WEIGHTED NORM INEQUALITIES AND VECTOR VALUED INEQUALITIES

Jos~ L. Rubio de Francia

in

these boundedness properties hold, namely

(i) Weighted norm inequalities

IITf(x) lPu(x)dx ~ c I If(x) IPv(x)dx

(2) Vector valued inequalities

E ITfjlS)i/Sll ~C ICI IfjlS)l/Silp IIr j P j

It is known (and it is a simple consequence of H~ider's inequality)

that a "good" inequality of type (1) gives an inequality of type

(2) . For instance, the vector valued inequalities for the Hardy-

Littewood maximal function were obtained by C. Fefferman and

Stein [7] from the estimate

Ilf*IP u ~ Cp I IflP u* (i < p < ~)

In the same way, a new proof of the vector valued inequalities for

singular integrals due to Benedek, Calder6n and Panzone [21 was

given by C6rdoba and C. Fefferman [4] by using the inequality

lITflP u Cp s I1flP uS)* lJs el< p s <

which holds when T is a singular integral operator of a certain

kind.

I wish to show here that inequalities of type (1) and (2)

are to some extent equivalent. Some applications of this

The basic operators in Fourier Analysis are known to be bounded

L p for certain values of p . Sometimes, stronger forms of

equivalence will be presented.

87

i: TWO @eheral theorems

Let ~ be a family of sublinear operators T:Lq(~) ---~ Lr(~),

where ~ and 9 are arbitrary measures, and consider the

following vector valued inequality which may or may not hold:

(*) II (~ ITjfjlp)I/PIIr ~ eli( 7 Ifjlp)i/p[lq (Tj e~) J J

where 0 < p,q,r <

T h e o r e m A: L e t p < q , r , a n d d e n o t e ~ = q / p , B = r / p T h e n

( ' 1 h o l d s i f a n d o n l y i f , f o r e v e r y u e L (~1 , t h e r e i s ~r

U e L+ (~) s u c h t h a t

I " ....llull~, and )IITflPud~ ~ C p IflPud~ (T e~)

T h e o r e m B: L e t q , r < p , a n d d e n o t e ~ = p / q , ~ = p / r . T h e n

(*) h o l d s i f a n d o n l y i f , f o r e v e r y u e L+ (~) , t h e r e i s

U e L @ ' / @ ( ~ ) s u c h t h a t +

lul l~, /B ~ I lu l f~ , /~ a n d

I T f ( x ) I P U ( x ) - I d ~ ( x ) ~ C p ~ If(x)IPu(x)-Id~(x) ITem,)

I have to point out that both theorems are little more than

restatements of Maurey's results on factorization of operators

(see [13]). Their proof is based on results from minimax theory.

The following particular case if [ii] , th. 2, will suffice:

M i n i m a x T h e o r e m : L e t A , B be c o n v e x s e t s i n s o m e v e c t o r s p a c e s ,

a n d a s s u m e t h a t B i s c o m p a c t f o r a c e r t a i n t o p o l o g y . L e t F be

88

a r e a l v a l u e d f u n c t i o n on A x B w h i c h i s c o n v e x on A a n d

c o n c a v e a n d u p p e r s e m i - c o n t i n u o u s on B . T h e n

i n f max F ( a , b ) = max i n f F ( a , b ) aeA beB beB aeA

Proof of theorem A: The "if" part is very easy. Given Tj e

and f3" e L q , there exists u(x) -~ 0 with I lu] IS , = 1 such that

[1 ( ~3 ]TjfJlP)I/Pl Ix = { I ~j ITjfj Ipud~}I/p L_

-- C {] 3~ ' IfjlPud~} I/p -~ cl [ (~3 IfJlP)I/Pll

Conversely, if (*) holds and we assume for simplicity C = 1 ,

define

A = {a = (ao,a I) , a O = ~. Ifji p , a I = ~. ITjfj Plfj e Lq(~), 3 3

T. e~ } 3

B = {b e L a ' (p) ] b ( x ) a 0 , ] ] b ] I~ , L 1}

Both are convex sets, and B is weakly compact. Now, given

8' _L 1 we define on A x B the function u ~ T,+ (~) with ] lul ]8, '

F(a,b) = ~aob dp - lalUd~ = ~j ( ~ ,fjlPb d~- ~ITjfjlPud~)

which is convex (actually linear) on A and weakly continuous and

concave (actually affine) on B . Thus, the minimax theorem

applies, and since

max F(a,b) ~II ( ~ IfjlP) Ila-l] ( ~ ITjfjlP) lIB ~ o b~B j j

there exists U e B such that F(a,U) A 0 for every a e A . This

ends the proof.

