Analysis Mathematica, 24(1998), 41-58
M a x i m a l f u n c t i o n s w i t h r e s p e c t to
differential bases m e a s u r i n g m e a n o s c i l l a t i o n
A.K. LERNER
Abstract.
In this paper we study maximal sharp functions associated with
arbitrary differential bases. The definition of these functions goes back to the papers
by F. John (1965), and by C. Fefferman and E. M. Stein (1972), where the classical bases
consisting of cubic intervals were considered.
We obtain conditions imposed on the basis, under which inequalities, known earlier
in the case of a basis of cubes, are valid for the considered maximal functions. The
main results are formulated in terms of noniucreasing rearrangements. In the capacity of
applications, we obtain estimates of the rearrangements of subadditive operators acting in
BMO. In particular, the estimate for the Hilbert transform, obtained earlier by C. Bennett
and K. Rudnick, follows.
w
Introduction
We remind the reader that a function
bounded mean oscillation (f E BMO) if
f
E
, , f J J . - s QcR,,
up
1 /Q If(Y)- :ql dy <oo,
T-~
L~oc(I:tN)
is said to have
1 /Q :'
/q = -~I
the supremum extends over all cubes Q c p N. This definition was introduced by JOHN and N'IRENBEI~.G in [12], in which the main estimate for a
function of BMO was proved: for any flmction f E BMO(R N) and any cube
Q C R g we have
A
(1.1) J { x E Q : ]f(x)-fQ] >A}I < c l l Q l e x p (
c2JJfJl.
) (0<A<oo),
where cl, c2 depend only on the dimension N.
After that JOHN [11] and STROMBERG [20] considered the condition:
there exist numbers c~ < 1/2, A > 0 such that for any cube Q c R N
(1.2)
t{x e Q : If(x) - cQI > A}I < ~lql
Received January 15, 1997.
0133-3852/98/$ 15.00
(~) 1998 Akad~miai Kiadd, Budapest
42
A. IC Lerner
(cQ is a constant depending on Q). At the first sight, condition (1.2) is
weaker t h a n (1.1). Yet in papers [11], [20] it has been shown that (1.2) is
equivalent to that f belongs to the class BMO.
T h e definition of BMO generates the notion of the sharp function
introduced by FEFFERMAN and STEIN [6]:
f (x) = og~
sup -~[
1 s If(Y) - fo1 dy.
(1.3)
Obviously, Ilfll. = ]If ~ll~. In turn, the equivalent, approach to the definition
of BMO expressed in condition (1.2) naturally reduces to the maximal
function
(1.4)
M~f(x) =
= sup inf inf{A > 0 : I{Y 6 Q : If(Y) - el > A}I < c~lVl}
Qg:e c6tl.
-
(0 < c~ < 1/2).
-
This function is less known. It was introduced in [11], [20] and studied in
detail in [10]. The h~llowing inequality is true [20]:
(1.5)
6~llM~f[l~ _ Ilfll. -< cllM~fll~
(0 < & < 1/2),
where c depends only on the dimension.
It is easy to see that for all x E R N
f~(z) _< 2Mf(:~.),
M~f(x) <_ i6tf : ( x )
(0 < ~ _< 1/2),
where
(1.6)
M f (9x )
= Q~
sup ~-T
1 fQ If(y)[ d,.,]
is the Har(iy--Littlewood maximal function (the s u p r e m u m in (1.3), (1.4),
(1.6) extends over all cubes Q c R N containing the point x 6 R N). At the
same time, the following statements are true.
Theorem
f 6 LP~
(see FEFFERMAN and STEIN [6]). If
N)
and
.f~(x) E U ' ( R :v)
(1 < p < ,>=. 1 < P0 _< P),
then ~ t f ( x ) e L p and
(1.7)
IIM.W[I~, < AvHf~IIj,.
