Generalized Carleson inequality on spaces of homogeneous type

GENERALIZED CARLESON INEQUALITY ON SPACES OF HOMOGENEOUS TYPE

YOON JAE Yoo*

1. Introduction

The purpose of this paper is to generalize the Carleson inequality, which is known to play important roles in harmonic analysis. The result given here is a generalization of Coifmann, Meyer, Stein [CMS].

Asimilar result is shown by Deng[D].

2. Preliminaries

Let (X, d,p.) be a space of homogeneous type, i.e.,X is a topological space, p. isa positive Borel measure onX, and disapseudo-metric on Xj more precisely, we assume that there are constant A and K so that for all x, y,z E X and all 8>^0,

i) d(x, y) ~ 0;d(x, y) = 0 ifand only if x= Yj ii) d(x,y) = d(y,x)j

iii) d(x,z):5 K[d(x,y) +d(y,z)], iv).V(B(x,28) :5 AI'(B(x,8)),

where B(x,8) = {y EX: d(x, y) < 8}. We also assume that the collection {B(x, 6)} forms a basis for the topology ofX. Property iv) is referred to the doubling property of p.. For the details, see [CW].

Let X+ = X X(0,00), which is an analogue of a generalized upper- half space over X. Let r(x)'; ((y,s) E X+ :x E B(y,s)}. This is an analogue of nontangential or conical region. For any set E eX, the tent over E is the set

T(E) = {(x,r) EX+ :B(x,r) c ^E}.

Received August 12, 1994.

Key words: Carleson measure, Carleson inequality.

*This researchissupported bythe research grant of KOSEF 1992.

It is. then easY tov:erify that T(E) = IUxiE^reX)]c. For a measurable functionI defined onX+, fot xEX,p;::: 1 and a ER,let

1

[ 1

i .

Tp,,,,(f)(x) = ^sup (B(t5)) I/(Y, t)IP tOl '

xEB(6)p. T(B(6»·

where B(8) is a ball with radius 8> 0. Let M(f) denote the Hardy- Littlewood's maximal function

M(f)(x) = sup (;(8» ^f I/(y)ldp.(y).

xEB(6)p. JB(6)

For a positive Borel measureAonX+ = X x [0,00),if there exists a constant C» 0 such that A(T(B(8»)) :::; Cp.(B(8)) fdr any <5>0, then Ais called a Carleson measure. For a measurable function I on X+, we define a p-areafunctio~ Ap,OIas follows:

1

Ap^,01(f)( x) = [ ( I/(Y, t)IPd~~~dt] ^{P, if 1:::; p<00, a E R}

11'(x)

and

Aoo,OI(f) = sup I/(y,t)J, if p=00.

(y,t)eI'(x)

For a nonnegative measurable function I on X+, if there exists a

constant C such that .

1 (Idp. < C inf I(x) p.(B(8» 1B(6) - xEB(6)

for all 8 > 0, then I is said to belong to the class Al [M].

3. Generalized Carleson inequality We begin with lemmas.

LEMMA 1 (COVERING LEMMA OF VITALI-WIENER TYPE). Let X be a space of homogeneous type. Let E c ^{X be a} bounded set i.e., E is contained in some ball. Let r(x) be a positive number fot each x E E. Then there is a sequence of balls B(Xi, r(Xi», Xi E E, such tbat the balls B(xi,4Kr(xi» coverE, wbere K is the constant in tbe triangle inequality. Furthermore, every x EE is containedin some ball B(xj,4Kr(xi» satisfying r(x) :5 2r(xi)'

(See Theorem 1.2 on p.69 of [S])

PROPOSITION 1. Letp~ 1. Tben Tp,a is of weaktype (p,p).

Proof. Fix A>O. Set

{lA = {x EX:Tp,a(f)(x) >A}.

Assume first that !lA is bounded. Apply lemma 1 to obtain a sequence of disjoint ball B(xi,ri) so that !lA C UB(Xi,4Kri) and

1 f If(y,t)lpdJl(y)dt > AP.

Jl(B(Xi, ri» IT(B(xi,r;}) t^a

Then from the doubling property of Jl it follows that Jl(!lA) :5 2:Jl(B(Xi, 4Kri»

:5 C2:^Jl(B(Xi,ra)

:5 ~ f If(y,t)IPdJl(~)dt

A IT(B(z;,r;)) t

:5 ~

L+

If!lA is not bounded, fix a E X and R > O. Then!lAnB(a, R) is a bounded set, so we can apply the above argument to obtain

Jl(!lAnB(a,R»:5 ~ f If(y,t)IPdJl(y)dt.

/\p Jx+ t^a

Letting R ^{--+ 00} we obtain the same estimate. This completes the proof.