89

The proof of theorem B is quite similar. To use the

minimax theorem one has to choose A as above and

8' B = {b e L+ (~ :IIbl 1 8, & i} ; then F is defined as

F(a,b) = ao u-ld~ - I alb-Bd~

where u(x) ~ 0 is given with ] lul I~./~ ~ 1 o

To see how all this fits into Maurey's theory of factorization,

let us assume that ~= (Tk)ke N consists of a countable family

of linear operators, and define the spaces B = I p ,

E = L~(D) , and the operator T on E

Then

Tf = (Tkfk)ke N

(*) may be written as

f = (fk)ke N e E

I( ~ I ITfjl I~)r/PdP ~ cr(~l [fjl I~) r/q fj e E) ] ]

Now, [13] th. 8 can be applied and what we obtain is just the

difficult part of theorem B .

Finally, I wish to observe that theorem B can be obtained

from theorem A by duality if q,r > 1 and if the operators of

are linear or linearizable (e.g.: maximal operators, etc.)

2: Singular Integrals

Let Tf(x) = p.v.

in R n whose kernel K(x)

(see [18] ) :

(I) TK(x) I _L Clxl-n

(2) I~(x) I -~ C

K * f(x) be a singular integral operator

satisfies the standard conditions

90

l (3) ~ IK(x-y)-K(x) Idx ~ C for all y e R n

ixt~21yl

When E = I p , 1 < p < ~, we denote by ~ the vector valued

e x t e n s i o n o f T

~f(x) = p.v. I K(x-y)f(y)dy = (Tfj(x)ij~ N

defined for the E-valued functions f(x) = (fj (x))je N for which

r 1 < r < ~, as a it makes sense. Then, ~ is bounded on L E ,

consequence of more general results due to Benedek, Calder6n and

Panzone [2] , and theorem A con be applied to give:

Coro l lar~ I: Let I < p,a < ~ be f i x e d . Then, f o r every

u e L+~(R n) t h e r e e x i s t s U e L~(R n ) + such t h a t l lUI Is ~- I I u i l

and

ItT CxJtPu xldx C IfCx i UCxldx

The reader can easily apply also theorem B , obtaining a

result which, by duality (since T is self-adjoint), is equivalent

to Corollary 1 .

The operator ~ satisfies also the weak type inequality

(4) l{x:J~f(x) VE > x}l ~ Ax -I ~ If(xllEdx (Vf)

but instead of using it in this form, I shall derive from it the

following

P r o p o s i t i o n : Let w(x) = (1+Ixjn) -I

I l f x)1 w x) dxl ljq cq, r

I f O < q < 1 < r ,

f I f ( x ) l E W ( X ) d x

then

91

Proof: We decompose ~ into two pieces

Tlf(x) = p.v. K(x-y)f (y)dy Tyi-~21xl

T2f(x) = lYl-~21xi K(x-y) f(y)dy

Then, by using (i)

I IT2f(x) I E lyl___2ixl (l+iyl-n) if(y) IEw(y)dy &

IL~ (l+Ixl-n) C11fi (w)

Since (l+Ixt -n) e Lq(w r) , the estimate for ~2 is proved. On

the other hand, we shall prove that ~i is of weak type (i,i)

with respect to the measure w(x)dx (this is enough, since

w(x)rdx is a finite measure and w(x) r ~ w(x)). If $(x)

denotes the characteristic function of the unit ball

B = {x:ixi~ i}, then

w(x) ~ Wo(X) = ~ 2-kn~(2-kx) k=o

Therefore

I{l~ifl>l} w(x)dx g C ~ 2-knl{x e 2kB:l~if(x) ]E > I} I k=o

If x ~ 2kB and fk(y) = f(y)~(2-k-ly) , it is clear that

Tlf(x) = Tlfk(x) , and since inequality (4) holds for T 1

T 2 separatedly

I c~ 2-knf { 1~if1>X} w(x)dx ~ ~ k=o Ifk(x) IEdx -

and

z 2Cl [ I f(x) IEW o(x)dx

and the proof is ended.

92

Now we can solve for the operator T the following two

questions (see E143) : Find all u(x) > 0

such that

I i f x TPu<x dx I If<x IPv<x dx R n R n

for some v(x) > 0 (resp. for some u(x) > 0).

