(I.2) s~(f) = v / ~,L~ ~f~, s(f) = sup s~(f). The Burgess Davis inequalities via Fefferman's inequality*

The Burgess Davis inequalities via

Fefferman's inequality*

ADRIAN0 ~r GARSIA

I n t r o d u c t i o n

L e t ( / 2 , 7 , P ) b e a p r o b a b i l i t y s p a c e a n d l e t ~ 0 C ~ i c . . . c ~ n c . . . b e a s e q u e n c e o f a - f i e l d s s u c h t h a t 7 = U~=0 % .

F o r a r a n d o m v a r i a b l e f e L~(~2, 7 , P ) w e s h a l l s e t f , = E ( f t ~ , ) , Afn = f~ - - A - I , f * = m a x lf, I, f * = s u p tf~i,

~ n n

s~(f) = v / ~,L~ ~f~, s(f)

= sup

s~(f).

W e also i n t r o d u c e t h e s p a c e s

~)~p = {f: E ( [ S ( f ) ] ' ) < m } (I.1) w i t h n o r m

F u r t h e r m o r e , w e l e t

w i t h n o r m * *

[Iflf% = [E([S(f)]P)] lIP (P ~ 1).

B M O = {f: s u p t i E ( I f - -

L_~l~l~.)iI~

^< ~ }

n_>l

(I.2)

[Ifi[BMO = SUp It ~/~7( I f - - fn--1 ]U

i~n)ll~.

a>_I

* This work was supported b y the Air Force Office of Scientific research under Grant A F - ~ O S R 1322-67.

** Strictly speaking, these functionals are norms in the usual sense only when restricted to {f: E(flVo) --= 0}.

230 A D R I A N O ~VI. G A : R S I A

The reason for this notation lies in the fact t h a t with an appropriate choice of (I2, ~Y, P) and {~}, the spaces in (1.1) can be identified with the classical spaces of function theory, while at the same time the space defined in (I.2) can be identified with the class of functions of

Bounded Mean Oscillation

introduced b y J o h n and Nirenberg [6].

This given, Burgess Davis [3] proved the following remarkable inequalities

:T E(f*) ~ E(S(f)) ~ C2E(f* )

1 (V f:

E(fl::o)

---- 0), (I.4)

where C1 and C 2 are universal constants.

More recently, Charles Fefferman [4] showed t h a t there is a universal constant C8 such t h a t for any two martingales f , =

E(ft~,, ),

Fn = E(F]~n) with f0 = 0, we have

f fnq~,dP] ~ C31lftly(,II~IIBMO,

(I.5) o

Actually, in [4] Fefferman only states the corresponding inequality in a classical function theoretic setting, however, we understand t h a t Fefferman had also proved (I.5).

The object of this paper is to show t h a t both sides of (I.4) follow in a rather natural manner from (I.5).

We hope t h a t our arguments here, in addition to throwing some new light on this matter, will actually provide w h a t is perhaps one of the simplest ways of establishing the Burgess Davis inequalities.

F o r sake of completeness, at the end of this paper we shall include a proof of (I.5) with C a = V / 2 , this will yield (I.4) with C t ~ V ~ and C 2 = 2@%/5-.

Perhaps it is worthwhile to include some historical remarks, l~irst of all, it was Burkholder, G u n d y and Silverstein [2] who made it apparent t h a t the ~ spaces of classical function theory could be viewed in a most natural manner in a martingale setting.

This has become even more apparent now after some recent work of Getoor and Sharpe [5].

Burkholder and G u n d y [1] conjectured the Burgess Davis inequalities for general martingales after proving them for the class of regular martingales*.

The space BMO as in (I.2) was introduced (after John-Nirenberg's paper) b y R. G u n d y a few years ago. We also understand t h a t after learning of Fefferman's analogous functional theoretical result, R. Gundy and C. Herz worked together on a proof of (I.g). Indeed, the proof of (I.5) we shall present here is essentially a simplified version of Herz's proof we learned from Gundy.

* For the definition see [1].

TiHE BUI~GESS DAVIS I N E Q U A L I T I E S VIA F E F F E I ~ A N ~ S I N E Q U A L I T Y 231 I n this connection, we wish to acknowledge here our gratefulness to the Mittag- Leffler institute for making possible during the s u m m e r of 1971 our mathematical exchanges with Burkholder, Fefferman and Gundy which provided us not only the stimulus b u t also the information without which this work could not have been carried out.

