Weighted Orlicz space integral inequalities for potential maximal operators

WEIGHTED ORLICZ SPACE INTEGRAL INEQUALITIES FOR POTENTIAL MAXIMAL OPERATORS

YONG MAL KIM AND YOON JAE Yoo

ABSTRACT. We characterize a condition forMepto be of weak type (Ch,ep2) in terms of Orlicz norms.

1. Introduction

Given a function f in Rn, we define a function Mf in _R~+l - {(x, s) : x ERn,s ~ O} by setting

Mf(x,s) ^=sup_{I~I

h

It is well known that this maximal operator M controls the Poisson integral defined by, for x E Rn and s ~0,

P(J)(x,s) = r f(y)P(x - y,s)dy,

JRn

where

ens

P(x,s) = (lxl²+s2)n+l/2

is the Poisson kernel. For a given positive measure v onRnX[0,00),un- der what condition onv can we assert thatM is bounded from £P(Rn) into weak-£P(Rⁿ x [0,00),v)? Carleson([3]) showed that this was equiv- alent to the Carleson condition and later Feffermann-Stein([5]) found a sufficient condition, and Ruiz([13]), Ruiz-Torrea([14]) unified all these

Received January 15, 2000.

2000 Mathematics Subject Classification: 42B25.

Key words and phrases: maximal operator, Orlicz norm.

This research is supported by Kyungpook National University Fund, 2000.

614 Yong Mal Kim and Yoon Jae Yoo

results. Recently, Gallardo([6]) and Chen([4]) obtained characteriza- tions in terms of the Orlicz norm andin[8], we obtained a characteriza- tion for the fractional maximal operator. Inthis paper, we characterize a condition for Mep to be of weak type (~b ~2) having four weights in the Orlicz norm. In the next theorem, we shall assume that cp is essentially nondecreasing, Le., there exists a positive constant p for which

cp(t) ~ pcp(s), t ~ s and

lim cp(t) = o.

t-+oo t Our resultis as follows.

THEOREM 3.1. Let Mep! be the potential maximal operator on lR~+l. Mep! is defined by

(1) Z(Q) ~s,

where Z(Q) denotes the sidelength ofQ. Letu,v be weights on ]Rn, W

be a weight on _]R~+l and J.L be a nonnegative measure on _{]R~+l. ~1} and ~2 areN-functions with complementary N-functions W1 and W2, respectively. Assume further that_~20~1¹ is convex. Then weak type boundedness, Le.,

(2)

~21[ ^[ ~2~~~(x,S))dJ.L(X'S)] ~ ~11[1n ~1(CI!(x)lu(x))v(x)dx]

J{(x,s)EJR+ :M<pf(x,s»>..} JJR

holds if and only if

(3) [ ['Y(A,Q) CP(IQI)]

JQW1 CAu(y)v(y) IQI v(y) dy < ,(A,Q) ^<00

holds for each cubeQ, where (4)

'Y(A,Q) = ~1 o~21

[,4

When rp(IQJ) = 1, the Hardy-Littlewood maximal operator is ob- tained. The fractional maximal operator Ma(O < ^0: < ⁿ⁾ is given by rp(IQJ) = _IQI~· Maximal operators connected to the Bessel poten-

~

tial operator are defined by rp(jQI) = fJQ'ⁿ 'I/J(s) ds, where 'l/J is the derivative ofrp.

2. Preliminaries

DEFINITION 2.1. Let <P : [0, (0) ^---t IR. be a function satisfying (i) <p(s) > 0 for all s~ 0;

(ii) lims-+o<p(s)js= 0;

(iii) _lims-+oo<p(s)js= ^00.

Then <P is called an N-function. Each N-function has the integral representation: <p(s) = f; <p(t) dt, where <p(s) > 0 for s > 0,<p(0) = 0 and <p(s) ^---t 00 as s ^---t 00. Further, <p is right-continuous and nonde- creasing. <p is called the density function of<P. Define p : [0,(0) ^---t IR.

bypet) =sup{s : <p(s) ~ t}. Thenpis called the generalized inverse of

<p. Finally, define

'11(t) =

it

and'11 is called the complementary N-function of<P. For further details, see [10].