(resp. all v(x) > 0)

(u

C o r o l l a r y 2:

{6)

t h e r e e x i s t s

G i v e n I < p < ~ and v ( x ) > 0 s u c h t h a t

I v ( x ) - P ' / P ( 1 + l x l n ) - P ' d x <

u ( x ) > 0 such t h a t (5) h o l d s . M o r e o v e r , i f

s < p ' / p i s f i x e d , we can f i n d u (x ) w i t h

l u ( x ) s ( 1 + q x l n ) - P ' d x <

C o r o l l a r y 3:

(7)

t h e r e e x i s t s

s < 1

G i v e n I < p < ~ and u (x ) > 0 s u c h t h a t

l u ( x ) ( 1 + I x t n ) - P d x <

v ( x ) > 0 s u c h t h a t (5) h o l d s . M o r e o v e r , i f

i s f i x e d , we can f i n d v ( x ) w i t h

I v ( x ) s ( 1 + I x l n ) - P d x <

Concerning the necessity of the conditions on the weights

u(x) and v(x) , it is easy to show for instance that, if (5)

holds when T is one of the Riesz transforms, then v(x) must

verify (6) and u(x) must verify (7) (see [16J ).

Proof of Corollary 2: Let w(x) be as in the Proposition, and

take q < 1 such that q/(p-q) = s , and 1 < r = (p-q)/(p-l).

93

Given v(x) , define ~(x) = v(x)-lw(x) so that (6)

equivalent to: ~ e L p'/p (w(x)dx). By theorem B ,

s exists ~ ~ L+(w(x)rdx) such that

~iTf(x) IP ~(x)-lw(x)rdx _z I If(x) IP~(x)-lw(x)dx

Then u(x) = ~(x)-lw(x)r satisfies everything.

By duality, corollary 3 is equivalent to corollary 2.

is

there

Remarks: It is well known that corollary 1 holds for every

operator T which is bounded on LP(w) for all weights w(x)

in the class A , but this is not the case for the general kind P

of singular integral operator considered here (see [i0]).

For singular integral operators which fall under the scope of

the A theory, corollary 2 is proved in [161 (by the same P

methods used here) and in [3~ (by a different argument), but no

information is given about the size of the weight u(x) . A

particular case of corollary 2 (for the conjugate function operator,

in the periodic setting, and for p = 2) was previously proved by

Koosis [9~ by using complex methods.

3: Positive Operators

If T is a positive linear operator which is bounded from

L r to L q (or L~ = weak-L q) , its vector valued extension

~(fj)j~N = (Tfj)jeN is bounded from Lr(/p) to Lq(/p) (or

L~(/P)) with the same norm as T , where 1 & p ~ ~ Therefore,

theorems A an B can be directly applied in this case. We

illustrate this first with the fractional integral operators in

Rn:

94

I f = Ixl a-n * f (0 < e < n)

It is known that I is a bounded operator from Lr(R n) to

Lq(R n) with i/q = i/r - ~/n , 1 < r < n/~ , and there is the

corresponding weak type result when r = 1 . As in the case of

singular integrals, we decompose

I f(x) = ~ f(y) I x-Y1~-ndy + lyl-~21xl lyl-~21xl

f(Y) Ix-Yl~-ndy =

= I'f(x) + I"f(x)

and by the same arguments of the Proposition in section 2 , one

proves that

ii,,f(x) I _L CI ifl iLl(w ) (l+ixi~-n)

I l~ :Ll(w ) > n n/(n-~) (w)

where w(x) = (i+ Ix I )-n , w (x) = (l+ Ix I )~-n . Combining both

things, we get:

-~ cllflIT I (~ > 0, q< n____) (8) ] I I f l ILq(wl+ E) (w) n-e

C o r o l l a r ~ 4:

(9)

a)

b)

Le t I < p <

~ I I ~ f ( x ) I p u ( x ) d x = I I f ( x ) I P v ( x ) d x

Given u (x ) > 0 , (9) ho lds f o r some v ( x ) <

i f

(10) ~ u { x ) ( 1 + I x l ) ( a - n ) P d x <

Given v ( x ) > 0 , (9) ho ld s f o r some u(x ) > 0

i f

be f i x e d , and c o n s i d e r t h e i n e q u a l i t y

(Vf)

i f and o n l y

i f and o n l y

(11)

95

I - p ' / p Le-n~ ' v(x) (1+Ixl) Pdx <

Proof: The necessity of (10) is easy. Take f ~ L~(V) , f ~ 0.