1. C o n s t r u c t i o n of s o m e B M O f u n c t i o n s

Our arguments will be based upon two results t h a t should be of independent interest.

L~,~IA 1.1.

Let

{0,}

Then, the function is

BMO

and indeed

be a sequence of random variables satisfying

~7=~ [0~! ~ 1. (1.1)

E([~ -- ~,_ll~l~,) ~_ 5, V n ~ 1. (1.3)

Proof.

Clearly

To prove 1.3 set

~o E L 1 since

E([~I) _< ~ E(I0~[)_< 1.

E(0~I ,).

This given, from (1.2) we get

~,-1 = E(~lg,-1) ---- ~7-~ E(0~IJ~) + E(~oi~,_I).

So

and, using (1.1), we obtain

E(I ~ - - ~ , _ ~ i 2 [ ~ , ) =

E ( r

2 E ( r )E(r _1) +

_ 2 ~

+ [E(r 2 < E ( r .) + 3.

On the other hand, again using (1.1),

Combining (1.4) and (1.5) we obtain (1.3) as asserted.

Our next result can be stated as follows:

(1.4)

(1.5)

232 A D t t I A ; ~ O !VI. O A R S I A

L E ~ A 1.2.

Let

{7~}

be a sequence of random variables such that for each ~ > O,

~(Y~) ^{C ~}

and set

y* : sup~>o ]Y~I.

Then the function

q~ = ~%~ 7,-1{E(1/7 * t~) --

E(1/Y* ]~-1)} (1.6)

is

BMO

and indeed

E(I~ ^{- -} ~,_al'~lS,) _< 2. (1.7)

Proof.

Set for convenience % -- E(1/y*I~+), A% • % - - % - r W e t h e n h a v e

E(I~ 0 ~ _ ~ 1 2 [ ~ ) = ~ + = E 2 2U _ = E ' , 2 A 2 ~ . where

y*

: m a x !y,].

O<t~<_v

Now, n o t e t h a t

1 < y * _ I E ( 1 / y * ] ~ _ I ) < *

- - _ - - _ 7 , , _ 1 A ~ . ~ y * _ ~ E ( 1 / y * [ ~ . ) < 1,

T h u s

O.s)

~ ] . 2 * 2

~_13~v, < 1.

(1.9)

F i n a l l y

~ N n + I E , . , 2 A 2 7 - E , : , 2 r 2 2

< ~ : . + I E , *2 2 E ' ,2 2 7 - _ _ E , _ , 2 2 7 " _ _

L e t t i n g n - + + a n d combining w i t h (1.8) a n d (1.9) our i n e q u a l i t y (1.7) im- m e d i a t e l y follows.

These t w o l e m m a s provide some r a t h e r general m e t h o d s of generating BMO functions. I n fact, i t can be shown t h a t all BMO functions can be o b t a i n e d in t h e m a n n e r given b y L e m m a 1.1.

Before d o s i n g this section, it is w o r t h w h i l e to p o i n t o u t t h a t these two l e m m a s are s o m e w h a t related.

I n d e e d , suppose we use L e m m a 1.1 w i t h 0~ = ( 7 * - 7"-~)/7" where 7*, 7"

are d e f i n e d as in L e m m a 1.2. T h e n we get t h a t

is BMO. However, s u m m a t i o n b y parts yields

~vn--_ 1 y*_~{E(1/y* I~:) - -

E(1/7*

I~:_x)} =

= E ( r , / y , I ~ ) _ E ( y , o / F , I ~ o ) _ ~ . = ~ E ( 7 * - - 7 * _ I ) y .

~Y . (1.1O)

T H E B U R G E S S D A V I S IIWEQUALITIES V I A FEI~FERIVIAN'S I N E Q U A L I T Y 233

So passing to the limit as n--> 0% we obtain

7 , ~ , y * _ , { E ( 1 / r * [ % ) - - E ( 1 / r * i ~ , _ , ) } = ' - - E ( r * / y * l ~ o ) - - ~ . (1.11) Since a n y bounded function is clearly BMO, and the BMO norm of the left h a n d side of (1.11) is (as we have seen in t h e proof of L e m m a 1.2) larger t h a n t h a t of the function in (1.6), we can see t h a t ~mssentially~> L e m m a 1.1 implies L e m m a 1.2.

