DELGADO, M.

GUERRA, P. JIMENEZ

Delgado, M.

Guerra, P. Jiménez

Osaka University Knowledge Archive

Orlicz classes of Hardy-Littlewood maximal functions

Osaka City University Repository

Delgado, M. and Guerra P.J.
Osaka J. Math.
35 (1998), 1-14
ORLICZ CLASSES OF HARDY-LITTLEWOOD
MAXIMAL FUNCTIONS
M. DELGADO and P. JIMENEZ GUERRA
(Received July 29, 1996)
0. Introduction
It is well-known that for a locally integrable function / : Rn —> R the Hardy-
Littlewood maximal function Mf is defined by
Mf(x) = sup (ϊ^y / \f(t)\dt :xeQ\ (xe Mn),
I I V I JQ )
where the supremum is taken over all bounded cubic intervals Q of Rn containing
x and |Q| denotes the Lebesgue measure of Q.
Assuming that f has support in the unit ball B a well-known theorem of Hardy
and Littlewood [8] states that if / G £(log+ L) then Mf G Ll(B}. Later E.M. Stein
[11] proved that the converse of the above mentioned theorem also holds. Also N.A.
Fava has proved that if f is again a function with support in the unit ball B and a
is a non negative number then the function (M/)(log+ Mf)a is integrable on B if
and only if / G L(log+ L)α+1.
M. Delgado and P. Jimenez Guerra [3] have defined the Hardy-Littlewood
maximal functions Mkf of order k G N (for a measurable function / : Rn —» R) by
iteration of the Hardy-Littlewood maximal operator, proving, among other results,
that Mkf G Ll if and only if / G L(l + log+ L)k.
The main object of this paper is to establish different characterizations of the
Orlicz classes of the Hardy-Littlewood maximal functions of arbitrary orders and to
state some relations between these classes according to its orders, and also between
these classes for a given Young function and the corresponding ones for its associated
function and its truncated function (also of different orders). Thus, many of the
known results about these questions are improved. In particular, Corollary 13, which
states that if Φ is a Young function satisfying the Δ2-condition and / G L(Φ) then
Mkf G L(Φ(<5)) if and only if / G L(Φf*) for every 6 > 0 and k G N, contains the
above mentioned results of Stein, Fava and Delgado-Jimenez Guerra.
For every measurable function / : Rn —> R, we will denote by Mf the maximal
function obtained from / applying the Hardy-Littlewood maximal operator associ-
ated to the differentiation basis formed by the collection of all the bounded cubic
2 M. DELGADO and P.J. GUERRA
intervals Q of Rn, i.e.,
Mf(x) = sup ||1- j \f(t)\dt : x G QJ (x G Rn),
where the supremum is taken over all the bounded cubic intervals Q of Rn containing
x and IQI denotes the Lebesgue measure of Q.
Following [9] we will say that Φ is a Young function if
= / φ(
s
)ds (t> 0),
Jo
with the function φ : [0, -hoc) —> R having the following properties:
i) φ(ΰ) = 0,
ii) φ(s) > 0 if s > 0,
iii) φ is right continuous at every point s > 0,
iv) φ is nondecreasing on (0, +oc),
