous computation yields

when

(4.7)

To estimate maximal function.

J Jf dx

>

N •

we need some inequalities for the Hardy-Littlewood

FE L1(Rn)

For

1

n(p- 1)

(MF)(x) =

sup xEB

write

1

B

J

IFI dx ,

I I B

B denotes a ball.

where

Suppose

LEMMA.

1

n

F E L (R )

.!!!!! ~ F .!!. supported .!!! .! ball B • Then O
l!22!· when

A homogeneity argument shows that it suffices to consider the case

J

IFI

=1

and

B

weak

1- 1 estimate

IBI

=1

•

Then the inequality in (a) follows from the

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn)

~

I jx ERn: MF(x) ;:::: et!l

CCt-l

j'

29

IFI dx

B

and (b) follows from the sharper estimate

See [S, p. 23] • Fix

k E

F(x) = 0

z

and define

otherwise.

Then

F(x) = f(x)

if

J 1 (x) ~ C MF(x)

-

xE~=jx:2

for

x E

~

k-1

•


For

k+2

l,

0
the lemma gives

< C2nk(l- p)(Jl -

k-

1

+ uf + uf k

k+

1

) •

Hence

(4.8)

Theorem la now follows from (4.6), (4.7) and (4.8). Assume next that lxl

<1

and (4.3) for

0
and that

1

n(p -

1) = N •

We use (4.2) for

lxl < 1 •

Thus

+ j' lxl > 1

J:

dx

+ j' jxl

<1

J;

dx) •

BAERNSTEIN AND SAWYER

30

Now (4.8) is still valid, so the first term on the right is suitably bounded.

(4.10)

Next,

J

J~ dx:=;c lxj>l

"'

E

J

k=O

~

J~ dx:=;c

"'

m~2jNp2k(n- p(N+n)]

110

r:

E

k=O j=k+l

j-1 . 110 E E mp 2JNp = C E j m~ 2jNp < C j=l k=O j j=l J a>

= C

J

I! fliP

•

Y(p)

Similarly,

(4.11)

J

Ixl > 1

J~ dx :=; C

= C(

Also, when

"' k+ 2 E E mp 2j(N+ l)p 2k(n- (n+N+ l)p] k = 0 j = -110 j

1 110 110 E E + E j = -= k = 0 j = 2 k

a>

m~ 2j(N+ l)p 2-kp)

E

=j

- 2

J

1 n(p - 1) = N ,

(4. 12)

(4.13)

Theorem 4b follows from (4.8)- (4.13) • Now assume

p= 1 •

In proving Theorem lc we may assume that

1\fll

1 = 1 . L

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn)

31

By (4.2) and (4.4) we have

(4.14)

computations like the ones above give

(4.15)

co

J

(4. 16)

Ixl > 1

J

J 2 dx~C

=1

Ixl > 1

lf(x)l(l+loglxl)dx~cl!fll*' y

0

Ixl > 1

J 3 dx

• CD m. 2J + E m.) j = -.., J j = 1 J

for

~ CJn

~ c ( E

Next, define again F(x) = 0

jmj~cj

E j

x f.~ ,

=

F(x) where

x E

f(x)

for

k~O

is fixed.

By part (b) of the lenuna, we have, recalling

~

R

=

I £1

~ C llfll *

•

Y

{2k-l~lxl ~2k+ 2 1

Then

J 1 ~ C MF

on

~

llfll 1 = 1 , L

s1

Let for

k E

s1

=

{k~O

~ e

-k

~

I , s2

= {k~O

:J_ 1£1 <e -k I ~

,

J ~

Since

:J_ 1£1

X

1

J 1 dx

log;~ X

1/2

~

C

J_ I fj ( 1 + log+ I fl + k) dx ~

for

0<x<1 ,

we have

•

.

Then,

32

BAERNSTEIN AND SAWYER

<J_

lfl dx)log r

.,~

-\

So, for

iI

~

f dx

e-k/ 2 '

k E s2

if

•

k E s2 ,

Adding up, and using

\lf\1 1

=1

we find

,

L

(4.17)

Finally, to estimate if

lxl ;:::2 •

Then

J6

we let

J 6 (x) ~ C MF(x)

F(x) for

= f(x)

lxl

<1

Ixl

if

<2

,

F(x)

=0

and, by part (b) of the

LeDIIla,

J

J 6 dx ~

lxl <1

cJ

(MF) dx ~ C lxl <1

where the last estimate

r +( IF(x) IIC I \]) IF ( x) IL 1 + log J

J lxl <2

uses

1 = \lf\1 1

:5

\lf\1

lxl <2 F.

* .

y

L

Hence

(4. 18)

J

lxl

<1

(J 6 (x)

+

1) dx

:5

C \lf\1

*. y

dx

EMBEDDING AND MULTIPLIER THEOREMS FOR Bp(F 0 ) Theorem lc follows from (4.14)-

{4.

33

18) •

I t remains to prove (4.2)- (4.4).

Write

Then

g(x,t) = Jf(y) R(x,y,t) dy •

We need the following estimates for

f

(4.19)

If

t

<

(4.20)

If

t

> zlxl

( 4. 21)

If

t

>

1

Recall that

f

lxl

N

E

1Yl <4t ,

and

I Yl 2:_4t

IRI

then

then

IRI

~Ct-n(¥)N+ 1

~Ct-n(¥)N

•

Suppose for simplicity that

Then

1 J.

Since

IRI = lvx,tl

PROOF OF (4.20).

and

aj

Yxz t (O) I Yl j = t

7i'

j=O

PROOF OF (4.19).

I Rl ~Ct -n

Vx,t(y) = t-n v<x:y) •

y =
Hence

then

lxl

R •

-n

oYj 1

supp

N

E

~-12j t-j

j=O

v c
we have

j:

PN vx,t(y) = 0.

S Ct-n.

By Taylor's theorem,

t

-n

BAERNSTEIN AND SAWYER

34

since

Since

PROOFOF(4.21).

PROOF OF (4.2).

supp

Suppose that

we have

;c!lxl
1

t>2 lxl •

If

t

1

< 2 lxl

x,

t(y)

~

0 •

Hence

Using (4.20) and (4.21) we obtain

t

since the second integral is dominated by

•

>

1 21 xj

'

C J 2 (x) •

then (4.19) shows that

which, with the previous estimate, proves (4.2).

PROOF OF (4. 3).

Suppose that

by definition (1.10)

0


,

1 N = n(p - 1) •

If

f E Y(p)

then

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn) and (4.3) follows by replacing

PROOF OF (4.4).