LEMMA 2. Let (Xl,JL1) and (X2,JL2) be measure spaces and T be a sublinear~pei'ator that transfoI'i:ds measurable· fupctionsIDeX^1,JL1)

into measurable functionsin(X₂,JL2). HT is of weak type(p,q), then

[L\T!(y)\rdJL2]·1^{r :5}CJL2(K)::-tllfllp

holds forall r,O < r < q, and for any subset K of finite measure.

THEOREM 1. Let f EL^I( dradt

),0:5 € < 1. Then TI,a(f)E E Al for anya E R.

Proof. Fix B(6_{0 )} andXo EB(6_{0 ).} Put

A₂ = {B(6) :^Xo E B(6),6 > 6_{0 }.}

Then we have

(1)

where

and

1

l

TI,a(f)(xO):5 sup (B(6)) If(y, t)l ^ta

B(6)€A1 J.L T(B(6»

1 1" If( t)ldJL(y)dt

+ sup y, a

B(6)€A₂ 1-'(B(6)) T(B(6» t

:=A{x(}.)+B(xo},

1

l'

A(xo)= sup' (B(6)) \f(y, t)1 t_a ' B(6)€A1 JL . T(B(6»

To estimate B(xo), suppose B(o) E A_{2 •} Since B(o)nB(oo) :F ^{4> and}

0> 00, there exists a constant C ~ 1 such that B(Co) :) B(oo). Hence

1 I dp.(y)dt

p.(B(o» 1T(B(6» If(y,t)1 t^Q

< p.(B(Co» 1 I ^{I f ( )}Idp.(y )dt - p.(B(o» p.(B(o» 1T(B(6» y,t t^Q

~ C inf T1cif)(x) :rEB(C6) ,

~ C inf T1,a(f)(x), :rEB(60)

and so

(2) B(xo) ~ C inf T1,a(f)(x).

:rEB(60)

As forA(xo),note thatB(Coo) :) B(o)for some constantC. Indeed, since B(oo)nB(o) :F ^4> and 0:500, B(o) c _B(Coo)for some constant C by the property of the pseudo-metric. Thus T(B(o» c _T(B(Coo»,

and so ifwe put h(y, t) =!(y, t)XT(B( Cc5o»(y,^t), then we have (3) A(xo)= sup 1 I I!(y,t)1dp.(y)dt

B(6)eA2 p.(B(o» 1T(B(6» t^Q

< sup 1 I If (y t)1dp.(y)dt - B(6)EA² p.(B(o» 1T(B(6» 1 , t a

~ T1,Q(fd(x).

Since 0 ~ ^f <1,

(4) T1,Q(f)(xo)f ~ A(XO)f +B(XO)f.

Hence it follows from (1),(2),(3), and (4) that p.(;(0» j ^T1,a(f)(xO)fdp.(xo)

(B(60»

(5) ~ (B~O» ^I A(XO)fdp.(xo) +C inf T1,a(f)(xr

p. 0 1B(60) :reB(60)

~ (B~O» ^{I T1},aUd(xrdp.(x) +C inf T1,Q(f)(X)f

p. 0 1B(60) xeB(60)

By lemma 2, the first term. of the right-hand side of (5) is less than

[

1 f ^dP.(Y)dt]E

C p.(B(So» 1B(06

0) If(y,t)1 tOt and so (5) is reduced as

which proves theorem.

LEMMA 3(WHITNEY DECOMPOSITION). LetX bea space ofhomo- geneous type. Let 0 c ^{X be} an open set. Then there are positive constants M,Cl,_C2,Ca, and^asequence {B(xi,6i)} of balls such,that

(i) Ui^{B{XhSi) = 0}

(H) B(Xi'C16i ) c 0 andB(Xi, C26i)nQC=I- cP,

(iii) the balls B(Xi,CaDi) are pairwise disjoint;

(iv) no pointin0 liesin more than M of the ball B(xi,6i).

(See [eMS].)

THEOREM .2. Let i ^{+ ~ = 1, 1 ~ p ~ 00.} Suppose there exist constants A > ^O,B > 0 such that A6n ~ p.(B(x,6» _~ B6n for all S> o. Then there exists a constant C such that

{ dp.(y)dt (

1x+ f(x, t)v(x, t) tOt _~ C1xAp,n_I+Ot(f)(x)Tq,Ot(v)(x)dp.(x)

for all !,v.

Proof Assume first that 1~P<^00. For each integer k, set

and 1

. nk=^{{x EX:} M(Xnk)(X) >'2}.