Unfortunately, it does not seem possible b y this approach to derive as good an estimate as in (1.7).

Alternatively, suppose the functions 6, in L e m m a 1.1 are of the form 0, = (Y, -- ~',-,)lY* (Yo = O, ~b',) c ~,).

Then the i d e n t i t y

~ ( / r

1

~-,)}

Y,,=, E(0,1~,) -- E ( r n / r * l ~ ) -- ~ = , y,,,{E(1/~*lT,) -- E 1 *

shows t h a t in this case, L e m m a 1.2 essentially implies L e m m a 1.1 This seems to suggest t h a t perhaps L e m m a 1.1 m i g h t still remain true if the condition 1.1 is slackened to

s u p

I~,%, ~,1 -<

1.

n

However we shall have to leave this as an open question.

2. T h e ~easy~, side

Although in Burgess Davis paper the proof of the two inequalities is somewhat symmetrical in f * and

S(f),

historically the left h a n d side of (I.4) has been easier to prove.

Our proof here should be reminiscent of H. Weyl's ~>n(x),> method. Indeed, set for v = 1 , 2 , . . . , n

E, = {~o:f*_l

<f*,f* =f*}.

We then have, assuming fo = 0

fz=>.o, f,z.,=r.=, fo, i,

E v

where we have set

Since

i t

--- f fn ~,:1 E(O,I~,)]dP,

O~ = XE~ sign (f~).

~,%110~] = 1,

(2.1)

234 A D ~ I A ~ O / ~ . G A ~ S I A

f r o m L e m m a 1.1 a n d (I.5) we get

* ~_ C~E(S,(f))

3. The hard side

T h e idea h e r e is t o s t a r t f r o m t h e e s t i m a t e

E(S.(f)) <_ ~r E--~) ~r E(S2.(f)/f *)

(3.1) w h i c h is a simple c o n s e q u e n c e o f S c h w a r z ' s i n e q u a l i t y . This given, we use t h e i d e n t i t y

S~.(f)

----

f.~

- - 2

~:=lf,_IAf,

a n d w r i t e t h e r i g h t h a n d m o s t t e r m o f (3.1) in t h e f o r m

E(S2,(f)/f *) = E(fd/f*) -- 2 ~ E(Af~f,_~/f*).

W e t h u s get

E(S.(f)) ~_

~r

%/ E(f*) +

2]Q]. (3.2) w h e r e

Q

= ~,"=~

E(Af~f,_i/f*)

-~ ~,"=~

E(Af,[E(f~_~/f* [Sz,)

- -

E(f,_Jf* I~,_0]).

Since for / ~ v ~ - 1 we h a v e

E(Af.. f~_~[E(Uf* )

ST,) - -

E(1/f*

]~,-1)]) = 0 we c a n w r i t e

w h e r e

Q = ff. .dP

.Q

=

1%) -- E(1/f* [%-0].

So b y L e m m a 1.2 w i t h y, = f , a n d (I.5) we get Iq[ -<

CaE(S.(f)) %/2.

S u b s t i t u t i n g in 3.2 we g e t

E(S.(f)) ~_ ~

V ~ f * ) -F 2 ~/2 QE(S.(f))

a n d this is easily seen to y i e l d

E(S,,(I)) ~ (~/2C3 + ~r + 2C~)E(I*).

TtIE BURGESS DAVIS Ih~EQUALITIES VIA FEFFEI~C~A~'S INEQUALITY 235

4. A proot ot Fefferman's inequality

L e t f , = E ( f l ~ . ) , ~0, = E(~[W,) w i t h f e 9 ( 1 a n d ~ E BMO. L e t f0 - - % ---- 0 a n d set as b e f o r e

A L = L - - L - l , /Jc~t~ ~ - qgn - - ~ n - 1 , S = S . ( f ) = ~,,%~ [ A L ] ~=.

N o t e t h e n t h a t , since

t h e p r o d u c t f . + . is i n t e g r a b l e V n > 1 a n d we h a v e

f ^fo %tiP -~ ~= , ^f %dP.

(4.1)

t2 Q

A v e r y clever idea d u e t o C. H e r z is t o w r i t e t h e r i g h t h a n d side o f 4.1 in t h e f o r m

a n d use S e h w a r z ' s i n e q u a l i t y t o o b t a i n

f f~+flP l ~-- +-A ~'-B

(4.2)

w h e r e

f f

F2

As we shall r e a d i l y see this leads t o a r e m a r k a b l y simple p r o o f o f F e f f e r m a n ' s i n e q u a l i t y .