If

*

N

f E Y c L1

N- 1

by

35

in the argument above.

then

g(x, t) "'Jf(y) '+'x t(y) dy ,

' and we obtain

sup t>O

Ig(x, t) I

~ sup

Ig(x, t) I +

t~l

~ sup Ct-n t

<1

"' C(J6(x)

sup

Ig(x, t) I

t~l

+

J I y - xl U£0

lf(y)l dy + C


1) • L

Jn R

lf(y)l dy

5.

BEST POSSIBLE NATURE OF THEOREMS lB AND lC

In §1 we gave simple examples illustrating the sharpness of Theorem la. Here we consider the case when p E (0,1]

and

1

n(p - 1) E Z

Throughout this section

NEZ will be connected by 1 N = n(p - 1) ,

In theorems 4a and 4b

e (k) ,

k 2::.0

stands for an arbitrarily pre-

scribed sequence with

e(k) 2::. e(k+ 1) ,

lim e(k)

0 •

k-+m

THEOREM 4a.

E£!: each N 2::.0

~

satisfying

!!!, .! function

(5. 1)

(5.2)

1 1 E Kn(p- a),q

fl

a

(5.3)

THEOREM 4b.

E£!: each N2::,0

~is.!

36

function

f2

satisfying

37

EMBEDDING AND MULTIPLIER THEOREMS FOR HP(Rn)

-1 I: k= -a>

.ID!!

the distribution defined

THEOREM 4c. L(O) 2:::, 1

!!x

f2

Y,!! (1.10)

Given.! .!!.2.!!-decreasing function

does .!!2!, belong.!£

L(x) ,

x>O ,

Hp •

satisfying

and

L(x) = o(log x) ,

x-t.., ,

satisfying

f 3 (x)

=0

PROOF OF THEOREM 4a.

for

Ixl > 1

Write

,

e 1 (N)

with the following properties:

supp f 0 c

Aa ,

and

(1,0, ••• ,0) ERN

Jf 3 = 0 ,

and choose

f 0 E La>

BAERNSTEIN AND SAWYER

38

where

dcr

denotes surface measure.

For

N =0

the vanishing moment

condition is vacuous. Next, let which

E

S

be an infinite subsequence of

14.(,: .(, E

z,

.t2:;2!

for

e(k)<..,.

kES Define

k Es , 0 ,

otherwise

where

(5.4)

The lacunarity of (5.1) and (5.3). Choose

supp

implies (5.2),

and the reader may easily verify

We shall prove

...

• E .c

t c

S

with the properties

(I xj < 6)

,

t(x)

lxl

1 ,

<5 '

and define

Suppose that

k E S

and

kSJ S2k- 4 •

Then for

x E A.

J

we have

EMBEDDING AND MULTIPLIER THEOREMS FOR HP(Rn) g(x,

I =J IX)

f 1 (y)(-l) N ylN

IX,-N- n

39

dy

~

=

c a:kP

c

where

C>O •

from

p(N+ n) Hence, for

k

2

kn

-1

The last equation follows from (5.4) and the next-to-last one n • k ES ,

J k 2


so that Jlg(x,lxl)lp dx

PROOF OF THEOREM 4b.

Let

=..,

f0

2klg(x,lxl)lp dx ~ C'

and f 1 f. HP

and

S

be as in the previous proof, and

define

( -k)

X

0 ,

otherwise •

Es ,

E Azk '

(-k) E S ,

40

BAERNSTEIN AND SAWYER

The proof that

f2

has the desired properties proceeds as above, the

only change being in the definition of

t

where y=O,

is as before, and

g(x,t) .

denotes

PN _ 1

This time

(N- l)st

Taylor polynomial at

'fx,t(y) = t-n 'f(x~ Y) •

PROO.r' OF TiiEOREM 4c.

Choose numbers

xj > 2

2n

j=l,2,3, .•• ,

,

such that

(5. 5)

and

(5.6)

Define integers

k(j)

by

1 1) 2nk(j) xj 1ogxj E [ 2'

(5. 7)

Then

k(j + 1)

~

g(x) =

k(j) - 2

l

and

if

X

k(l)

~

-3 •

E ~(j) '

. Define

j=l,2, ••• ,

xj '

0 '

otherwise

It follows from (5.5) and (5.7) that

Jg L(g)

Notice that b .:: 1"",

supp b c

g

< ao

•

is supported in

,

and Jb

Let

= J-g

•

If

gl = g+ b '

!ls 1!1 ~

where

1 define

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn)

that

1

supp b 1 c
Take

Jv dx

= 1.

'f ECfZ>

'f~O,

with

Jb 1 =

o,

and

'f=1

on

lxl

Us 1 +


b 1 ll

= 1.

N(x) =

sup

IY- xl

!...:..X.

t) t

-n

suppyc
dy,

lu(y,t)l • $

t

then

I

N(x) ~ u(O, xl) ~ C

Ixl-n J IYI

Write

m.{,

= J

g •

g dy - C • 1
then

A.{.

J

Ixj

<1

N dx ~ C

~

-2 i: .{, = -..,

~

C j

By (5.5) and (5.7) we have

=1

(j-!,1 -

jk(j)j

l)m.{. - C

X.

J

then

1

Define

u(x,t) =Jf 3 (y) y(

41

2nk(j)

- C •

BAERNSTEIN AND SAWYER

42

Since

L(x.) ;:::, 1 , J

it follows that

log x.

J

< C njk(j)j •

Hence

J

... N dx;:::, C

j xj < 1

where we have used (5. 7) • in (FS, p. 151] •

E j

Thus

=1 N

E.

xj(log x.) 2nk(j) = .., , J

1 L ,

and

by Theorem 4

6.

PROOF OF THEOREM 3

Our proof makes use of the atomic decomposition of a

HP •

The function

is said to be a p-atom if

supp a c B ,

for some ball

Jxt3 a(x) dx

0 ,

B

c Rn

(6. 1)

Recall that These are

N

E s'

f

"'

1

when

Hp

if and only if

...

are constants with inf tiAjjP,

the

inf

E IAjlp 1

Hp

s'

<..,

f

for general

proved

admits a representation

where the Moreover,

n ,

aj

II f!l p

Hp

are p-atoms and the is comparable to

being taken over all possible such representations.

To prove that the operator ously into

I t3l :=: N

and Latter [L],

with convergence in

J

:=:

1)] •

n= 1

belongs to

f=I;A.aj,

0

atoms in the notation of [TW].