Observe that Ok ~.nk+landnk :) fik^{+1 for each k. Further U~_oo}

T(nk) contains the support off. By Lemma 3, there exist collections

of disjoint balls B(c5k,j) and B(tS'k) such that fh C UjB(ctSk,j) and Ok C UjB(cc5k,j). Also we can choose a constant cso that T(Ok) C UT(B( cc5k,j)) and thatT(flk)^C UjT(B(cc5'k,j))' PutIk,j =T(B(cc5k,j)) and Jk,j = T(B(cc5'k)) for simplicity. Then it follows from Holder's inequality that

li+

< f: ^f f(y, t)v(y, t)dJ.l~~)dt

k=-ooIT(O;l\T(O;+I)

(1) < ,t;oo

t,

00 00 1

:S k~OO; ^[lk,j\u._Jk⁺1

,. If(y, t)IPdJl~~)dt] ^p

[j_{Jk,j\u.Jk+1 ,.} ^{Iv(y,t)19dJ.l~:)dt] q.}1

Put

and

IT we show

and

then "we have

I"( ^fey,t)v(y, t) dp.(:)dtI·

lx+ ^t··

(4)

Since Tq,a( v)E AI, it follows from proposition 1 that

and so

(5) L~^{Tq,a(v)dp. $ 4}

L

$ C

L

= C ( Tq,cr(v)(x)dp(x).

lnk "

Therefore by (5), (4) can be rewritten as

ILJ(y, t).(y,^t)_dP~~)dtI^{S c},=f;~^2'fa: T".(')dp

(6) $ C f ^2k

1

k=-oo n~

=C (·lxAp,n-I+a(f)Tq,~(^{v )dp(x).}

So ifwe show (2) and (3), then (6) holds for 1~p <00.

To see (2), observe that (7)

f Ap,n-l+a(f)(x)Pdl-t(X)

1B(c6k,j \u, B(c6.+1,,)

= f ^dp(x) f If(y,t)IPdl-t~~:dt

lB(c6:,j \u,B(c6Hl,,) lr(x) t

=

1

[B(c6k,j)\u,B(c6.+1,'»)xX+ t

~ f If(y,t)lpdp~~~dt f xr(x)(y,t)dp(x).

1J.,j \u, J.+l" t 1B(c6;,j )\u, B(c6Hl,,) For any fixed (y, t) E Jk,j\ ^UII Jk+l,8' we have

(8) B(y,t)n(Ok+l)C :I ^<1>.

In fact, suppose B(y,t) C 0;+1 == UjB(c6k+l,/)' Then (y,t) E T(B(c6kH^,j))' for some j, that is, (y, t) E Jk+l,j, which leads to a contradiction. Hence (8) holds. So we can choose a point P E B(y,t) so that M(XO_H 1)(X) _~ !' which implies

(9) _P(Bt t))_{y, lB(y,t)}f XO.+ 1(x)dl-t(x) ~ ^-21,

and so by (9), we have

(10) 1 f Xr(x)(y,t)dl-t(x)

I-t(B(y,t))1B(c6Z,j)\u,B(c6.+_1,,)

~ ¹ f XB(y,t)(x)dl-t(x)

I-t(B(y, t)) 1B(c6j"j)\u,B(c6.+1,,)

~ ¹ f XB(y,t)(x)dl-t(x)

I-t(B(y, t))1^{B(c6i.,j)\u,B(c6.+}1 , , )

= (Bt t)) f [1- XB(y,t)nU,B(c6.+1,,)(x)]dl-t(x) I-t y, 1^B(y,t)

= (Bt t)) f [1 - XB(y,t)nOHl (x)]dl-t(x) I-t y, 1^B(y,t)

>-1 - 2'

In the second inequality of (10), we used the fact B(y, t) C B(cbk,j).

Thus

( Xr(x)(y,t)dp(x) ~ ^Cp(B(y,t»

J^{B(coi.,j)\u.B(cbk+l,.)}

>Ctⁿ

- ,

which implies (2). To estimate (3), note that

This shows (3).

Finally, assumep = 00. Then

I( fey, t)v(y, t) dp(:)dtI

Jx+ . t

00

1

~ L If(y, t)v(y, t)! Pt:

k=-oo T(Ok)\T(Ok+t>

f f ^2k+l

1

k=-oo j=1 Ik,j

~ ^C f ^2kf

1

k=-oo j=1 B(COk,j)

:5 C

L

This completes the proof.

References

[CMS] Coifman, R. R., Meyer, Y., and Stein, E., Some new junction spaces and their applications to harmonic analysis,J.of Func. Anal.62 (1985), 304-335.

[CW] Coifman, R. R. and Weiss G., Extension of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645.

[D] Deng, D. E., Ona generalized Carleson inequality, Studia Math. 78 (1984), 245-251.

[M] Muckenhaupt,8., Weighted norm inequalities for the Hardy maximal func- tion, Trans. Amer. Math. Soc. 165 (1972), 207-226.

[8] Sueiro,J., On maximal junctions and Poisson-Szego integrals, Trans. Amer.

Math. Soc. 298 (1986), 653-669.

Department of Mathematical Education Teachers College

Kyungpook National University Taegu 702-701, Korea