v) lim
s
^+00 φ(s) = -foe.
The function φ is called a density function of Φ and L(Φ) will denote the
corresponding Orlicz space of the real measurable functions defined on Rn.
If φ : [0, +00) —* R is a function verifying i), iv), v),
ii7) there exists a > 0 such that φ(s) > 0 for every s > α,
and
iii7) φ is right continuous at every point s > 0,
then the function
Γ
t
o(s)ds (t > 0)
will be called a Young function in wide sense.
As it is well known, for every Young function Φ, there exist <50, <$ι > 0 such
that Φ~l(t) > t for every t £ (0,<50) and Φ(s) > s for every s > δι. From now on
we will refer to <50 and δι in this sense.
DEFINITION 1. Let Φ be a Young function with a density function φ verifying
the following conditions:
1.1. φ satisfies the Δ2-condition.
1.2. f Q ( φ ( s ) / s ) d s < +00 for every t > 0.
Then, the associated function to Φ will be the function Φ
α
 : [0, +00) —> [0, +00)
defined by
HARDY-LITTLEWOOD MAXIMAL FUNCTIONS 3
If Φ : [0, -hoc) — > [0, +00) is a Young function with a density function Φ and
δ > 0 then the truncated function of Φ in 6 will be the function Φ(<5) : [0, -hoc) -» R
defined in the following way:
,„ ( 0 if 0 < M < 6
V(δ)u = < ~ ~
if u>δ.
Clearly the function Φ^) is a Young function in wide sense since the function
ί 0 if 0 < s < δ
Φ < 6 > ( s ) = < - ~[ Φ(s) if s > δ
verifies the conditions i), ii'), iϋ')» iv) and v), and it is a density function of
Moreover, for every measurable function / : Rn — > R we have that
\f\>6
if and only if
and
r
u
 φ(*)(s)
Φ^)(u) = u I — ds
for every u > 0.
Proposition 2. //*Φ is <z Young function with a density function φ verifying 1.1
#«£/ 1.2, ί/ze« ί/ze associated function Φ
α
 /5 αfao α Young function having a density
function satisfying the ^-condition.
Proof. Clearly if the function φ
a
 : [0, +oc) —> R is such that
if u = 0
+ <^(w) if M > 0
V ^o ύ"
then
Φ«
Γ
a
(u) = / (^α(
Jo
4 M. DELGADO and P.J. GUERRA
for every u > 0 and Φ
α
 is a Young function having φ
a
 as a density function. Let us
prove now that φ
a
 satisfies the Δ2-condition. In fact, since the function φ satisfies
the Δ2-condition then there exist /c0 > 0 and M0 > 0 such that φ(2u) < k0φ(u) for
every u G (M0, +00) and therefore,jf MO
MO
Jo L Jo
and φ
a
(2u) < koφ
a
(u) + c for every u > MQ with
-L -ds.
Since \imt^+00φa(t) = +00 then there exists M > M0 such that ψa(u) > c and
φ
a
(2u) < (/c0 + l)φa(u) for every u > M. D
Proposition 3. Let f : Rn — > IR be a measurable function and Φ a Young
function. If there exists δ > 0 such that
(3.1) / Φ(|/(t)|)dt<+oo
J\f\>δ
then
(3.2) Φ(|/(ί)|)dt<+oo.
J\f\>6
Proof. Clearly it can be assumed that 6 < δι since in the other case the result
is trivial. It follows from the properties of the Young functions that there exists
h > 0 such that Φ(u) > h for every u e (<5, δ\). Therefore
and it is deduced from (3.1) that
\f(t)\dt < δl\{x eRn:δ< \f(t)\ < δ,}\ 4- / Φ(|/(t)|) dt\f\>δ J\f\>δl