(p,~,N)

Coifman [ C] , that

1 = [n(p-

for

f-+ (mf)

it suffices to show that

We may assume that the ball

mE 1"'

ll<manl Hp ::: c

takes

HP

continu-

for all p-atoms

B of (6.1) has center at the origin.

a.

Further-

sup !lm0 llx is the same for all dilations of m to 0 obtain Theorems 3a and 3b it will be enough to consider the case when B is more, since the number

jxj

<1 Thus, our problem is reduced to proving that

(6.2)

43

BAERNSTEIN AND SAWYER

44 1

where

n(-- l),p X = K1 P for

0

,

X = K(w)

for

p= 1 ,

and

:r:"'

-·

v<s2-j)

=1

,

t =1

a

is a

"unit p-atom" satisfying

II xl < II ,

supp a c

I a(x) I

(6. 3)

Ja(x)x

c"'0

v E

Take

supp.

c

:;i 1

13 dx = 0 ,

o ::: I sl ::: N

with

1

12 ::i

lsi

::i 21,

•

For instance, we can let

t<s>

where

-t E Then

c..0

,

tlJ=t

supp

t c lz1 ::i

t
Is I ::i 21

0 ::i 'i

,

::i 1

on

and

.., m(s) i(s)

where

i 6 (s)

I: m(S} 11(2-j S) i(t;) tjl(2-j If:)

= !(6t;)

t(t;) •

Write

(m .)"

f

2J

Then

... (6.4)

f(x)

I: 2nj (£ .*b.)(2j x) J

J

=

(mi)" •

15 . 85 ::i Is I ::i B

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(R 0 ) We need some estimates for the

LEMMA

1.

Let

r >0

be given.

bj

Recall that

a

45

satisfies (6.3) •

Then

(a)

if

(b)

The constants

depend only on

C

j~O,

p,n, and r

This lemma, and the next

one, will be proved at the end of the section. To prove Theorem 3a we need another lemma. 1

LEMMA 2.

Suppose 1.h!!,

Q E L\Rn)

... E k

C

,

j ~0 '

r>

n(p - l),p g E K1 ,

n

p'

=j +2

dx .s 1 ,

<J

ls*Qjdx)p2kn(l-p).scllsiiP

1

Kn(p - 1), p 1

Ak

'

depends only £!l r, p, and n •

PROOF OF THEOREM 3a.

(6.5)


satisfies

JIQI

where

0

Assume that

sup 11&6 11

li >0

0

1

•

n(-- l),p Kl p

We may assume that

1 •

~

1.h!!,

BAERNSTEIN AND SAWYER

46 This implies that

11£0 11 1 ~ 1

for every

llmll.., ~ 1

and hence that

& ,

L

and also

II f -II 1 ~ II f -II

(6.6)

J L

<

1

n(-p- l),p-

J

-.. <j<
1 ,

Kl

We shall prove that when

(6.7)

a

satisfies (6.3)

ilfll

< c

1

n(-- l),p-

Kl p

1

Recall that in

Lp ,

tion of

f = (m!)y •

Since

this will give (6.2) with Hp

Kn(p- l),p 1

LP

is continuously included

in place of

HP •

The characteriza-

in terms of iterated Riesz transforms will then enable us to

finish the proof. To prove (6. 7) ,

llmll,. ~ 1 •

where we have used Next, assume

(6.9)

dx

we first observe that

k;:::2 •

~

E

j

=-CD

From (6.4) we have

JA

1f

.*b.l J J

-k+ 1 E

dx =

j=-
+

0 E

j=-k+2

+

E

j=l

k+j

By (6.6) and Le11111a 1 we have j < 0 •

Hence

llfjll 1 L

S

1

and

llbjll 1 :S C 2j(N+ l) L

for

47

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn) -k+l ak ~

E

II fj*bj II L

-CD

1 :S c

CD

"'

CD

aP 2 kn(l- p) < C E 2 -kp(N+ 1) + kn(l- p) :S C , k=2 k k=2 E

( 6. 10)

N+ 1

since

-k+l 2 j(N+ 1) :S C 2 -k(N+ 1) ~

1

> n(p- -

1) •

Next

~

~

E

f3p 2 kn(l-p)< E 2 kn(l-p) k=2 k -k=2

0

a>

E

I:

J =-CD

2kn(l-p)

k=-j+2

<J '\.+j

jf.*b.j dx)p J J

0 E j

=-CD

By lemma 1, the function of le11111a 2 (for

j = 0)

when

c 2

- j(N+ 1)

j :SO

and

c

b.(x) J

satisfies the hypothesis

is small enough.

So, le11111a 2

and (6.6) give

0

E

(6. ll) j

2-nj(l- p) 2jp(N+ 1) :S C

=-CD

Similarly

By lemma 1, small and

j;:::, 0 •

c b /x)

satisfies the hypothesis of Lemma 2 when

Lemna 2 and (6. 6) give

c

is

48

BAERNSTEIN AND SAWYER OD

OD

E y~2kn(l-p):::; E 2 -j(l-p)n :::; k=2 j=l

(6.12)

c

From ( 6. 9) - ( 6. 12) we see that

. E

k=2

<J I fj

dx)P 2kn(l- p) :::; C

~

Which, with (6.8), shows that (6.7) holds. In particular, we have proved that when (6.5) holds

II (mll)"'!l

(6.13)

P :::; C , L

for every p-atom

a .

We want this inequality to hold with Fix a multi-index

t3

and a number

Hp

e>O.

Let

norm instead of

Lp

norm.

R be the function with

Fourier transform

Then

R*g ,

an iterated Riesz

g E L

2

,

tran~form

is a constant multiple of the Poisson integral of of

g •

Write

f = (mil)"

We have

R*f

Write ; (s)

( 6. 14)

= R(s)

m(s) •

= ( (Rm) ll)"'

We claim that

again •

49

EMBEDDING AND MULTIPLIER THEOREMS FOR HP(Rn)

Then, using (6.5) and (6.13) applied to IIR*fll P

:S

C(f3) •

; ,

From the characterization of

it will follow that

Hp

in terms of iterated

L

Riesz transforms [FS, §§8, ll] , it follows from this that

llfll P

:S

as

C.

H

required. To prove (6.14), note that

cp E

where

c~

satisfies

cp= 1

on

i :S lsi :S

4 ,

1

cp= 0

B~

off

ld

~ 8

Thus

Q(t;)

where

independently of cQ

that 6

and

e

R(8~) .

= cp(S) c

p =1

6 •

So, for fixed

y

we have n

r

> P'

satisfies the hypothesis of Lemma 2 (with

\\D Y &II.., ~ C( y, 13) ,

there exists j = 0) ,

c

such

for every

(6.14) now follows from Lemma 2.