M a x i m a l functions and m e a n oscillation
43
In [10, p. 221] the following inequality was proved: for all x E
R N
clMM~f(x) < f~(x) < c2MM~f(x),
(1.8)
where f is any function from L loc~
1 rRlv~j, cl, c2, a depend on the dimension
N. By the Hardy-Littlewood maximal theorem [8],
I[Mf[[p ~ [[f~[Ipx JIM, flip
(p > 1),
where ~ depends on N. In the papers of BENNETT, DEVORE and SHARPLEu [1, 5] inequalities involving the nonincreasing rearrangements of f(x),
Mf(:c), and f~(x) were obtained.
The nonincreasing rearrangement of f is defined to be the function
f*(t) that is nonincreasing on (0,+oo) and equimeasurable with If(z)l.
We shall consider the rearrangement which is left-continuous. It is defined
uniquely by the equality
(1.9)
/*(t)=
sup
EcrtlV
jnflf(x)l
iEi--t
(see [13]), or can be expressed in terms of the distribution function
= I{x e r t N : If(x)l >
as follows
f*(t)
= inf(A >__0: /~f(A) < t}
(0 < t < oo).
In papers [1, 5] the following inequalities were obtained: for any function f E L t o1c ( R N ) and any c u b e Q C _ R N , w e h a v e
(1.10)
( f - X Q ) * ' ( t ) - ( f . XO)*(t) < c(f~)*(g)
(0 < t <
-
(1.11)
6/'
(Mf)*(t) <_c(f~)*(t) + (Mf)*(2t)
(0 < t < oo),
where
/** =
_It fo ~/ * ( r )
dr,
c depends only on the dimension. From inequalities (1.10), (1.11) one can
deduce (1.1) and (1.7), respectively (see [1, 5]).
If in (1.3) and (1.6) the supremum extends over all parallelepipeds with
sides parallcl to the coordinate axes containing x, then the strong maximal
functions Q(x), M~f(x) will be obtained. For the strong maximal function,
the following inequality is proved in [14]:
(1.12)
(M~f)*(t) < c(f~)*(t/2) +/*(2t)
(0 < t < oo).
A question arises: for what basis different from parallelepipeds is true
an inequality of type (1.11)? In particular, in many questions bases fl'om
44
A . K . Lerner
parallelepipeds are important whose sides are parallel to the coordinate axis
and related with a specific relation. For instance, integral means over such
parallelepipeds arose in a natural manner by studying the local properties
of anisotropic Sobolev spaces. The basis B4 in R 3 consists of all parallelepipeds whose ribs are parallel to the coordinate axis and whose dimensions are given by s x t • ~(s, t), where s, t > 0, and the function ~(s, t)
is positive, continuous and monotonically increases in each variable. The
properties of this basis were studied by Cdrdoba [4].
The above considerations bring about the following general problem,
the s t u d y of which comprises the main purpose of this work: to select a class
of differentiation bases, for which the associated maximal functions satisfy
an inequality of type (1.11).
The usual approach to estimate cubic maximal functions is based on
covering theorems. Considering the general case, we reveal those geometrical properties of a basis, which pl~\v the main role in sinfilar estimates.
These properties are formulated in the condition (A) given below. Roughly
speaking, condition (A) signifies that the basis nmst not be to() "saturated"
a n d t o o "rare".
In w some definitions from the theory of differentiation of integrals
in R N are given; then a class of bases is introduced, which satisfies condition (A) and it is shown that this class is sufficiently wide. In particular,
any density and invariant ibr homothetic basis satisfies condition (A). The
Cdr(loba basis B . also satisfies this condition.
In w maximal hmctions 1 I{<,.~J(:~:) of type (1.4) are studied, which are
associated with an arbitrary basis 93. Note that these functions are more
flexible to characterize the local properties of the function than the function
of type (1.3). The main result of !i3 is the following inequality: if the basis
~3 satisfies condition (A), then for anv meas,n'able flmction f we have
f*(t) < 2(M,~.,~J')*(t) + f*(2t)
(0 < * < ~ ) ,
where oe depends only on ~3 (in the case when 93 is a basis consisting of cubes,
inequality of this type in terms of dist:ributi(m functions was o})tained in
[10]). From this inequality a number os corollaries are deduced, in particular,
it is proved that some strengthening of inequalities (1.11) and (1.12) is true.