Then I f(x) ~ C(l+Ixl) ~-n , so that (l+Ixl) ~-n ~ LP(u) . Now we

shall prove the sufficiency of (ii) , and the rest follows by

duality. Consider the /P-valued extension of the inequality (8),

n and take q < min(p, ~ ) , so that theorem B can be applied.

Then (9) holds for some u(x) > 0 provided that

v(x) = ~(x)-lw (x) , with w ~ LP'/P(w ) , but such functions

v(x) are precisely those ones verifying (ii).

In the case p = 1 , condition (ii) becomes:

v(x) ~ C(l+Ix l)~-n - , and this is necessary and sufficient for the

existence of u(x) > 0 such that (9) holds (with p = i) ~ The

necessity is easy and the suficiency is contained in (8) . On

the other hand, for the analogue of Corollary 4 (a) when p = i,

we can only prove the following:

C o r o l l a r ~ 5: G i v e n u ( x ) > 0 , i n o r d e r t h a t

c121 fcxllucx)dx I IfCx)Ivlxldx Cvfl

h o l d s f o r some v ( x ) < ~, i t i s n e c e s s a r y t h a t

(13) l u ( x ) ( 1 + I x l ) a - n d x <

and i t i s s u f f i c i e n t t h a t , f o r some s , t w i t h 0 < t < I < s <

(14) l u ( x ) S ( 1 + I x l ) t ( a s - n ) d x < J

Proof: It is known ([15J) that, if i/q = i/p-e/n and

w e Al+q/p,

cllfllLq llqfll~pr r

96

use this with w(x) = (l+ixl) tnq/p' , and q' = s , and We

consider the ll-valued extension of this inequality. Then,

theorem A gives (12) for some v(x) < ~ provided that

u(x) = ~(x)w(x) , with ~ e Lq' (w) , and this is equivalent to

(14). This proves the sufficiency part, and the necessity is proved

as in Corollary 4(a).

Remarks: The size of v(x) in cor. 4(a) and cor. 5 , and the

size of u(x) in cor. 4(b) can be estimated by using the

inequalities from which both corollaries are obtained and the full

information provided by theorems A and B. The result of cor. 4(b)

has also been proved by a different (constructive) method in [i~ .

The next example is the Bergman projection in the unit disc

D = {z ~ c:rzl < 1}

Bf(z) = I (l-z~)-2f(~)dm(~) (f e L1 (D)) D

where m denotes Lebesgue measure on D . Though B is not a

positive operator, its positive majorization

Pf(z) = I I l-~I b2f (~)dm~ ) D

is bounded on Lr(D,dm) , 1 < r < ~ , and of weak type (i,I).

The same estimates will be satisfied by the /P-valued extensions

of P and B (1 ~ p ~ ~). Therefore, for the weighted norm

inequality

(15) I IBf(z)IPD u(z)dm(z)~ IDI f(z)Pv(z)dm(z) (u

we get the following results:

C o r o l l a r y 6:

t h a t (15)

a) L e t I < p < o0. T h e r e e x i s t s u ( z ) > 0 s u c h

h o l d s i f and o n l y i f v - p ' / p e L I (D) . I n t h i s c a s e ,

97

g i v e n s < p ' / p , we c a n f i n d u ( z ) s u c h t h a t u - s e L I ( D ) .

b) L e t I < p < ~. T h e r e e x i s t s v ( z ) < ~ s u c h t h a t (15)

h o l d s i f and o n l y i f u e L I ( D ) In t h i s c a s e , g i v e n s < I , we

can f i n d v ( z ) s u c h t h a t v s e L I (D)

c) L e t p = I . I n o r d e r t h a t (15) h o l d s f o r some

i t i s n e c e s s a r y t h a t u e L I ( D ) and i t i s s u f f i c i e n t t h a t

u e Lr(D) r>1

v ( z ) <

Proof: The necessity is easy in all cases. The sufficiency of

(a) follows from

m{( ~ IBf]P) I/P > ~} ~ CX-I I (~ IfJ(z)IP)I/Pdm(z) j J j

by theorem B . Part (b) is equivalent to (a) by duality, since

B is self-adjoint , and (c) is obtained by application of

theorem A to the inequality:

I ( Z IBfj (z) l)qdm(z) ! C I (~" IfJ (z) l)qdm(z) 3 3

where 1 < q = r' <

R e m a r k : By u s i n g t h e known A t h e o r y f o r t h e Bergman p r o j e c t i o n P

( s e e [ I ] ) one can o b t a i n , f o r e a c h u e LI+~(D) a f u n c t i o n

v e LI+~(D) s u c h t h a t (15) h o l d s ( f o r a f i x e d p , w i t h

I < p < ~ ) . T h i s i s a l s o a c o n s e q u e n c e o f t h e b o u n d e d n e s s o f B

i n LP(1 r ) , I < r , p < ~.