PROOF OF THEOREM 3B. Now

and

For multi-indices

We continue with the notation of the preceding proof.

and we shall assume

(6.15)

,

... -2 I: w(k) 1

< .., ,

and normalize by

1 .

Then

The 1-atom

a

is assumed to satisfy (6.3).

as before, so we only need to show that

The estimate (6.8) follows

BAERNSTEIN AND SAWYER

50

dx

( 6. 17)

~

C •

By (6.4) we have

0 E

... E

+

= A(x) +

B(x)

j = 1

j =-<XI

Now

0

J

( 6. 18)

I:

IACx>l dx:S

l!f.*b.l! 1 j = -oo J J L

Rn

0

:sc

j

E

2j(N+ 1)

j + 2

If J*b .1

= -oo

:sc,

by (6.16) and Lemma 1. Next,

(6.19)

J I xl

>4

I B(x) I dx ~

... E j =1

J

I xj > 2

...

...

E j=1

Fix

j

and

k

with

k ::::_ j+2,

dx

J

s

I

E f.*b.l k=j+2 ~ J J

j::::_1 ,

and suppose that

x E

~.

Write

f.*b.(x) = J

J

rIYl < 2k - 1

..

f.(y) b.(x-y) dy + (f.x J

J

J

+J

where

~

= ~ _1

U

~

U

~+

1 •

Take

r

=

~

)*b.(x) J

IYl > 2

k+ 2 f.(y) b.(x-y) dy J J

n+ 1

in Lemma 1 •

Then

,

EMBEDDING AND MULTIPLIER THEOREMS FOR HP(Rn)

51

IY,1>2k+2

•

Hence, using (6.16),

*b.)(x)l l(f.J J

~ cz-k(n+l) + If jX- * ) b (>I jx'

t\

and so

= cz -k

+ c

k+ l t J(.t,,mlb.n 1 .{,=k-1 JL

where

J(k,j)

=J

t\

lf.l J

•

Thus, by (6.19),

J Ixj > 4 Since

w(k)t ,

"'

"'

!B(x)ldx~C+C

I: j

=1

I:

k

=j + 1

J(k,j)llb.ll 1 J L

it follows from (6.16) that

~ 1 I: J(k,j) ~w(j) k=j+l

1 I: w(k) J(k,j) ~w(j), k=j+l m

j~O.

BAERNSTEIN AND SAWYER

52 Hence

.

""

~C+C

1 ) 1/2 (E llb.ll2 )1/2 ( E-j=lw(j) 2 j=l JLl

We claim that

(6.20)

With the previous inequality and (6,18), (6.17), atoms

a

and all

this will give for all

m satisfying (6,15)

(6.21)

The proof of (6,20) is postponed temporarily. H1

go from (6.21) to the Let

R(s) =

2J.. e-elsl

inequality we need. where

e >0

and

Is I

Then

IIR*all 1 ~ C

We will show now how to

j E

P, ... , n}

is fixed.

By the atomic decomposition, there exist 1-atoms

H

and constants

A.j

with

EIA.jl

~c

and

""

R*a = E A.. a. 1 J J

with convergence in

Since

mE 1""

s'

.

Formally, we have

and Fourier transformation is an isomorphism on

s' ,

the

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn) series on the right converges to

R*f

in

But

S' •

53

II (mij)"ll

1

!::

C ,

by

L

(6.21),

L1 •

so in fact the series converges in

... \IR*fll 1 !:: L

Hence

1\f\1 1 !:: C ,

c

E

1

Ift)

Moreover

!:: c

by the Riesz transform characterization of

H1

H

This completes the proof of Theorems Ja and Jb, modulo Lemmas 1 and 2 and (6.20).

PROOF OF ( 6. 20).

where

(6.22)

Also

where

Recall that

supp a c
< 1)

and

I a(x) I !:: 1 •

We have

54

BAERNSTEIN AND SAWYER

Hence

Combined with (6.23), this gives

so that

(6.24)

= C

n

n

E i .. 1

I:

2.t

Jlx.l .t= 0 ].

2

la(x)l

dx~C,

t (x- 2j

y) dy .

which, with (6.22), gives (6.20) •

PROOF OF LEMMA 1 •

We have

(6.25)

(a) Suppose that formula gives

b. (x) = J J

j <0.

IYl

a(y)

<1

Let

r>O

be given.

For fixed

x

Taylor's

EMBEDDING AND MULTIPLIER THEOREMS FOR HP(Rn)

~ (x-

When

IYl


,

E

2j y) =

IPI :SN

the facts that

t

55

ct3 /' + R(y)

has rapid decrease and

j:::! 0

to the estimate

From (6.25) it follows that

Ib.<x> I J

IJ

a(y) R(y) dyl

I Yl

:S C 2j(N+l)(1+lxl)-r,

<1

as required. (b) Suppose that

j

;:::o •

b.(x) = 2-nj J

Since

Ia(x) I :S 1 and

lit II

1

Rewrite (6.25) as

r oJ

IYl

a(2-j y) t<x-y) dy.

< 2j

:S 1 we have

L

suppose that

since

t

1xl

> 2J + 1

has rapid decrease.

Then

lead

56

BAERNS TE IN AND SAWYER

PROOF OF LEMMA 2 •

j ~0 •

Fix

( 6, 26)

We may assume

11811 K n(lp

1 •

1),p

1

k~j + 2

Suppose that

8*Q(x) =

k-2 E

J

x E

and

~

•

Write

Q(x- y) 8(Y) dy + (8

x.... ) * Q(x) A.c_

A.e,

.f.= _..,

CD

+

where

~ = ~ _ 1

U ~ U ~+ 1 •

IQ(x-y)l ~ 2-r(k- 1 ) , Write

J(.(..) =

J

If

181 dx •

x E ~ ,

.f,~k+2

while i f

E .(..=k+2

J

Q(x- y) 8(Y) dy

A.c,

.f, ~ k - 2 ,

y E A.(, ,

then

then

IQ(x-y)l ~ 2-.(..(k- 1) .

Then

A.f.

I8*Q(x)l::; 2-r(k-1)

J

k-2 "' E J(.(..) + 1<8 X- )*Q(x)l + !: 2-r(t-1) J(.(..)' t= -... ~ .(..=k+2

I 8*QI dx ::; C 2nk- rk +

~

::; c

where we have used

using (6,26),

2

!1811

nk-~

1::;

L

k+ 1 "' I: J(t) + C I: 2°k- r.(.. J(t) t=k-1 t=k+2

+ c

"' nk r• I: J(t} 2 - ..., .f,=k-1

11811 n(l _ 1),p K p 1

1

and

,

n

r>p->n.