In w some estimates of the rearrangements of suba(lditive operators
acting in BMO are proved. The idea to use the flmction M~.f(x) in these
questions first occurred in paper [10J.
It should 1)e noted why the reduced proofs look sufficiently simple,
is mainly due to condition (A), whose check fi)r a specific basis requires
nontrivial reasonings.
Maximal functions and mean oscillation
w
45
Differentiation bases and condition (A)
For each point x E a N, let fiB(x) be a family of bounded, measurable
sets of positive measure containing x such that there exists {Rk} C fiB(x)
for which diam Rk --~ 0. T h e family
fiB= U fiB(x)
z6R N
is called a differentiationbasis [7, p. 43].
The basis, containing all cubes in a N and the basis BN, formed by all
possible parallelepipeds, whose ribs are parallel to the coordinate axes are
the most important examples of differentiationbases.
By virtue of the classical Lebesgue theorem, almost all points of the
measurable set E C a N are its density points and almost all points of the
supplement are its rare points, i.e.,for almost all x 6 R N,
lim [ E A Q[ =
,Q,-o
Iql
-eQ
XE(X).
This t h e o r e m motivates the following definition [7, p. 65].
A basis fib is called density (or possesses the property of density) if
for each measurable set E and for arbitrary sequences {Rk} from fiB(x)
converging to x for almost all x E R N, we have
lim IZ n Rk] _
k-oo
IR I
,~Z(X).
T h e Saks theorem about strong density [15, p. 196] asserts that BN is a
density basis.
A basis fib is called invariant for homothetic if for each homothetical
set R 6 fib with any coefficient and any center of homothetic enters into fiB.
A basis fiB, formed with open sets is called Busemann-Feller basis (BFbasis) if from conditions R E fib and x E R follows that R E fiB(x).
Let fib be a differentiation basis. If in (1.3), (1.4) and (1.6) the suprem u m extends over all sets R E fiB(x), then the corresponding functions will
be called maximal functions associated with the differentiation basis fib and
they will be denoted by f ~ ( x ) , M~.,U(x), M ~ f ( x ) . If fib is a BF-basis,
then it is easy to see t h a t these maximal fimctions are measurable [7, p. 44].
Consider two functions, which characterize the behaviour of a BF-basis
fiB. Set for u > 1
[~[{
1}[
qa(u) = sup
x: I'VI~XE(X ) > u
IEI>O
46
A.K. Lerner
(see [7, p. 1631),
~b(u) = inf > 01T~_T { x : M~.'(E(X) > -IEI
u1}
(the s u p r e m u m and the infimum are taken over all measurable sets E of finite
positive measure). It is clear that the functions ~(a), g,(u) are nondecreasing
and
r
<
It is also easy to sec that if !~ is a density basis, then g,(u) > 1.
We shall say that a BF-basis !t} satisfies condition (A) if
(2.1)
~a(u) < cx~
and r
---, cx~.
By the Busemann-Feller theorem [7, p. 65], the property of density is a
necessary condition for ~(u) < oo, and it is sufficient if the basis is invariant
for homothetic. Consequently, the property of density is necessary to have
condition (A). The requirement of an unbounded function .w/ leads to one
more necessary condition.
Proposition
2.1. If condition (A) holds, then tile basis ~ consists
of sets of arbitrarily large diameter.
P r o o f . Suppose the contrary: diameters of all sets R E ~ do not
exceed some constant c > 0. Let Q0 be an arbitrary cube of the unit
measure. T h e n for any u > 1
Thus, the function r
is bounded and the basis ~3 does not satisfy condition (A).
From this proposition it follows that only the property of density does
not guarantee the fulfillment of condition (A).
Proposition
2.2. Any density and invariant for homothetic basis
satisfies condition (A).