Other examples of positive operators which can be treated by

this method are the maximal function or Hardy-Littlewood (see

[16~) or its analogue for martingales, and the Poisson integral,

r n+l , p:Lr(R n) > L (R+ ,~) (where U is a Carleson measure).

98

4: Bochner-Riesz Multipliers

In general, a linear operator T is bounded in LP(~) for

(p/2)' a certain p > 2 , if and only if, for each u e L+ (~) ,

there exists U ~ L~P/2) ' ( ~ ) - such that

I lul I , and IITfI2ud~ ~ C I If[2~d~ fluff(p/2), ~ - (p/2) J J

(u

This results from theorem A and a well known result of Marcin-

kiewicz and Zygmund [121 . For some multiplier transformation T

in LP(R n) , U can be explicitely found as U = Mu for a certain

maximal operator M (see [5] ) .

We shall be specially interested in Bochner-Riesz multipliers

in R2:

^ 2 . 2 ~ ~(~) (f ~ (R2)) (s~f) (~) = (1-1~l /R ~+ ~2

The theorem of Carleson and Sjolin asserts that (S~)0<R< ~ are

LP(R 2 ) 4 z z bounded operators in , ~ - p - 4 , for every ~ > 0 . More

than this actually holds, since C6rdoba and L6pez-Melero [61

recently proved the following:

e fj12)i/2 ~z6) l lc~ IsR l] J 3

p~ c~,p11~ Ifj]2)i/2T1p 3

4 ~ (~- p - 4;~ > 0)

Of course, for each fixed e > 0 , a wider range of p's is

allowed (as in the Carleson-Sjolin theorem). By theorem A , we

get

2 C o r o l l a r y 7: G i v e n ~ > 0 , f o r e v e r y u e L+(R 2) , t h e r e e x i s t s

2 U e L+(R 21 s u c h t h a t IIUI 12 L_I I a l l 2 and

99

I I ~ R f ( x ) ] 2u(x)dx L- c~ I I f (x ) I 2U(x)dx (0 <R < oo).

A reasonable conjecture stated in [19] is that U(x) could

be obtained as a suitable maximal operator applied to u(x). Since

Bochner-Riesz multipliers are radial, if u(x) = Uo(IX I) is radial,

so can be chosen to be U(x) = Uo(iXl). Moreover, we can dilate

everything to obtain

(17) IISRf(x) 12Uo(tlxl)dx _x C I If(x)12U~

(t,R > 0)

Since Uo(t).t ~ L2(R+) (where we consider R+ provided with its

Haar measure dt/t), if we multiply both sides of (17) by

h(t -I) and integrate over R+ , where h ~ L2(R+) is arbitrary,

we obtain

(18) IIs f(x) 12 (Ixr)ixl-ldx % IIf(x) r2rxl-ldx (R > 0)

for every function g(t) with I ig] ILl(R+) ~ 1 (denoting by

the Fourier transform of g in the group R+) . Taking an

approximate identity (gn)neN in LI(R+) and passing to the

limit in (18) we see that (~R) 0<R<~ are uniformly bounded

operators in L2(ixl-ldx) . Simple interpolation and duality

arguments extend this to the following:

Corol lary 8: Given ~ > 0 , i f 0 ~- lal < I+~ , then

I' 4f(x)' 2, X' adx -L C , a [' f(x)' 2. X, adx (0 < R < ~).

It is suprising that this result, including the case ~ = 0

100

(which cannot be treated by our method) was proved by Hirschman

[8] by a rather curious method, much earlier than the theorem of

Carleson-Sj~lin was known.

These dilation invariance arguments can be applied to obtain

weighted norm inequalities with weights of the form Ixl a for

other operators (for instance, Pitt's inequalities for the Fourier

transform). This will be discussed in detail in a forthcoming paper.

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101

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Facultad de Ciencias

Zaragoza (SPAIN)