Hence,

.

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn)

J

1: ( k=j+2 ~


..

E

2k(n-pr) + C

k=j+2

~ C+ C

.

E

.(, = j+ 1

..

t+l

t=j+l

k=j+2

E

J(t)P 2tn(l-p) ~

E

c

J(t)P ln-rtp

57

7.

BEST POSSIBLE NATIJRE OF THEOREMS 3

We begin with two lemmas.

satisfies

~

(7.1)

lxl

>

2 •

g E Ko:,q 2

LEMMA 2.

e(k) ,

suppose k2_1 ,

!l!!!.

n

r>-p,

satisfies

(7.2)

O
e(k) 2, e(k+ 1) 2, 0

llgll

K(e,p)

~ C(r,n,b,p)

K(e, p)

"' llsll

1

L

+

~

.!£!!. ~

e (2k) 2, b e (k) ,

lls*QII

Q satisfies(7.1),

b>O •

llgll K(e,p)

I: k=O

58

and

EMBEDDING AND MULTIPLIER THEOREMS FOR HP(~ 0 )

~~

The proofs follow the lines of the proof of lemma 2 in §6.

In the

proof of this lemma 2 one also uses the inequalities

for some M > 0 ,

which follows from (7. 2) , .(,

ko E 2 k=l

(7.3)

t, M

\k) :S

and

o.t C(M,o) 2

M>O ,

,

i
which follows from the inequalities

li >0 ,

i<2 k-

if

The following theorem illustrates the sharpness of Theorem 3a.

THEOREM Sa.

Suppose that

sequence with h E HP

such

O
0

:S

< p < 1 .!!!!,!! e(k) •

~

e (k)

There~

m

and

.!.!!.!!.

for every

(7.4)

(7.5)

sup

o

By §2,

ME L"'(Rn)

1\!\1\

<

<»,

K(e,p)

an equivalent formulation of (7.4) is

and by (1.6), the function

m also satisfies

q>p ,

.!!!!,!!

BAERNSTEIN AND SAWYER

60 sup

q>p

•

6

PROOF OF THEOREM Sa.

f ( ) 0 X

Define

f0

by

= 2 -kn/p

k-1/p ,

=0

otherwise

,

Q E C,.,

Take

Then Q(O)

k = 2, 4, 6, ••• ,

>0 ,

Q E c""

1\Q\1 1 = 1 ,

and

supp

II xl ~ t1

> n(~ +

t>

IQ(x) I ~

satisfies

Q

C

Ixl-r

For fixed

for some

C ,

and so

c >0 •

Hence

satisfies the hypothesis of Leoma 1 for some

=e

f(x)

ix 1

1

(fo*Q)(x) E

unless

l .

m=

1

8 ~6 ~8

•

Then

supp me

n

Let

for every

Q

= cl],

h = ~ •

CX> 0 ,

~ 21

n(p - 2),q

K2

i t follows that

1

K2

The inverse Fourier transform of 1 1

These functions form a bounded set in Leoma 1 with

1} ~lsi

and

m6 (g)

m(!lg)

is

h E

n

by Corollary 1 of §2.

We have

.,

= m = f •

6

8 ~6 ~

m satisfies (7.4).

p>O

= m(6E;) -n

1

when

Then

so

cQ

n(p - 2),q

q>p

Define

Q2:,0 ,

.

Qc

L

r

satisfying

8

T)(S)

f(6

-1

=0

x).

From

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn) Since

f E L

Take

x E

(f0 *Q)(x)

1

~

to show

,

,

it

even.

k

k 2:. 2

J

=

f ~ HP

Then

~

<2

Q(x- y) f 0 (y) dy

+J

k+2Q(x-y)fo(y)dy.

IYI Since

IQ(x)l

SC

lxl-r,

r >.!!

Since

the middle integral is

it follows that for all sufficiently large

p

(f 0 *Q)(x) 2:. c 2

fo*Q ~ Lp ,

Hence

and

-nk/p

f ~ LP ,

I:

e (k) < .., •

k

-1/p

2:.

-nk/p -1/p c 2 k •

k

,

as required.

Next, we construct the function for which

>2

the first and third integrals are

Q(O) > 0

Since

f ~ LP

suffices to show

k _ 1 Q(x- y) f 0 (y) dy + J

IYI

61

Take a subsequence

M •

S c 2 z+

Define

k ES

2-nk/p

X

Q be as in the construction of

e(k)

above show that If does.

e (k)

s ,

m and define

M=g.

g(x)

If

k E

E ~

otherwise.

0 ,

Let

'

satisfies (7.2), then Lemma 2 and an analysis like the one M satisfies (7.5) but that

(Mh)¥

~ Hp

does not satisfy ( 7. 2) we replace it with a sequence

We may suppose

e ( 1)

=1

•

Define inductively integers

.t j

,

whi.:l: :

z: ~

62

by

BAERNSTEIN AND SAWYER

t0 = 0 ,

and

tj + 1

1 ~(2tj) ~(2tj+l)~-2 ~ ~

Then Construct

B(k) ~

De f"1ne

and

M using

is the smallest positive integer for which

B(k)

k~1 ,

8 (k) ,

bY

satisfies ( 7. 2) •

8(k) •

Also,

8(k)

~

e (k) •

Then

sup I!M6 11

6>0

~ sup IIM6 11 6>0

K(e,p)

<• , K(8,p)

and the proof of Theorem 5a is complete. Now we consider the case

p

=1

and demonstrate the sharpness of

Theorem 3b.

THEOREM 5b.

Suppose

(7.6)

lh!l

k~

w(k) ,

l~w(k)t

as

1 ,

kt ,

satisfies

w(2k)

~

C w(k) ,

...

.!!!!!, t w(k) - 2 1

sup

6 >0

\1616\1

<.., K(w)

,. .. 1 (mh) ~ H •

Moreover,

for every

q

>0 .

EMBEDDING AND MULTIPLIER .THEORE~5 FOR Hp(~n) As we pointed out in §3,

2: w(k)

with m =0 •

-2

= "'

63

there exists a nondecreasing sequence

such that the only

m

satisfying

Thus, some supplementary hypothesis such as

11m :1

sup

6 >0 " 6' K(w)

S C w(k)

w(2k)

w(k)

'"'

is

is needed

for the theorem to hold.

PROOF.

Choose a set

S = \k 1 , k 2 ,

="',

lim (ki+l- ki)

~ 3 , .•.