P r o o f . From the Buscmann-Feller theorem [7, p. 65] it follows that
p(u) < oo. We shall show that ~(u) --+ ~ . I f % is a basis formed with cubes;
then for the Hardy-Littlewood maximal flmction the following inequality
due to HERZ [9] and STEIN [17] (see [1, p. 56]) holds for all f E ( R N) and
all u > 0:
CN(U)/MS>~/,}lfla{
< I{Mf > 1/u}l.
Maximal functions and mean oscillation
47
Hence it follows that if f is the characteristic function of a measurable set
E of finite positive measure, then
cNulEI <
I{z: MXE(X)> l/u}l.
So, for a basis of cubes
r
> cNu
(u > o).
If the basis ~3 is invariant for homothetic, it is easy to see that for any cube
Q there exists a set R E ~ such that
Q c R
and
[R[ < ciQ[,
where c depends only on ~ . So
Mr(x) < cM~f(x)
and, consequently
r
> c'~.
Thus, for any basis ~ invariant for homothetic the condition ~(u) ---*
holds.
Note t h a t to be invariant for homothetic is not a necessary condition
for (2.1). Let us consider the C6rdoba basis B e (see w
It is clear that
this basis is not invariant for homothetic. Nevertheless, the following is true.
Proposition
2.3. The C6rdoba basis B~ satisfies condition (A).
P r o o f . Let us consider the differentiation basis B~,, containing all
open rectangular parallelepipeds, the base of which is a square with a side
s and the height is equal ~ ( s , s ) , where s > 0. Since Be, C B e C B3, we
have
MB~,f(x) < M s . f (x) < Ms3f(x)
and, therefore, it suffices to show that r
--+ oo for the basis B~>, and
~o(u) < oo for B3. The latter follows, for example, from the JessenMarcinkiewicz-Zygmund theorem [7, p. 178]:
I{x e rt3: Ms3f(x) > A}I <__
If(x)l,x(1 -4- (log + li(x)l) ) dx
(0 < ,~ < ~).
Further, by virtue of the monotonicity of ~9, it is easy to see that parallelepipeds for Be, possess the same property as cubes: if
(2.2)
R1, R2 E Be,, R1 N R2 ~ ~ and IRll < IR2[,
then
Rt C 3R2
(let 3R be the parallelepiped concentric with R but with rib length three
times larger). Let us take an arbitrary set E of finite positive measure; we
48
A . I':. L e r n e r
may suppose that E is compact. Then, by virtue of the continuity (I), we
have
{x: Mn+,XE(x)> l/u} = 0
RT,
i=1
where
Ri
1
~ B~,,
~IRil
IRi n El >
and
1
13~, n
El _< ~I3R, I.
Using (2.2), in the standard way (as in the case of cubes) we can pick out
from R 1 , . . . , R,~ pairwise disjoint parallelepipeds R , ~ , . . . , R ~ such that
Ri C U
i=I
3R~.
i=1
Since E C Uin=l IV{i, we get:
k
IEI= ENU3R,,~=,
k
27 k
27
u
IR,.,I <_ - 9=
1 k
<_~IEnaR~,I<-~I3R,.,I=,=,
tt
{
x:
u .=
M,~+,x.(=)
1
> "lt
}.
So '~b(u) _> 'u/27, as required.
Let us deduce one more sufficient condition for the unboundedness of
the function r It is related to the question of the behaviour of the function
~o(u) as u --+ 1 + 0. This question was dealt in SOLYANnCs paper [16], where
it was proved that for the basis BN we have
qo(u) ---+ 1
as u --+ 1 + 0
and it was claimed that the same is true for an arbitrary density basis. First,
let us prove the auxiliary lemma, which is contained in essence in [7, p. 181].