<τ ί Φ(\f(t)\)dt+ (
n
 Jδ<\f\<δl J\f\>δι
< +00. Π
HARDY-LlTTLEWOOD MAXIMAL FUNCTIONS 5
Lemma 4. For every n e N, there exist c
n
, c* > 0 such that the inequalities
(4.1) \{χ e Rn : Mf(x) > λ}\ < —*L- / \ f ( t ) \ d t(i-p)A7|/ |> p λ
and
(4.2) \{x G Rn : Mf(x) > μ}\ > ^  f \ f ( t ) \ d t
I* JMf>μ
hold for every measurable function f : Rn —> R and all the real numbers λ, μ > 0
andQ < p < 1.
Proof. It is enough to proceed as in the proof of the similar results stated in
[7] for integrable functions. D
Theorem 5. Let f : Rn —> R be a measurable function, Φ a Young function
with a density function φ verifying 1.1 and 1.2, and 0 < <52 < <5o such that
(5.1) / Φ(|/(f)|)dί<+oo.
J\f\>δ2
Then for every δ G (#2, <$o) the following assertions are equivalent:
5
 !
5.2.
Proof. Let us consider <5 G (^2^0)- Clearly
Φ(Mf(t))dt = - λdω(X)
with ω(X) = \{t G Rn : Φ(M/(ί)) > λ}|, and so integrating by parts we obtain that
Λ+OO
/ Φ(M/(t))dt < [-;
/»+oo
- lim λω(λ) + / α;(λ)dλ.
Since α (λ) = \{x G Mn : M f ( x ) > Φ"1^)}], it follows from Lemma 4, Proposition
3 and (5.1) that
6 M. DELGADO and P.J. GUERRA
In a similar way,
lim λω(λ) < 2c
n
 lim / | /WI*
= 2c
n
 lim / |/(£)| <i£M-+OO 2μ ym>μ
and since the function φ satisfies the Δ2-condition we can find k > 0 and K > 0 such
that Φ(2μ) < kΦ(μ) is verified for every μ > K, and considering the nondecreasing
function η(μ) — Φ(μ)/μ we have that
η(2μ) I Φ(2μ) ^ fe
2 Φ(μ) ~ 2
for every μ > K and then it is deduced from (5.1) that
lim λω(λ)<c
n
 lim kη(μ) \f(t)\dt
- + oo μ-+oo J\f\>μ
<c
n
k lim \f(t)\η(\f(t)\)dt
-c
n
A: lim / Φ \ f ( t ) \ d t
μ
^+°°J\f\>μ
= 0
and therefore 5.1 holds if and only if
r+oo
/ ω(X)dX < +00.
Jδ
Let us suppose that 5.1 holds, then
/»+oo
/ ω(X)dX
Jδ
r+oo -^
+OO
|/(ί)| / ^dμdt
HARDY-LITTLEWOOD MAXIMAL FUNCTIONS
α(μ)l'/(ί)l
at
and since it follows from Proposition 3 and (5.1) that
j ^ |/(ί) |dt<+oo,
we have that
j
 ι
 Φ
α
(|/(*)|)dί<+oo
and 5.2 holds because
l/ l>*
<h
a
\{xeEn:δ<\f(x)\<φ-1(δ)}\ +
<ha
P
< + 00,
where ft
α
, p > 0 are respectively an upper bound of Φ
α
 and a lower bound of Φ
α
on [δ,Φ~l(δ)].
Let us suppose now that 5.2 holds. It follows from Lemma 4 that for every
0 < 7 < 1 we have that
Λ+OO H-oo j^  Λ
/ ω(X)dλ<—^— _ 1 / λ λ / |/(t)|dtdλ
Λ i -7 Λ φ (Λ) 7ι/
/
+OO Λ f
- /
.φ-Hί) M/|>M
c /• / |/(ί)
= T^ / I / W I /
*-~Ύ jf>
Ί
Φ-
ι
δ J^Φ-ι
w V 7
C
n
 ΦαίΦ""
1
' """
1-7
8 M. DELGADO and PJ. GUERRA
If 7 e (1/2,1) and 7Φ~1(<5) > δ, then it follows from Proposition 3 and (5.1) that
(5.2) j
 ι
 |/(ί) |A<+oo.
Moreover we have that
Λ
 (*\ ήv(*)j [2X
M-) ! / -T *^ ! /\7/ _ J- Jo s i JQ
φ(s)
Φ.(λ) 7
[2
h
-ds
s)ds
, iΓ^
IP
/.2>
±Jχ_
I φ(s}ds
Jo
\
φ(s)ds
7 /*/ φ(s}ds
Jo
1 Φ(2λ) ^
 7
7
for every λ > K, where /c and K are the constants obtained before from the fact
that φ satisfies the Δ2-condition. Therefore,
7Φ-i(6)<|/|<x
e Rn : TΦ-1
/ι>«
I/I
^^ i^m /7Φ H0) ΛΦ-1