!c

z+

with

ki < ki+l,

and

i-tm

2:

w(k)

-2

= .., •

kES

k 1 ?. 10

We also assume that

~ E c~

~

,

'!

0 ,

with

supp

Yc

I: k ES

Then k E S

m

and

Define

.

WrLte

li

(S) = 0

unless

2k- 3 soS2k+ 3

f 0 = ..

k

6 = 2 e ,

k k m(2 (sli- 2 el))

is

F(x)

!lsi< 1\ .

for

i ?_1

Take

Define

1 w(k)

2k- 3 S li S 2k+ 3

for some

k ES •

For fixed

we have

and fix

where

ki + 1 - ki ?. 10

and

1

r

>n •

B S e S8

Then

f0

satisfies

The inverse Fourier transform of

BAERNSTEIN AND SAWYER

64 which satisfies

IF(x)l ~ C 2

(7.7)

C which depends on

for some

-2kn

r

all x

,

but not

E •

An argument like the one used to prove Lemma 2 of §6 shows that the

F*~

functions

also satisfy (7.7)

1 ,.. (F*T]) w(k)

¥

m~

Since

= --

v

it follows

that

< 1 6\~(x)l v -

(7.8)

C _1_ 2-2kn w(k) '

I616 (X) I <- C _1_ w(k)

all

2 2k(r- n)

x ,

lx 1-r

,

Hence

"" E t=l

(7.9)

<J

1£~1 dx) w(.f,) ~ C

A.r,

v

2k E t=l

w(.f,) 2-Zkn 2n.f, w(k)

"" I:

+ C

~ 2 2k(r-n) 2 (n-r)t

.f,= 2k+ 1

The hypotheses (7.6) imply that

(7.10)

for some C >0 •

w(.f,) S cw(k)

t\M w(.f,) S C ( k) w(k) ,

for

l~tS2k

t>k ,

Thus, the right hand side of (7.9) is majorized by

C + C 2 2k(r- n)

~

(i)M

.f, = 2k \k;

•

w(k)

2 (n- r)t

and also

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(R 0 ) 0(2(n- r) 2 k).

The last sum is easily shown to be side of (7.9) is

0(1) ,

uniformly in

5 ,

11£511

sup 5 >0

65

Hence, the left hand

and we have shown that

<

ao •

K(w)

Also, from (7.8) it follows that

for every

since

>0 ,

q

w(k) ;::: 1 It remains to construct the function

h •

Choose

k 1 < k 2 <...

such that

1S

E

w(k)

-2

<2 ,

j=l,2, •••

kES(j)

where

S(j) = !k ES: kj Sk
Define

cp , A. : S ... z+

by

j

cp(k)

-1

if

k E S(j)

_KU

A.(k) - w(k)

Then

E A. (k) k ES

Take Define

g 0 E c~

with

2

<• ,

supp g 0 c

U!0..w(k) -

E k ES

II sI < 1J

,

go =

'"' •

1

1

on

IsI < 2 .

in

S

BAERNSTEIN AND SAWYER

66

h =

g.

Then

for every

CX>O ,

1 h E H ,

so

by Corollary 1 of §2 •

We have

m(~) g(~)

= (mg)"(x)

f(x)

where

f

0

=t

•

f(x)

(7.11)

Fix

Write

k ES •

'E

.tES .t
+

E

+

.tES .t=k

t

.tES .t>k

= f 1 (x) +

f 2 (x)

Notice that

(7. 12)

Here

c>O

Next, since

where

k"

since

+f,

V E C~ and

0 •

lf 0 (x)l ~ C,

denotes the successor of

k

in

S •

Hence

+

f 3 (x)

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(R 0 )

J~ I f31

(7.13)

I f 0 (x) I

Finally, since

dx

< -

b7

C L(U 2n(k- k") w(k)

~ C I~~-r

for

I xl

k ES

6>0 ,

>1 ,

x E ~

if

then

(7 .14)

We claim that, for any

and

E A.!..Q 21\.L < C(li) A.ill 21\k .f- E 8 w(.(.) w(k)

(7. 15)

.(.~k

Assume (7.15) for the moment. predecessor of

k

in

S •

l fl(x)l

<

-

Return to (7.14) and let

Then, with

o= r

- n

and

k'

k'

be the

in place of

k

C ~2-rk 2(r-n)k' w(k') '

Hence

(7. 16)

r .J

~

If I dx < C ~ 2(n-r)(k-k') 1 w(k')

Using(7.10)andthefactthat

ki+l-ki .....

easy to show that the right hand side of (7.16) is (7.13) it follows that

So, from (7.11) and (7.12) it follows that

for

kiES,

A.ill

o(w(k))

itis

From this and

68

BAERNSTEIN AND SAWYER

k ES ,

for some

c>0 •

r:

f

~

1

L ,

PROOF OF (7.15).

t..ill

S

= ...

'

and therefore

If \l)(k) = 1

follows from (7,10) and (7.3). element of

large,

Since

k E S w(k)

we see that

k

then

;\(.t) = w(.t)

Assume

\1) (k)

-1

1

S2.

for Let

.tSk

k0

such that

Since cp (k)

so that for some

-1

- cp (k0 )

-1

S (k- k0 ) ,

we have

C = C(M, 6)

( 7. 17)

Here

M is the number in (7.10) •

.!.{ll_ -2 (k'2M 6(ko-k) w(k)2 ~ c w(k) ko) 2

(7.18)

Using

From (7.17) follows

~

S 1,

(7,10)

and

(7.3) ,

we obtain

and (7.15)

be the largest

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn) I:

liU

.C.ES

w(.C.)

26-C. =

L: .C.ES

.(.~k

2~-e.

p(.C.)2 w(.C.)

=

.(.~ k

~

°+

6k C w(k0 ) -z 2

~ C w(k)

.C.~k0

2

k

+

I: -L=k0 +1

cf.'P (k) w(k) - 2 z 6 k

6 ko

-2 .- k, ZM

\k)

I:

69

+

2 C f.'P(k) w(k)-

6k 2

•

0

By (7.18), the first term in the last expression is A(k) w(k)-l

z6k,

and the proof of (7.15) is complete.


=

8.

LOWER MAJORANT THEOREM

We shall prove the following result.

For

p= 1 ,

case (See [Z,

n= 1

this is a theorem of Hardy- Littlewood in the periodic

p. 287]).