Lemma
2.1. Let R 1 , . . . , R , , E ~B. Then for any ~ > 0 .from
R1,...,R. we can choose some R , , . . . . ,R.~ such that
(2.3)
*:
1+~]
~-~IR,,,I <
~
k
]
U R,, ,
i=1
i=1
%
k
i=1
{=I
Maximal functions and mean oscillation
49
P r o o f . Let us take R~ 1 = R1 and suppose that R ~ I , . . . ,R~k_1 has
been chosen. T h e n R ~ will be the first set after R.k_~ in the sequence
R 1 , . . . , P ~ , for which the inequality
k-1
1
i=i
-
I+
holds. It follows that
{~: M~u::, R~,(~) > r;--~}'
I
U R~ ~
i=l
and, therefore, (2.4) also holds. We set
j--1
i----1
It is easy to see that the sets Ej are pairwise disjoint,
k
k
U E ~ . - - Uj=t
n,,,,
I@1 >-
j=1
{
I +
In,,,I.
So,
k
1+~ k
~IR.~I<--~IEsI~=~
~
The lemma is proved.
1+~1~U ~ _ 1 + ~ I Uk R~ 9
i
j=~
~
j=x
~
j=x
We shall say that the basis ~ possesses the property of lacunarity if
there are constants cl, c2 > 1 such that for any R E ~B there is a set R' E
containing R and
cllnl < ln'l < c21nl.
It is clear that any invariant for homothetic basis POSSesses the property of
lacunarity. By virtue of the continuity of ~, the basis B e also possesses this
property.
L e m m a 2.2.
lacunarity and
Let the density BF-basis f~ possess the property of
~a(u)~ l
as u - - . l + O.
Then ~ ( u ) --+ o~.
P r o o f . Let E be an arbitrary set of finite positive measure, K a
compact subset of E, and u > 0. Using properties of density and lacunarity,
we can choose sets R 1 , . . . , R n E ~ such that
(2.5)
K \ 0 R,I = o and
su
< IR~ n K I <
s
u
50
A.K. Lerner
It is clear that
i----1
Choose ~ = 1/V'u and apply Lemma 2.1 to R i , . . . , R n .
We get sets
R , , , . . . , R,~ satisfying (2.3) and (2.4). Using (2.3)-(2.6), we have:
k
k
k
0
i=1
i:1
i=1
i=1
IKI= zcnUR., + K\UR,., _<~IKnR~,I+ I
c
i=1
,_
IR~,I+
"=
k
R~\U~, <
i=I
i=1
i=1
_< (c l + x / ~ I / v ~ + p ( I +
XE(X )> 1/u}l.
)v/~
1 _I)[{x:M,
Since the set It" is arbitrary, we get
IEI <
(c 1 + V1/v'c~
~
+ ~(1 + ~1)
- 1)[{x : M~3,yE(X) > 1/u}[.
The statement of the lemma follows, because the expression in the brackets
becomes zero as u --~ oo.
Using Solyanik's result and Lemma 2.2, we can extract the following
Let ~ be an arbitrary subbasis of the basis B~.
possessing the property of Iacunarity. Then f13 satisfies condition (A).
Corollary
(a).
2.1.
In particular, Proposition 2.3 follows one again.
In the sequel, we shall assume that ~3 is a BF-basis. satisfying condition
w
Estimates
of maximal
functions
Note that by means of rearrangement, we can write the definition
M,~,~f(z) more (:ompactly:
(3.1)
M~,,~f(x) =
sup
i n f ( ( f - c)xR)*(cqR[)
R6~B(x) c 6 R
(0 < a < 1/2).
If in (3.1) we set c = 0, then wc get the function
m~,,~f(x) =
sup (f" XR)*(~IRI)
R6~(x)
(0 < ct _< 1/2).
M a x i m a l functions a n d m e a n oscillation
51
Let us denote
IEI>O
It is clear that for all u > 1,
qo(u) < ~(u) < lira ~o(u + e).
- -
- -
e---* + 0
Since the function qo(u) is monotonic, it follows t h a t for almost all u,
~(u) = ~(u).
T h e next two lemmas show that the integral properties of the functions
f and m ~ . ~ f and, by an appropriate choice c~, also the function M~,~f
coincide.
3.1. For any measurable function f ,
Lamina
:,.,>
-<
for 0<t<oo, 0<o~_<I/2.