^ f
1-7 7|
rΐ /1
- 7 7 /
Φ
α
with ft
α
 an appropriate bound, and 5.1 holds since it follows from (5.1) and
HARDY-LITTLEWOOD MAXIMAL FUNCTIONS 9
Proposition 3 that
/ \f(t)\dt<+oo. D
J
Ί
φ-i(6)<\f\<K
Corollary 6. With the notations of Theorem 5, //(5.1) is verified andδ" > δ2,
then the following assertions are equivalent:
6.1. fMf>δ,,Φ(Mf(t))dt<+oo.
6.2. f\f\>δ,fΦa(\f(t)\)dt<+oo.
Proof. Let δ" > δ0 be and suppose first that 6.1 is verified. Then for every
δ G (#2, £o) we have that
Φ(Mf(t))dt = I Φ(Mf(t))dt+ I Φ(Mf(t))dt
Mf>δ Jδ">Mf>δ JMf>δ"
< Φ(«") \{x G Rn : δ < Mf(x)}\ + / Φ(Mf(t))dt
JMf>δ"
and it follows from Lemma 4 and Proposition 3 that
<Φ(<$")7—\^~ ί I / W I * + / Φ(Mf(t))dt
Mf>δ M _ £?. ] g J\f\>δ2 JMf>δ"
\ 6 )
< -hoc.
Now it is deduced from Theorem 5 that 5.2 holds and so 6.2 is verified.
If it is assumed that 6.2 holds, then for every δ G (<52, <$o) we have that
< Φ
α
(«") \{x € Rn : δ < \f(x)\ < δ"}\
\f\>δ \f\>6"
\f\>δ2 \f\>6"
< +OD
and it follows now from Theorem 5 that 5.1 is verified, and therefore 6.1 also holds.
D
Corollary 7. Let Φ : [0, -hoc) — > [0, -hoc) be a Young function satisfying the
Δi-condition, f : Mn — > R α measurable function and δ,δ2 > 0 swc/z ίΛ«ίO < δ2 < δ
and
(7.1)
10 M. DELGADO and P.J. GUERRA
Then
(7.2)
if and only if
(7.3) Φ (!/(*)!)*< +00,
Corollary 8. Let Φ : [0, +00) — » [0, +oc) Z?e a Young function satisfying the
Δ<2-condition and 6 > 0. Then for every f e I/(Φ) we /zαve ίλαί M/ e L(Φ^) //
DEFINITION 9. From the Hardy-Littlewood maximal operator, the k-iterated
maximal function Mkf (k = 2, 3, . . .) is defined in the following way:
(9.1) M k f ( x ) = sup I — / M fc_ι/(t)dt : x G Q ^ (x G Rn).I I V I «/Q J
Sometimes we will write MI/ instead of M/.
In a similar way, for a Young function Φ with a density function φ, δ > 0 and
k — 2,3,..., we will use the following notations:
φ(t) if 0 < t < δ
^'(0=: .. . „.-,. .. _,
-ds it t > o,
s
Φ(t) if 0 < t <
if ί>«5,
if
*fc* (*) =
and
>f-1*m if
Proposition 10. Lei Φ : [0, +00) -* [0, +oc) be a Young function, f : Rn
HARDY-LlTTLEWOOD MAXIMAL FUNCTIONS 1 1
a measurable function, k G N and <5, <52 G M such that k > 2, 0 < δ2 < δ,
(10.1) / \f(t)\dt<+oo
J\f\>δ2
and
(10.2) / Φ(Mkf(t))dt < +00
JMkf>δ
Then
(10.3) / Φ(Mif(t))dt < +00
JMif>6'
for every ί e N and δ' e R sw/z ίλαί z < fc #«;/ δ2 < δ' < δ.
Proof. Let us suppose first that k = 2. If δ2 < δ
1
 < δ then, since Mf < M2f
a.e. and the function Φ is nondecreasing, it follows from Lemma 4, (10.1) and (10.2)
that
Φ(Mf(t))dt= ί Φ(M/(ί))dί+ / Φ(Mf(t))dt
' Jδ'<Mf<6 JMf>δ
< Φ(δ) \{t € IT : M f ( t ) > δ'}\ + / Φ(M2f(t))dt
JM2f>δ
<ψ^ ί !/(*)!*
0 - β2 Λ/Ί><$2
< +00.
Suppose now that the result holds for 2 < k and let us prove it for k +1. Then,
we have that
(10.4) / Φ(Mkf(t))dt < ί Φ(Mfc+1/(ί))dt < +00
JMkf>δ J Mk+ιf>δ
and therefore, (10.3) is verified for every ί e N and δ' e R such that z < /c and