DEFINITION2.2. An N-function<P is said to satisfy the .6.₂-condition

. [0 )'f <1>(2s)

m ^{,00 1} sUPs>o <1>(s) < ^00.

REMARK 1. If ijJ is the density function of <P, then <P satisfies the .6.2-condition if and only if there exists a constant ^0: > 1 such that s<p(s) < o:<P(s), for anys> o.

REMARK 2. If'11 is the complementary N-function of <P, then st ~

<p(s)+'11(t) for all s, t ~ o. Futher, = holds if and only if<p(8-) ~ t ~

<p(s) or else p(t-) ~ s~ pet).

616 Yong Mal Kim and Yoon Jae Yoo

DEFINITION 2.3. Let(X, M,f-L)is a (I-finite measure space and~be an N-function. Then the Orlicz space£,ip(df-L) and£,4.(df-L) are defined by

£,ip(df-L) = {f:

Ix

and

£,4.(df-L) = {f : fg E £'1(df-L) for all 9 E £,\f!} , where Wis the complementary N-functionof~.

Keeping these definitions and notions, the following properties about the Orlicz space will be used in the proof of the Theorem 3.1.

PROPOSITION 2.4.

(i) The Orlicz space £,4.(df-L) is a Banach space with the Orlicz norm

IIfllip=sup { /Ifgldf-L :9 ES\f! } ,

where S\f! = {g E£,\f! :I W(/gl) d/-L _~ I}, or with the Luxem- burg norm

IIfll(ip) =inf{A> 0: / ~('{I)^df-L ~ I}.

(ii) (Holder's inequality) Hf E £,4.(df-L) and9 E £,; (df-L), then

(5) IIfgllip _~ 211fll(ip)lIgll(\f!)'

(iii) (Young's inequality)

(6) ab~ ~(a)+^{W(b) for all a, b}>O.

LEMMA 2.5. Let~ be an N-function with complementary function W. Let x and y >^O. Then

(7) ~(x) ~ x<jJ(x) ~ ~(2x),

(8) and (9)

~(x) +~(y) ~ ~(x+^y)

~ [W~x)] ~ ^W(x).

LEMMA 2.6. Suppose that f is an integrable function in _~n. For each A>0, let EA = {(x, s) E ]R~+1 ^: Mcpf(x, s) > A}. Then, ifEA is not empty, wehave

(10)

(11)

whereQj is the family of nonoverlapping maximal dyadic cubes satis- fying

~ < <p(IQjl) r ^{f(y) dy·}< _~

4ⁿp IQj! }Qj - 2ⁿ

for each integerj. Furthermore, wehave that

Proof. Following [7](p.160), we let CA = {Pj} be the family of the dyadic maximal nonoverlapping cubes satisfying the condition

A< <p(!Pjl) r ^{f(y) dy.}

!Pjl }Pj

To show that there is such a familyCA' observe that

<p(jQI) r f(y) dy ---. 0

IQI }Q

as Qt ]Rn, since f is integrable and since limt-+oo<p(t) / t = O. If, for some dyadic cube Q,

then Q is contained in dyadic cubes satisfying this condition, which are maximal with respect to the inclusion. Thus, there is a family of maximal nonoverlapping dyadic cubes{Pj } yield

(12)

618 Yong Mal Kim and Yoon Jae Yoo

where Pj denotes the only dyadic cube containing P_{j .} From this dis- cussion, it is clear that

{x E:!R^{n :} M;f(x) >>'} = UP^{j •}

j

Let (x, s) E EA; by definition, there is a cube R containing x with l(R) 2: s such that

>. < cp~~I)

L

Let k be the unique integer such that 2(k+l)n < IRI ~ ^{2- kn .} There are some dyadic cubes with side length 2-^k, and at most 2ⁿ of them, {Ji :i = 1, ... ,2ⁿ} meeting the interior ofR. It is easy to see that, for one of these cubes, say Jl,

~ < cp(IRI) ( f(y) dy.

2ⁿ IRI JRnh

Now, since IRI ~ IJ1¹< 2ⁿIR!, cp(IRI) _~ PCP(IJil) and

4:IJ1¹<

2:

JRnJ1 JJ1

Hence,

~ < cp(!J11) ( f(y) dy.