P r o o f. Let us show that
(3.3)
(z : M93x{~:1f(y)l>~)(z) > c~} C
C {x: m~,~f(x) > A} C {x: M~x{y:l.r(,)l>~)(z ) > c~}.
Let
x E {x: M~3x{y:lf(y)l>~}(x) > ct}.
It means that there is a set R such that
n e ~(x)
and
]{Y e R : If(Y)l > ~)1 > ~Jnl.
It follows that
( f . ,~R)*(c~IRI) > A,
and therefore,
m~,~f(x) > )~.
T h u s the left-hand side of (3.3) is proved. The right-hand side can be proved
analogously.
By (3.3) and the definition of ~(u), @(u), we get:
(3.2) follows immediately from these inequalities if the definition of rearrangement in terms of the distribution function is used.
In the sequel, we shall use an easily verifiable property of a rearrangement: for any measurable functions f , g,
(3.4)
(f + g)*(tl + t.2) <_f*(t:) + g*(t2)
(t:,t2 > 0).
52
A.K. Lerner
3.2. For any measurable function f and any point x C
Lemma
m ~ , ~ f ( z ) < 2 M ~ , ~ f ( x ) + m~3,1-~f(x)
(3.5)
R N,
(0 < a < 1/2).
Proof.
Let R be an arbitrary set from ~3(x) and c an arbitrary
constant. Using (1.9) and (3.4), we have:
( f ~ n ) * ( ~ t n I ) < ( ( f - c)~n)*(~lnl) + Icl <
< ((f
-
c)xR)*(alRI) + inf(Ic - / ( Y ) I + If(y)l <
--
y E R
--
< ((f - c)XR)*(~IRI) + (l(f - c),~RI + If" ~RI)*(IRI) ___
< 2 ( ( f - c),~R)*(~IRI) + ( f " XR)*((1 -- ~)IRI).
Passing first to the lower b o u n d over all c E R , then to the upper b o u n d
over all r E ~3(x), we obtain inequality (3.5).
The next theorem follows from Lemmas 3.1 and 3.2.
T h e o r e m 3.1. Let a BF-basis f8 satisfy condition (A). Then there is
an a, depending only on ~ , such that for all measurable functions f ,
f*(t) <_ 2(tvI~,<,f)*(t) + f*(2t)
(3.6)
(o < t < oo).
P r o o f. Let c~ be such that
+(1)
1
Using (3.2), (3.4), (3.5), we obtain:
as required.
Corollary
3.1. Let a BF-basis ~ satisfy condition (A). Then there
is an o~, depending only on ~3, such that for" all measurable functions f with
f * ( + ~ ) = 0,
(3.7)
clllfllp M IIM~,<JIIp ___c211fll,
where cl, c2 depend only on p and c~.
(1 M p < oo),
M a x i m a l f u n c t i o n s a n d m e a n oscillation
Proof.
53
The right-hand side of (3.7) follows from (3.2). Further, by
(3.6),
oo
f*(t) < 2 ~(M~,~f)*(2kt) <
(3.8)
k=O
2
-< log 2
r
=
j2u-lt
(MkJ)*(r)2
r
.
r
Iog 2 Jr~2
r
dr.
From this and Hardy's inequality [19, p. 221] the left-hand side of (3.7)
follows.
R e m a r k 3.1. If ~ is a basis of cubes, then inequality (3.7) was
obtained in [10, p. 242] by other methods.
R e m a r k 3.2. Let ~ be a basis of cubes. By the Hardy-LittlewoodHerz inequality (see [3]),
clf**(t) <_ (Mf)*(t) < c2f**(t)
(3.9)
(0 < t < cxz).
Integrating inequality (3.6), we obtain
f**(t) <_ 2(M~f)**(t) + f**(2t)
(0 < t < c~).