Coifman and Weiss [CW,

decomposition to give a new proof for

H1 (R)

p. 584] used the atomic

Weiss [W,

p. 199] raised the

question of whether this "lower majorant property" would hold for Theorem 6, which states that for every

p E (0,1] ,

HP(Rn)

H\Rn)

has the lower majorant property

was proved by the authors during the early stages of

this project (See Abstracts of the AMS, 1(1980), p. 444).

A similar proof

was found independently by A.B. Alexandrov [A] • The analogous problem for

has an interesting history.

The interested reader is referred to [LS] •

PROOF OF THEOREM 6.

By the atomic decomposition (see §6) it suffices to con-

sider the case when

f

up to order

is a unit p-atom, that is

has vanishing moments

N and satisfies

(8.1)

supp f c

Let

f

Q = [0,1]

n

II xl

.S 1} ,

denote the closed unit cube.

70

For

~ E Zn

define

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp~hn) a(.t)

=

sup 1£<~+-t)l

71

.

sEQ

£

The key property of

is the. inequality

(8.2)

This inequality has been discovered independently by (at least) SleddStegenga [SS],

J. Stewart [St] , Alexandrov [A] and the present authors.

We also need the fact that (8.1) implies

(8.3)

f

To see this, note that variables.

is an entire function of

n

complex

Expand it in Taylor series around the origin

The vanishing moments condition shows that

c

f3 = 0

for

If31 :::

N ,

while

and (8.3) easily follows. Take 1

1

<2, · · ·, 2)

q> E C~

with

q>= 1

and sidelength

1) 2

supp q>C

on Q ,

23 Q

(the cube with center

Define

E a(.t) q>(!; + .t)

where the sum is taken over all the origin.

Take

contain the origin.

1jt

E C~ Let

n

.t E Z

such that

1jt

for which

=1

Q+ .t

does not contain

on all the cubes

Q + .t

which do

72

BAERNSTEIN AND SAWYER

where this

C is the same one as in ( 8. 3) •

Then

From (8.2) it follows that

for every CX>O , If for

and hence

l\G 1!1

N is odd then G2 E C~ 1

CX
above that

1!62 11

Choosing

P •

p ~ C,

by Corollary 1 of §2.

H

and (8.4) holds with 1

1

a E (n(p - 2) ,

G2 1

N+l+ 2n),

in place of

G1

it follows as

Thus, the conclusion of Theorem 6 holds, with

H

In case

N is even, one can use standard methods of Fourier analysis,

like those in [S, chapter 3] ,

to prove that

(8. 5)

so that (8.4) again holds for follows as in the case

1

CX
and the proof of Theorem 6

N odd.

Here is another proof for the case

N even, which avoids (8.5).

A be a constant such that

Define, with

C the constant of (8.3)

G3 (s)

n

= AC

I:

i

S· _l

= 1 Is I

N+l

S· l

y(g) •

Let

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn) Since

,.N_ +1 ':>1.

·~T E C000

>

73

it follows as

above that

ll<s~+ 1 t>"ll for that g =

i= 1 , ••• , n •

I\G 31l

p ~ C, H

Gl + G3 •

p H

~c

Since Riesz transforms are bounded on

Hp

it follows

so that the conclusion of Theorem 6 holds with

9.

Suppose satisfying

q> 1 , b1 = 1

ON A THEOREM OF PIGNO AND SMITH

I b j I~

and let

be a sequence of positive numbers

and

b.

1

--l±...!. > b. - q '

(9. 1)

J

Pigno and Smith [PS 1], see also [PS2], assumed that

q2;:2

and used

the method of Cohen-Davenport to prove the following theorem about Given an analytic function

f E H1 (n)

there are measures

~·

J

H1 (n)

E M(n)

satisfying

f(.t)

j = 1, 2, •.• '

.(, E Z ,

and

We are going to use the atomic decomposition to prove an analogous result for

Hp(Rn),

O
Suppose that

q>l

and that

jb

j

!..,- ..

is a

strictly increasing sequence of positive numbers satisfying (9.1) for every j E Z •

THEOREM 7. sequence

f.!?£ each f E Hp (Rn) , jfjJ:..,

in

Hp(Rn)

0 < p~ 1 ,

n 2: 1 ,

~ ~ .!

such that

f.

j EZ ,

J

74

EMBEDDING AND MULTIPLIER THEOREMS FOR HP(Rn)

75

and

C>=C(p,q,n)

where

depends.Q!l q

butnotQ_!!

!h.! • J

where

Examples of the form supp cp c ! IS I < 1} '

nHP

which belong to

by Corollary 1

O
1\f.!lr

for any

of §2,

suggest that

PROOF.

An easy argument shows it suffices to prove the theorem when

J Hp

r<2

f

unit p-atom, that is, one satisfying (8. 1) • Let

jO

be the integer such that

and write

For

j

> jO choose cpj

E C~

satisfying

cpj , 1

on

b j :;_I g I :;_ b j + 1 ,

and

(9.2)

where

does not depend on the

C(t3, n) Define

F., cp. J

J

£

and let

b .. J

Take

From (9.2) and Corollary 1 of §2 it follows that

s E

z+

with

is a

76

BAERNS TE IN AND SAWYER

Since C(q)

bj+l~qbj'

of the

Bj .

each

s

with

1sl >

1

2

can belong to at most

Hence

(9.3)

Next we consider the case

s

I I

<1

,

t c

supp

\1 sI < 2! •

j

~

jO - 1 •

Choose

t

...

E c0

with

t

=1

on

Define

., and let

fj = Fj •

shows that i f

for any

6Sl

The proof of Lemma lain §6 uses only that the Fourier transform

h

of

l(6g) t(s)

t E S

and

satisfies

r>O lf<s>l S CI s IN+ 1 ,

Also, since by (8.3)

Choosing

1

A

1

0: E (n(p- 2>,

n

N+1+ 2>

and

we have

r>O:+n,

Corollary 1 of §2

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(Rn) shows that

llhll P ::; C6

N+ 1

{,

•

H

Letting

6

=bj + 1

,

we have 1

llf.ll J Hp 1

where e = N+ 1- n(p- 1) >0.

n(l- -) =b. J

+1

p

llhll HP::; C beJ"+l'

Hence

(9.4)

2e ::; Cbj ::; C •

0

Theorem 7 follows from (9.3), (9,4),

and the choice

f.

Jo

=

f •

10.