4ⁿp IJ11 lh

By letting ^Ct/4np = {Qj}, we see that J1 C Qk, for some k, and x ERe Jp ^c Q~.

Onthe other hand, s ~ l(R) ~ l(QJ)' From this, we conclude that

EA c UQ;'

j

Finally, it follows from (12) that

for eachj, concluding the proof of the lemma. o

3. Main Results

THEOREM 3.1. Let Mepj be the potential maximal operator on R~+l. Mepj isdefined by

<p(IQI) r

Mepj(x,s) = ~~g IQI lQ Ij(y)1 dy, where l(Q) 2: s.

Let U and v be weights on Rn, w be a weight on R~+l and J.L be a nonnegative measure on R~+l. epl and ep2 are N -functions with complements q,1 and q,2, respectively. Assume further thatep2 0 epl 1 is convex. Then weak type boundedness, i.e.

holdsifandonly if

r ^['Y(A'Q) <P(IQI)] -

JQ q,1 CAU(X)V(x). IQI v(x) dx ::; 'Y(A, Q) < ⁰⁰

holds for each cubeQ, where

Proof. For the necessity, we follow the idea of [2]. Since \111;10) is increasing in c and has full range R+, for given A> 0, we can choose c such that

r ( ^{c ) v(y) IQI}

JEq,l u(y)v(y) -c-dy =2CA<p(IQI) , EcQ

1 ( c ) v(x)

j(x) = Cq,1 u(x)v(x) -c-XE(x).

620 Yong Mal Kirn and Yoon Jae Yoo

If(y,s) E Q, then yE Q ands::; l(Q). So

cp(IQI) (

M<p!(y, s) = =~g IQI J

Q1!(y)1dy 2: cp~~I)

k

cp(IQD {I ( ^{g )} v(y)

= IQI J_Q ^Cqrl u(y)v(y) -g-XE(y) dy

= 2A> A.

Thus if(y,s) ^E Q, ^then (y,s) ^E EA ={(x,r) ^E _{lR~+l :} M<p!(x,r) >

A}. Hence

,(A,Q) ⁼ <PI 0 <P2¹

[k

::; <PI 0 <P2^{1[ { <P2}(AW(X, s)) dp,(x, s)]

J{M,.,f>A}

::; L

::; L

IQI

= 2CAcp(IQI)^g.

The third inequality follows from (2). Put

.6.

-1

Then

1 (

.6.< 'l/J1 - d y

- E 2u(y)v(y) u(y)

1 (

::; 2 Eqr1 u(y)v(y) ^{- g -}dy

IQI

= 4CAcp(IQlf

Also

Thus

~ ^{> 4CAIQI} r^{'1J [( ,(A,Q) )}<P(IQI)] d

- <p(IQlh(A,Q) lE ¹ 4CAU(Y)V(Y) IQI v(y) y.

r [( ^{,(A,Q) )}<P(IQI)] -

lE'1J¹ 4CAU(Y)V(y) IQI v(y) dy ::; ,(A, Q) < ^00.

As E -+ Q, (3) holds.

For the sufficiency, we follow the idea of [4] and [11]. From Lemma 2.6, for each A> 0, let E>.. = {(x, s) E JR~+l : M<pf(x, s) > A}. Then, ifE>.. is not empty, we have

where Qj is the family of nonoverlapping maximal dyadic cubes satis- fying

~ < <p(lQjl) r ^{f(y)d <} ~

4ⁿp IQj! lQj ^{y - 2n}

for each integer j. Then 4~P < _<p~~~j) IQj ^{f(y) dy and {QJ} is a cov-}

ering ofE>... Hence it follows from (6) that

2,(A,QJ)::;

hj

r ^{n n ,(A, QJ)} <p(IQJI)

= lQj^{23 4} Cpjf(x)lu(x) CAU(X)V(x) IQJI v(x) dx

::; hj

r ^{(,(A, QJ)} <P(IQJI))

+lQj '1J¹ CAU(X)V(x) IQJI v(x) dx

::; hj

622

So

YongMal Kim andYoonJaeYoo

and thus

k~(<P2(AW(X,s)))dp,(x,s) ~^{<P20 <PII}[£j <PI (23n4nCplf(x)lu{x) )v(x)dx].