^2 < <5
;
 < δ. In particular, we also have that
(10.5) / Φ(Mfc_!/(ί))dί < +00
7M fc_1/>^
/
for every δ" £ ( δ 2 , δ f ) .
Therefore, it follows now from (10.4), Lemma 4, (10.5) and Proposition 3 that
Φ(Mkf(t))dt< ί Φ(Mkf(t))dt+ ί Φ(Mkf(t))dt
f>δ' Jδ'<Mkf<δ JMkf>δ
12 M. DELGADO and PJ. GUERRA
G Mn : Mfc/(ί) > ί'}| + Φ(Mkf(t))dt
JMkf>δ
^W, ί A**.!/ (ί)dt + I Φ(Mkf(t))dt
-° JMk.lf>δ" JMkf>δ
< -hoc,
and the proof is finished. D
Let us remark that Proposition 3 ensures that (10.1) holds in particular if
Φ(|/(t)|)dt<+oo.
Proposition 11. Let f : Rn -^ R be a measurable function, Φ a Young func-
tion, k G TV and δ > δ2 > 0 such that
(11.1) / Φ(|/(ί)|)dί<+oo
- / / > « 2
and
(11.2) / Φf*(|/(t)|)Λ<+oo.
^|/|>«2
(11.3) /
 t ΦS«*(|/(ί)|)dί < +00
for every i < k and δf G (#2? <$)•
Proof. In fact, if δ' G (#2,6) and i < k we have that
/ ΦJJ**(|/(ί)|)dί < Φβi*(δ) \{t G Rn : δ' < |/(ί)| < <5}| -
<^B/ Φ(\f(t)\)dt+ ί Φf*
> « / l f l > 6
< +00. D
Theorem 12. Let f :Rn -^ R be a measurable function, Φ a Young function
verifying the conditions 1.1 ##£/ 1.21 and 62 G (0,<50)
(12.1) / Φ(|/(f) |)dt<+oo.
J\f\>δ2
1This condition can be avoided using Young functions in the wide sense.
HARDY-LITTLEWOOD MAXIMAL FUNCTIONS 13
Then for every k G N and δ > δ2 the following conditions are equivalent:
12.1. "
12.2. flfl>6Φ*δk*(\f(t)\)dt<+00.
Proof. It follows trivially from Corollary 6 that it is enough to prove the
result for fe(> 2) assuming that it holds for k-1, thus let us make this assumption.
If 12.1 is verified then it follows from Proposition 10 that
(12.2) / Φ(Mk-lf(t))dt<+oo
JMk-ιf>6'
for every δ1 e (63,6) and so it is deduced from Corollary 6 that
(12.3) / Φ*
δ
l
*(Mk.lf(t))dt<+oo.
Moreover, it follows from (12.2) that
Φ*
δ
k
-
l
*(\f(t)\)dt<+oo
for every δf G (<$2,<5), and then Proposition 11 assures that
(12.4) / ΦS1*(|/(ί)|)Λ<+oo
J\f\>6'
for every δ1 G (62, δ). Now 12.2 is deduced from (12.4) and (12.3) (applying the
induction hypothesis to Φ^1*).
If 12.2 holds then it follows from Proposition 11 that
*\f\>6'
for every δ1 G (<$2,<5) and so
Φ(Mk-ιf(t))dt < +00
^f>δ'
for every δ1 G (62,6) and 12.1 follows immediately from Corollary 6. D
Corollary 13. Let Φ be a Young function satisfying the ^-condition, f G
L(Φ), δ > 0 and k G TV, fλέ?/ι Mfc/ G L(Φ^) // rarf only iff G
ACKNOWLEDGEMENT. The authors are greatly indebted to the referee for his
careful reading of the manuscript.
14 M. DELGADO and PJ. GUERRA
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M. Delgado
Dpto. Matematicas Fundamentales
Facultad de Ciencias
U. N. E. D.
c) Senda del Rey, s/n
MADRID 28040 (SPAIN)
P.J. Guerra
Dpto. Matematicas Fundamentales
Facultad de Ciencias
U. N. E. D.
c) Senda del Rey, s/n
MADRID 28040 (SPAIN)


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Orlicz classes of Hardy-Littlewood maximal functions

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