EXTENSION OF A THEOREM OF OBERLIN

A theorem of D.M. Oberlin [0] asserts that for

f E H1 (Rn) ,

.., E k

where

= -..,

dcr

sup 2k.::;, r.::;, 2k + l

J

IS I = l

lfl dcr(g).::;, cllfll 1 , H

is surface measure on

jxj = l •

We will extend this result in two ways - by replacing the n- l dimensional spheres with circles and the

THEOREM 8. ~

Suppose that

1.::;,q
and

.!&!

sk ,

the following property:

L1

norms by

Lq

norms,

~~constants

n;:::2 • k E Z ,

be

~

C(n,q)

sequence£.!. circles

ill Rn with center .!! the origin ,!m! ~ between 2k ,!!!.!! 2k+ 1 •

Then

( 10. l)

As a corollary, we obtain the inequality

E k= - ...

The theorem and corollary are false for

q

="" • A counterexample is

furnished by

...

~

k

=l

1

k

- q>(g- 2 e )

k

78

l

EMBEDDING AND MULTIPLIER THEOREMS FOR Hp(R 0

cp E C~ ,

where

supp cp c

cp(O) = l •

,

Corollary 1 of §2 shows

f E H1 •

that

p<1

For

a proof like the one of 'Theorem 8 leads to the inequality

But here the case

co

(10.2)

~

k

If


79

)

q

=co

holds too,

2 -kn(l-p)

=-co

The proof of (10.2) is left to the reader. It extends the inequality 1 n(ll' - 1) (t;) ~ P which we have used earlier. H

cl sI

I

II £11

PROOF OF THEOREM 8. satisfied.

f

is a unit 1-atom, so that (8.1) is

By H'older's inequality, we may also assume

lf<s>l

we have

We may assume

~

clsl ,

q2:2 •

By (8.3)

so that 0

( 10. 3)

~

k

=-co

To analyze the case

k2: 1

we introduce the unit cube

Q

and the

numbers

a(t)

as in §8. length

~

Let C(n) ,

.s:k

=

= It E zn so

sup sEQ

:

n

If <s + t) I ,

Q+ t

meets

tEZ ,

sk

l .

Each intersection has arc

80

BAERNSTEIN AND SAWYER

By

~older's

I: ( k= 1

inequality, with

1 1 q+ qt =

1,

J

sk

~ C(

!:

2 -kq I /q) 1/q I

k=l

Now each q;:::2 •

.{,

belongs to at most two of the

(

. !:

I:

k= 1

.t,Etk

tk ,

a(.(,) q) 1/q

and we are assuming

Hence

where we have used (8.2).

This inequality, with (10.3), proves Theorem 8 •

REFERENCES [A]

A.B. Aiexandrov, A majorization property for the several variable Hardy-Stein-Weiss classes (in Russian) Vestnik Leningrad. ~· No. 13 (1982), 97-98.

[C)

R.R. Coifman, A real variable characterization of Hp,

Studia Math.

51 (1974), 269-274. [CW]

R.R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. ~· Math. §££. 83 (1977), 569-645.

[CT]

A.P. Calder6n and A. Torchinsky, Parabolic maximal functions associated with a distribution, II, Advances i£ Math. 24 (1977),

101-171. [F)

T.M. Flett, Some elementary inequalities for integrals with applications to Fourier transforms, Proc. ~Math. 22£. (3) 29 (1974),

538-556. [FS]

c. Fefferman and E.M. Stein, Hp spaces of several variables, ~ Math. 129 (1972), 137-193.

[H)

C. Herz, Lipschitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms, l· ~· ~· 18 (1968), 283-324.

[Ja]

s.

Janson, Generalizations of Lipschi.tz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math l· 47 (1980),

959-982. [Jo 1] R. Johnson, Temperatures, Riesz potentials, and the Lipschitz spaces of Herz, f!££. ~ Math. ~· (3) 27 (1973) 290-316. [Jo2]

R. Johnson, Multipliers of Hp spaces, ~·

[L]

R.H. Latter, A characterization of HP(Rn) in terms of atoms, Studia Math. 62 (1978), 93-101.

[LS]

E.T.Y. Lee and G·I. Sunouchi, On the majorant properties in Lp(G), T8hoku Math. l· 31 (1979), 41-48.

[MS]

S. Minakshisundaram and 0. Szasz, On absolute convergence of Fourier series, ~· Amer. Math. 22£. 61 (1947), 36-53.

[M]

A. Miyachi, On some Fourier multipliers for HP(Rn) Tokyo 27 (1980), 157-179.

[0]

D.M. Oberlin, A multiplier theorem for H (R ) ,

1

M!!· 16 (1977), 235-249.

n

l· E!£. 2£1.

.f!2£.. Amer. tl!£!l .

.[££.

73 (1979), 83-88. [ P]

J. Peetre, ~Thoughts Dept., Durham, 1976.

~~Spaces,

81

Duke University Mathematics

82 [PT]

BAERNSTEIN AND SAWYER J. Peral and A. Torchinsky, Multipliers in Hp(Rn), O
kl!.!:.· 17 ( 1978)' 225-235. [PS 1] L. Pigno and B. Smith, A Littlewood-Paley inequality for analytic

measures,

~·

l:l!.!:.• 20 (1982), 271-274.

[PS 2]

L. Pigno and B. Smith, Quantitative behavior of the norms of an analytic measure, ~· ~· Math. 22£. 86 (1982), 581-585.

[Sj]

P. Sj~lin, An Hp inequality for strongly singular integrals, ~ ~· 165 (1979), 231-238.

[SS]

w.

[S]

E.M. Stein, Singular Integrals ~Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970.

[Stw]

J. Stewart, Fourier transforms of unbounded measures, 31 (1979), 1281-1292.

[Sz]

0. Szasz, Fourier series and mean moduli of continuity, Math. 22£. 42 (1937), 366-395.

[ T]

M. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean n-space, I, 1· Math. ~· 13 (1964), 407-480; II, ibid 14 (1965), 821-840; III, ibid 15 (1966), 973-981.

[TW]

M. Taibleson and G. Weiss, The molecular characterization of certain Hardy spaces, Ast~risgue 77 (1980), 67-151.

[W]

1

Sledd and D. Stegenga, An H multiplier theorem, (1981), 265-270.

~· ~·

G. Weiss, Some problems in the theory of Hardy spaces, 35, A.M.S., Providence, 1979.

£!a· 1· ~·

19

~·

Amer.

~· ~·

~Math.

[Wi]

J.M. Wilson, A simple proof of the atomic decomposition for Hp(Rn), 0 < p 5,1 , ~ ~· 74 (1982), 25-33.

(Z]

A. Zygmund, Trigonometric Series, 2nd ed., Cambridge University Press, Cambridge, 1968.

Washington University St. Louis, Missouri 63130 McMaster University Hamilton, Ontario L8S 4Kl