From this, (1.8) and (3.9) we obtain the estimate by Bennett, DeVore and
Sharpley (see (1.10)) in case when Q = RN:
(3.10)
f**(t) - if(t) <_2(f**(t) - f**(2t)) ___
<__4(M~f)**(t) <_c(f~)*(t) (0 < t < c~).
According to the Fefferman-Stein theorem (see (1.7)), for p > 1
Ilf~llp • IlfLIpFor p = 1 this is false, because, as is seen from (3.10), the function f~(x)
is not integrable if f ,~ const. The advantage of the function M~f(x) as
compared to f~(x) is that if f ~ BMO, then the global behaviour of the
function f~(x) is essentially the same as that of the function Mf(x). This
explains that (3.7) is true for p = 1.
Let f E L~oc(RN). Consider the question about the estimate M~f(x)
by f~(x). We shall show that inequalities (1.11), (1.12) admit a strengthening in general. We set
sup ((If]- IflR)XR)*(c~IRI)
Re~(~)
It is easy to the see that
-M~,~f(x) =
((If]- [ftR)~R)*(~IR]) <--~
(0 < c~ _< 1/2).
]]f(Y)ll- ]fiR dy <
54
A.K.
< alRI
Lerner
If(Y) - fR] dy,
and so
--ff[~,~f(x)_< -~f~(x).
~ ~
If f _> 0 then obviously
M ~ , ~ f ( x ) <_ -M~,~f(x).
(3.11)
T h e o r e m 3.2. Let a BF-basis ~ satisfy condition (A). Then there is
1 IRJV~J,
an a depending only on ~ such that for all f E L lo~
(3.12)
( M ~ f ) * ( t ) < (-H~,~f)*(t/2) + f*(2t)
(0 < t < co).
P r o o f . Since all functions involved in inequality (3.12) are defincd for
Ifl, we may assume f > 0.
Using (1.9), (3.4), we obtain
fR _< i~lf(]fR - f(y)] + ]f(Y)l) -<
<_ ( ( f - fR)XR)*(IRI/2) + ( f x n ) ' ( I R I / 2 ) .
So,
M ~ f ( x ) < -ff[~A/2f(x) + , n ~ , u 2 f ( x ).
By (3.13) and (3.2), we have
(3.13)
(3.14)
(]t/~f)(x) < (l-V[~,,/.2f)*(t/2) + (m,~,u.2f)*(t/2) <_
<_ ( M ~ A / 2 f ) (t/2) +
Choose o~ such that
Applying (3.2) and (3.5), we obtain
.
t
From this, (3.11) and (3.14) the statement of the theorem follows.
C o r o l l a r y 3.2. Let a BF-basis ~ satisfy condition (A) and let f E
L}oc(Rg), f*(+c~) = O. Then there is an a depending only on ~ such that
M ~ f E LV(R N)
if a,,d only if
!l----[~.~J'e LP(RN),
and
Cl[[Msf[[ v <_ [l~-i~,~f[[ v _< c2[[M,sf[[,
(1 _< p < oc).
Maximal functions and mean oscillation
55
P r o o f. The second inequality follows from the inequality
-M~,~f(x) < 4 M ~ f ( x ) .
o~
To prove the first inequality, we may assume again that f > O. Then by
(3.6), (3.11), (3.12) we obtain
(M~f)'(t) < c
~oo
(M~,~f)*(T) dT.
/4
7
It remains to apply Hardy's inequality [19, p. 221].
~4. S o m e a p p l i c a t i o n s
In [10] it is shown that the function M ~ f ( x ) may be used effectively
for the study of integral operators. We shall give some results in this way
by using inequality (3.8). Recall two well-known theorems: for all f E
L 1 + LCO(R N) we have
(4.1)
f=g+hinf([[gltl__/_+ [[h[]oo) = f**(t)
(0 < t < oo),
and for all f E L 1 + B M O ( R N) we have
(4.2)
l =inf
9 + h \(I}gllz
--~
+
Ilhll.)
x
(/")*(t)
(o < t <
Equality (4.1) is proved, for example, in [19, p. 240]. Inequality (4.2) is
obtained by BENNETT and SHARPLEY in [5]. Note that
(4.3)
M~,,~(f + g)(x) < M~,,~/2f(x ) + M~,,~/gg(x).