I n d e e d ,

= 5."=1 f

S~

S,

While, on t h e o t h e r h a n d we h a v e

f " S

B = ~,a,=l [zJVv] 2 ~ , u = l ( ~* - -

St*-i)dP

: , 1

~9

D

_ n f _ 2 E

- - ~ / ~ = 1 (S/z - - S._I)E(Iq~. - -

~._~3l~.)dP <

[[~l[~Mo

(S.(f)).

Q

(4.3)

..0

(4.41

236 A D R I A : N O ] g . G A R S I A

Combining (4.2), (4.3) and (4.4) we obtain

Y2

which yields (I.5) with C3 = ~ / 2 as asserted.

5. Further remarks

Before closing it migh be good to point out another interesting w a y of establishing the hard side of the Burgess Davis inequalities.

However, we shall only give an outline of the arguments here since some of the details are quite intricate and the resulting constant is not as good as t h a t obtai- ned in section 3.

The idea consists in establishing first the converse of L e m m a 1.1, namely LEPTA 5.1. There is a universal constant c ~ 0 such that each q~ E BMO with I]~[IBMO ~ 1 can be written in the form

with

--~ ~ 1 E(0~]~)

This given, when E(f*) ~ ~ and [I~]]BMo ~ 1 we can write q~ndP = ~=xE(O~l~)dP = ~=~ A~OflP ~_

And this yields

lfs.

IJ~IIBMO <-- 1

f2 f~

On the other hand, we have

I n other words

E(S (f)) -= ~ = 1 f [Af~]2/S. dP =

Y2

= ~=~ f Af,{E(Af~/S~ ]~) ^-- E(Af~/S n I~,_~)}dP.

(5.1)

T H E B U R G E S S D A V I S I N E Q U A L I T I E S VIA F E F F E R M A N ~ S I N E Q U A L I T Y 23r/

w i t h

E(S.(f))

=

f A dP

g2

m ⁼

~.=~{E(ALIS.I .) E(ALIS,,I~._~)}

N o w , it is e a s y t o s h o w t h a t t h e following result, s o m e w h a t a n a l o g o u s t o t h a t o f L e m m a 1.1 holds.

L E M ~ 5.2. L e t

then the f u n c t i o n

is B M O and indeed

{0,} be a sequence of random variables satisfying

~0 = ~ v = l { E ( 0 , ] % ) - - ~ E(0,1 ~,_1)} c7

This g i v e n we g e t

E(S.(f)) <_ v ~

sup

]I~ollBM 0 _< 1 a n d c o m b i n i n g w i t h (5.1) we f i n a l l y o b t a i n

I

^g~

^ffnvdP,

^(5.3)

E ( S , ( f ) ) < %/-ScE(f*~).

F i n a l l y , we s h o u l d m e n t i o n t h a t L e m m a 5.2 like L e m m a 1.1 h a s also a converse.

B u t we h o p e t o come b a c k on this m a t t e r in a f o r t h c o m i n g p u b l i c a t i o n .

Bibliography

1. BURKHOLDER, D. • GUNDY, R . , )>Extrapolation and interpolation of quasilinear operators on martingales)>, Acta Math. V 124 (1970) p. 249--304.

2. --~>-- -->)-- & SILVERSTEIN, M., )>A maximal function characterization of the class 9~P>), Am. Math. Soc. Transactions V. 157, June 1971, p. 137--153.

3. DAVIs, B., )>On the integrability of the martingale square function,, Israel Journal of Math.

V. 8, :No. 2 (1970) p. 187--190.

4. FEF~ER~'% C., >>Characterizations of Bounded Mean Oscillations>, A m Math. Soe. Bulletin V. 77, :No. 4, p. 587--588.

5. G E T O O R , I~., & SHn~aPE, M., ~>Conformal martingales* inventiones M a t h . July 1972.

6. J O H N , F., & NIRE~-BEIZG, L., ~>0n functions of b o u n d e d m e a n oscillations>, C o m m . P u r e &

Appl. M a t h . V. 14 (1961), p. 785--799.

Received August 8, 1972 A d r i a n o M. Garsia