Summing overj gives

By using that <b₂0 <b1¹ is convex, this last sum is bounded by

<b20 <b1¹[~

k

j

<b1(23ⁿ4ⁿCplf(x)lu(x) )v(x) dX]

::; <b2 0 <b1¹[C~Ln^<b1(23n4ncplf(x)lu(x) )v(X) dx

1

::; <b2O<b1¹[Ln ^{<b1 (Cn}lf(x)\u(x))V(X)dX]. D COROLLARY 3.2. From (3), put w = U = 1,<b₁ = <b2. Then (3) gives

(13)

[IL(Q)CP(,Q')]4>fcp(\QI) f 1/1(_C_)^dy ] ::;Cc, for each c> o.

IQI ~ CIQI JQ v(y) .

Proof We follow the proof of Corollary 3.2 of [8]. o

COROLLARY 3.3. Hip10 <b2^{1has thel:!,.' condition}and (3) holds for w = U = 1, then there exist constants C', C" > 0 such that, for any

c>0,

-1( ^(-))cp(IQD [C' ^f (C)] "

(14) <b10 <b2 JL Q IQI 4>1 1QIJ

Q1/11 v(y) dy ::; C c.

Proof. We follow the proof of corollary 3.3 of [8]. o

EXAMPLE 3.4. Let cp(IQI) = 1 and c= t. From Corollary3.2, (13) gives _A~ of [4], Le.

But (13) is weaker than A;, because an N-function <P of A; ^{in [4]}

satisfies the ~2-condition.

EXAMPLE 3.5. From Corollary 3.3, put dp,(x, t) = u(x)dx0 d8(t), where 8 is the Dirac mass on [0,00), concentrated at O. Also set

<Pl(X) = x; ^{and <P2(X) =} xq

q

, where 1 < p ~ q < 00. Then (14) gives (u, v) E A(cp,p, q) of [12], Le.

cp(IQI) IQlt-i[I~Ik^u(x)dx]t L~Ik^{v(x)-P!-l dxr-i ::;A for all Q.}

References

[1] S. Bloom and R. Kerman, Weighted£if> integral inequalities for operators of Hardy type, Studia Math. 110(1994),35-52.

[2] _ _ , Weighted Orlicz space integral inequalities for the Hardy-Littlewood maximal operator, Studia Math. 110 (1994), 149-167.

[3] L. Carleson, Interpolation by bounded analytic functions and the corona prob- lem, Ann. Math. 16 (1962), 547-559.

[4] Jie-Cheng Chen, Weighted and .cif>-boundedness of the Poisson integral opera- tor, Israel J. Math. 81 (1993), 193-202.

[5] C. Fefferman andE. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115.

[6] D. Gallardo, Weighted weak type integral inequality for the Hardy-Littlewood maximal operator, Israel J. Math. 61 (1989), 95-108.

[7] J. Garcia-Cuerva and J. L. Rubio De Francia, Weighted Norm Inequalities and Related Topics, Elsevier Science, North-Holland, 1985.

[8] Yong Mal Kim and Yoon Jae Yoo, Weighted Orlicz space integral inequalities for fractional maximal operators, Far East J. Math. Sci. 5 (1997), no. 2, 225- 236.

624 Yong Mal Kim and Yoon Jae Yoo

[9] B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 191 (1974), 261-274.

[10] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034 (1983).

[11] C. P'erez, Two Weighted Inequalities for Potential and Fractional Type Maxi- mal Operators, Indiana University Mathematics Journal 43(1994), no. 2.

[12] Y. Rakotondratsimba, Weighted Norm Inequalities for Maximal Functions from the Muckenhoupt conditions, Publications Matem'atiques38 (1994), 25- 36.

[13] F. Ruiz, A unified to Carleson measures and A_p weights, Pacific J. Math. 117 (1985), 397-404.

[14] F. J. Ruiz and J. L. Torrea, A unified approach to Carleson measures and A_p weights Il, Pacific J. Math. 117(1985).

Department of Mathematics Education Kyungpook National University

Taegu 702-701, Korea

E-mail: [email protected]