Indeed, for all ~ E R and all R E ~$(x),
inf((f + g - c)XR)*(aIR[) = inf((f + g - c - ~)XR)*(aIR[) <
I
Passing to the lower bound over all ~ E R, then to the upper bound over all
r E ~ ( x ) , we obtain (4.3).
Denote by A~0(R N) the space of measurable functions on R x. An
operator T : A40 ---* Ado is called subadditive (or sublinear) if for all f, g E
Ado and AER,
IT(f +g)(x)l < ITf(x)I + IZg(x)l,
IT(Af)(x)l =
I:,1 ITI(x)I
a.e.
56
A . K . Lerner
T h e o r e m 4.1. Let the subadditive operator T be of weak type (1,1)
and bounded from L ~ into BMO. Then for all f E L 1 + L ~ ( R N) with
(Tf)*(+cx~) = 0 the following inequality is true:
(Tf)*(t) <_ c(f**(t)+ ft ~ if(V)r dr)
(4.4)
Proof.
(0 < t < ~ ) .
Let
f = g + h,
g E L 1, h E L ~,
where
and a is such that (3.8) holds for M~f. It follows easily that
(4.5)
]]Tf(x)l - ITh(x)l I <_ IT(f - h)(x)] = ITg(x)l.
Using the left-hand side of (1.5) and also (4.3), (4.5), we obtain
M ITfI(x) < M / (ITfl- tThl)(x) + M /21ThI(x) <
<_ m~/2(lTfl = IThl)(x) + 21lIThlli, <
< m~/2Tg(x) + 4tlThli, _< m~/2Tg(x) + clIhN~.
By this and Lemma 3.1,
(M~lTfl)*(t) < (,z~/2Tg)*(t)+ clIhN~ <
_
.
_
--7-+llh
.ll )
Consequently, by (4.1),
(M~lTfl)*(t) <_ cf**(t)
(t > 0).
Applying inequality (3.8) to the flmction ITfl, we obtain the statement of
the theorem.
In the case when T = H, the Hilbert transfbrm, inequality (4.4) was
proved by BENNETT and RUDNICK [2]. It should be noted that a sufficiently
wide class of singular integral operators satisfies the conditions of Theorem
4.1 (see [18]).
Similarly to Theorem 4.1, by using (4.2) one can prove the following
T h e o r e m 4.2. Let the subadditive operator T is of weak type (1,1)
and bounded from BMO into BMO. Then for all f E L 1 + B M O ( R N) with
( T f ) * ( + o c ) = 0 the following inequality is true:
(Tf)*(t) <_ c f ~ (ff)*(r) dr
Jt
7-
(0 < t < co).
Maximal functions and mean oscillation
57
Remark.
I n this p a p e r t h e condition f * ( + o o ) = 0 o f t e n o c c u r r e d . It
is e a s y to s h o w t h a t for m e a s u r a b l e , a l m o s t e v e r y w h e r e finite f u n c t i o n s f
t h i s is e q u i v a l e n t to t h e following:
]zI(A) < oo
for all A > 0.
A c k n o w I e d g m e n t . The author is grateful to V. I. Kolyada, under
whose guidance this work was accomplished.
References
[1] C. BENNETT, R. DEVORE and R. SHAlZPLEY,Weak-L ~176
and BMO, Ann. of Math.,
113(1981), 601-611.
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c~ym<imJlX ol<a~MneHHn ~nyI ~H~c~epeHIIIaa:II,HI,IX 6acI4COB,
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A. K. Lerner: M a x i m a l functions a n d m e a n oscillation
58
[19] H . CTEHII i4 F. BE~C, BoeOenue e zap~onuuecuu'g ananua na eounuOoo~zx npoc m p a n c m o a x , M~Ip (MocI<Ba, 1974) - - E . M. STEIN a n d G. WEISS, Introduction
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